COUNTING CUSP FORMS BY ANALYTIC CONDUCTOR MAY 2017

FARRELL BRUMLEY AND DJORDJE MILICEVIC

Abstract. We prove an asymptotic formula for the number of cusp forms on GLn over a numberfield of bounded analytic conductor, sometimes called the universal family. For n > 3, we in factimpose an additional technical condition on the members of this family, that they be spherical atinfinity. We conjecture an explicit form of the Sato–Tate measure for the universal family. Ourmethods naturally provide uniform Weyl laws with explicit level savings on these groups.

Contents

1. Introduction 22. Equidistribution and Sato-Tate measures: conjectures 63. Discussion in the classical case of GL2 over Q 74. Outline of the proof 10

Part 1. Preliminaries 135. General notation 136. Local and global conductors 177. Local conductor zeta functions 18

Part 2. Tempered count 218. Preparation for the proof of Theorem 1.1 219. Spectral localizing functions 2310. Bounding the discrete spectrum 2511. Proof of Proposition 8.3 2912. Reduction to error estimates 3313. Boundary error term 35

Part 3. Trace formula estimates 3814. Bounding the non-central geometric contributions 3815. Estimates on local weighted orbital integrals 4316. Construction of test functions 5117. Controlling the Eisenstein contribution 57References 63

Date: May 25, 2017.2010 Mathematics Subject Classification. Primary 11F72; Secondary 11F66, 11F70, 22E55, 58J50.Key words and phrases. Automorphic representations, analytic conductor, families of automorphic forms, trace

formula, Plancherel measure, Weyl law.The first author was partially supported by the ANR Grant PerCoLaTor: ANR-14-CE25. The second author was

partially supported by the National Security Agency (Grant H98230-14-1-0139), National Science Foundation (GrantDMS-1503629), and through ARC grant DP130100674. He thanks the Max Planck Institute for Mathematics fortheir support and exceptional research infrastructure. The United States Government is authorized to reproduce anddistribute reprints notwithstanding any copyright notation herein.

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1. Introduction

Automorphic forms and their L-functions are among the central notions of modern numbertheory. While they can be notoriously difficult to study individually using analytic techniques,desired results can often be obtained by embedding them in a family of automorphic forms offavorable size. In this article, we address the question of size of the “universal” family consistingof all automorphic forms on GLn over a number field F when ordered by their analytic conductor.

We denote by F the set of unitary cuspidal automorphic representations π of GLn(A), subjectto a natural normalization that the central character of π, viewed as a Hecke character of F×\A×,

should be trivial on R+, diagonally embedded in A× as t 7→∏v|∞ t

1/d. This normalization has the

effect of eliminating continuous families. Any cuspidal automorphic representation π of GLn(A)can be brought into F by twisting by |det |s for an appropriate s ∈ C. Following Sarnak, Shin, andTemplier [39, 41], we shall refer to the set F as the universal family.

Motivated by questions in analytic number theory related to L-functions, Iwaniec and Sarnak [17]introduced the notion of analytic conductor of π ∈ F. This positive real number Q(π) is a globalinvariant of π; it can be expressed as a product of local conductors, each arising from the localfunctional equation of the standard L-function L(s, πv). One way to understand the significance ofthe analytic conductor in the analytic theory of L-functions is that it controls the effective lengthsof the partial sums appearing in the global approximate functional equation for L(s, π). In turn,the analytic conductor controls complexity in analytic problems including evaluation of moments,subconvexity, nonvanishing, extreme value problems, numerical computations of L-functions orautomorphic forms themselves, as well as the requisite number of twists in the known forms ofthe Converse Theorem, just to name a few applications. Since the appearance of [17], Sarnak hasrepeatedly emphasized the importance of understanding the statistical properties of the truncatedfamily

(1.1) F(Q) = π ∈ F : Q(π) 6 Q,the first among which is its cardinality.

In this paper, we prove asymptotics for |F(Q)|, with explicit logarithmic savings in the error term.Such a result may be termed a Weyl–Schanuel law, for reasons which will be shortly explained. Infact, we can only do this for n = 2, and for a more tractable subfamily for general GLn in which thearchimedean components are assumed to be spherical. See §1.3 for exact statements. In addition,we address related counting and equidistribution problems and prove uniform Weyl laws, estimateson the size of complementary spectrum, and uniform estimates on terms appearing in Arthur’strace formula for GLn.

1.1. Schanuel’s theorem. Let Π(GLn(A)) denote the direct product over all places v of theunitary duals Π(GLn(Fv)). Each Π(GLn(Fv)) is endowed with the Fell topology, and we giveΠ(GLn(A)) the product topology. Let Π(GLn(A)1) be the closed subset verifying the normalizationon the central character, and we give this the subspace topology. Then F embeds in Π(GLn(A)1)by taking local components, and an old observation of Piatetski-Shapiro and Sarnak [40] showsthat F is dense1 in Π(GLn(A)1).

In this set-up, the counting problem for cusp forms becomes reminiscent of the familiar problemof counting rational points on projective algebraic varieties. In particular, one can set up an ananalogy between counting π ∈ F with analytic conductor Q(π) 6 Q and counting x ∈ Pn(F ) withexponential Weil height H(x) 6 H. That one should consider F(Q) as analogous to x ∈ Pn(F ) :H(x) 6 H is best seen as an expression of the deep conjectures of Langlands, in which the generallinear groups serve as a sort of “ambient group”.

1As remarked to us by Sarnak, if we give Π(GLn(A)) the restricted product topology, the set F is discrete inΠ(GLn(A)). This is a point of view more adapted to computational problems of isolating and numerically computingcusp forms.

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To be more precise, let us recall some classical results on counting rational points in projectivespace Pn, where n > 1. For a rational point x ∈ Pn(F ), given by a system of homogeneouscoordinates x = [x0 : x1 : · · · : xn], we denote by

H(x) =∏v

max(|x0|v, |x1|v, . . . , |xn|v)1/d (d = [F : Q])

the absolute exponential Weil height of x, the product being take over the set of normalized valu-ations of F . Northcott [32, 33] was the first to observe that for any H > 1 the counting function

|x ∈ Pn(F ) : H(x) 6 H|is finite. For example, when F = Q, one is simply counting primitive integral vectors in Rn+1 lyingin a box of side-length H; clearly this is bounded by (2H + 1)n+1. The automorphic avatar ofNorthcott’s finiteness theorem can be attributed to Moreno [29], who showed that |F(Q)| is finitefor any Q > 0.

An asymptotic for the above counting function was given by Schanuel [42]. The leading constantwas later reinterpreted by Peyre [35], as part of his refinement of the conjectures of Batyrev-Manin [3]. Following Peyre, we write τ for the total volume of the Tamagawa measure of Pn. ThenSchanuel proved that

|x ∈ Pn(F ) : H(x) 6 H| ∼ τ

n+ 1Hn+1 as H →∞.

Schanuel in fact gave an explicit error term of size O(H logH) when n = 1 and F = Q and

O(Hn−1/d) otherwise.More generally, in the same spirit as the Batryev-Manin-Peyre conjectures for counting rational

points on Fano varieties, given a reductive algebraic group G over F and an appropriate notionof conductor, one would like to understand the asymptotic properties of the counting functionassociated to cuspidal automorphic representations of G(A) of bounded analytic conductor. Thisanalogy served as an inspiration and organizing principle throughout the elaboration of this article,where we address the setting of general linear groups.

1.2. The conjectural Weyl-Schanuel law. For any finite place v of F let

(1.2) ∆v(s) = ζv(s)ζv(s+ 1) · · · ζv(s+ n− 1)

Furthermore, let ∆(s) =∏v<∞∆(s) and write ∆∗(1) for its residue at s = 1. Let

‖ν(F)‖ =ζ∗F (1)

ζF (n+ 1)n+1‖ν∞(F)‖,

where ζ∗F (1) denotes the residue of ζF (s) at s = 1 and

‖ν∞(F)‖ =

∫Π(GLn(F∞)1)

q(π∞)−n−1 dµpl∞(π∞).

We make the following conjecture.

Conjecture 1 (Weyl–Schanuel law). As Q→∞ we have

(1.3) |F(Q)| ∼ 1

n+ 1∆∗(1) ‖ν(F)‖Qn+1.

We now explain the constant the leading term constant in this conjecture.The residue of ∆(s) at s = 1 comes from a global volume. To see this, first let µGLn,v be the

canonical measure on GLn(Fv) defined by Gross in [13]. It is given by µGLn,v = ∆v(1)|ωGLn,v|, and|ωGLn,v| is the measure induced by a canonical top degree differential form ωGLn,v. For example,at finite places v, the measure µGLn,v is the unique Haar measure on GLn(Fv) which assigns themaximal compact subgroup GLn(ov) volume 1. In view of this latter property, the product measure

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µGLn =∏v µGLn,v is well-defined. We continue to write µGLn for the measure on GLn(F )\GLn(A)1

given by the quotient of µGLn by the counting measure on GLn(F ). Then it is shown in [13] thatthe canonical volume of the automorphic space GLn(F )\GLn(A)1 is finite and satisfies

∆∗(1) = µGLn(GLn(F )\GLn(A)1).

This explains the occurence of ∆∗(1) in the statement of Conjecture 1, since the volume of theautomorphic space appears in the identity contribution of the Arthur trace formula.

The factor ‖ν(F)‖ comes from a measure on the ambient space Π(GLn(A)1). We fix a normal-

ization of Plancherel measures µplv on Π(GLn(Fv)) for each v by taking Plancherel inversion to hold

relative to µGLn,v. At finite places v the measure µplv assigns the unramified unitary dual volume

1. Next we define a measure νv(F) on Π(GLn(Fv)) by setting (for open A ⊂ Π(GLn(Fv)))

νv(F)(A) =

∫Aq(πv)

−n−1dµplv (πv).

Since µplv is supported on the tempered spectrum, so too is νv(F). In particular, since for GLn(Fv)

a tempered representation is automatically generic, it makes sense to write q(πv) in the integral. InLemma 7.1 we show that for finite places v the total mass ‖νv(F)‖ of νv is equal to ζv(1)/ζv(n+1)n+1.We deduce that the measure on Π(GLn(A)) given by the regularized product

ν(F) = ζ∗F (1)∏v<∞

ζv(1)−1νv(F) · ν∞(F)

converges; its total mass is precisely the factor ‖ν(F)‖ appearing in Conjecture 1.

1.3. Main results. The present paper consists of two principal components: a reduction of Con-jecture 1 to trace formula estimates, and the proof of these estimates in certain cases. Namely, inPart 2, we show that the Weyl–Schanuel law for GLn, as formulated in Conjecture 1, follows asa consequence of sufficiently strong estimates on the trace formula. Then, in Part 3 we establishthese estimates in certain situations. To describe these results, we first introduce some technicalnotions related to the trace formula.

A natural framework for counting automorphic representations is provided by Arthur’s non-invariant trace formula. This is an equality of distributions Jspec = Jgeom, along with an expansionof both sides according to primitive spectral or geometric data. Roughly speaking, the most regularpart of the spectral side of the trace formula Jcusp, coming from the cuspidal contribution, isgoverned by the most singular part of the geometric side Jcent, coming from the central elements.

To be more precise, for an function ϕ ∈ H(G(A)1) we let

J1(ϕ) = ∆∗(1)ϕ(1) and Jcent(ϕ) = ∆∗(1)∑

γ∈Z(F )

ϕ(γ)

be the identity and central contributions to the trace formula, and

Jcusp(ϕ) = tr(Rcusp(ϕ)) and Jdisc(ϕ) = tr(Rdisc(ϕ))

be the cuspidal and discrete contributions, where R• is the restriction of the right-regular repre-sentation of GLn(A)1 on L2

•(GLn(F )\GLn(A)1). Finally put

(1.4) Jerror(ϕ) = Jdisc(ϕ)− Jcent(ϕ).

We shall be interested in Jerror(ϕ) for ϕ of the form εK1(q) ⊗ f , where f ∈ C∞c (GLn(F∞)1)and εK1(q) is the idempotent element in the Hecke algebra associated with the standard Heckecongruence subgroup K1(q). The latter subgroup is known, by the work of Casselman [6] andJacquet-Piatetski-Shapiro-Shalika [19], to pick out the cuspidal representations of conductor di-viding q. One expects Jerror(εK1(q) ⊗ f) to be small relative to quantities involving q and f . Ifthis can be properly quantified, one can hope to deduce that a sharp cuspidal count modelled byJcusp(εK1(q) ⊗ f) inside Jdisc(εK1(q) ⊗ f) is roughly equal to J1(εK1(q) ⊗ f) inside Jcent(εK1(q) ⊗ f).

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Let h denote the function defined on the generic unitary dual of GLn(F∞)1 as h(π∞) = trπ∞(f).For our purposes, we would like h to localize around a given generic unitary representation inΠ(GLn(F∞)1), or a measurable family of such. (The term localize should not be taken too literally:the function h lies in a Paley-Wiener space and is therefore not of compact support. It will howeverbe of rapid decay outside of some fixed compact.) Moreover, we would like to have some controlover the error in the localization. In general this is given by the support of the test function f .Indeed, if suppf ⊂ K∞ exp(B(0, R))K∞, where B(0, R) is the ball of radius R in the Lie algebra ofthe diagonal torus, then the walls of the corresponding h, i.e., where it transitions to rapid decay,will be of size 1/R.

Let Ξ be a subset of the generic unitary dual of GLn(F∞)1. For a parameter R > 0 let volR(Ξ)be the Plancherel volume of the (1/R)-thickening of Ξ, as defined in §8.2. In Section 9, we showthat if Ξ is nice enough (and R > 0 is arbitrary) we may use the Paley-Wiener theorem of Clozel-Delorme [7] to deduce the existence of test functions fΞ

R ∈ C∞c (GLn(F∞)1) such that suppfΞR ⊂

K∞ exp(B(0, R))K∞ and hΞR(π∞) = trπ∞(fΞ

R) localizes about Ξ. We refer the reader to Section 9for exact statements. We will use these test functions in making the following definition.

Effective Limit Multiplicity (ELM). Let Ξ be a nice subset of the generic unitary dual of GLn(F∞)1.We say that Property (ELM) holds with respect to Ξ if there exist constants C, θ > 0 (independentof Ξ) such that for every integral ideal q of O, and all real R > 0, we have

Jerror(εK1(q) ⊗ fΞR) eCRNqn−θvolR(Ξ).

When Ξ is the entire generic unitary dual of GLn(F∞)1, one then says that Property (ELM) holds.

We shall make more extensive comments about Property (ELM) in §4.2. For the moment, weremark that the stated estimate is trivial in the Ξ parameter. It is rather the power savings inthe level which is of critical importance in our application. To get a better feeling for the variousranges of parameters, and corresponding savings, see §3.1.

Our first main theorem, proved in Part 2, is the reduction of Conjecture 1 to Property (ELM).

Theorem 1.1. Property (ELM) implies Conjecture 1 in the following effective form

|F(Q)| = 1

n+ 1∆∗(1) ‖ν(F)‖Qn+1

(1 + O

(1

logQ

)).

More generally, if Property (ELM) holds for a nice subset Ξ of Πgen(GLn(F∞)1), then writingAΞ = Π(GLn(Af ))× Ξ, we have

|π ∈ F(Q) : π ∈ AΞ| =1

n+ 1∆∗(1) ν(F)(AΞ)Qn+1

(1 + O

(1

logQ

)).

In our second main theorem, proved in Part 3, we establish Property (ELM) in certain cases.These are described in the following result.

Theorem 1.2.(1) For every n > 1 Property (ELM) holds with respect to the spherical part of Π(GLn(F∞)1).(2) For n 6 2, Property (ELM) holds.

Combining these two theorems yields the Weyl-Schanuel law for GL2 and for the GLn when thelatter is restricted to the archimedean spherical spectrum.

Acknowledgements. We would like to thank Nicolas Bergeron, Andrew Booker, Laurent Clozel,James Cogdell, Guy Henniart, Emmanuel Kowalski, Erez Lapid, Dipendra Prasad, Andre Reznikov,and Peter Sarnak for various enlightening conversations. We are particularly indebted to SimonMarshall for suggestions leading to a simplification of the proof of Proposition 10.1 and to PeterSarnak for originally suggesting this problem.

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2. Equidistribution and Sato-Tate measures: conjectures

In fact we conjecture that the universal family F(Q) equidistributes, as Q→∞, to a probabilitymeasure µ(F) on Π(GLn(A)1) that we now explicitly identify. This allows us to properly interpretthe leading term constant in the conjectural Schanuel–Weyl law and our main theorems. We expectthat our techniques can be leveraged to yield of proof of these equidistribution conjectures and planto address this in follow up work.

The universal family F(Q) gives rise, by way of the embedding into Π(GLn(A)1) via local com-ponents, to two automorphic counting measures

(2.1) ν countF(Q) =

1

Qn+1

∑π∈F(Q)

δπ and µ countF(Q) =

1

|F(Q)|∑

π∈F(Q)

δπ

on Π(GLn(A)1).Recall the measure ν(F) on Π(GLn(A)1) of Section 1.2, whose total mass enters Conjecture 1.

Denoting by Π0(GLn(F∞)1) the subset of Π(GLn(F∞)1) consisting of spherical representations, thestatement of our Theorem 1.2 verifies that

νcountF(Q) (A) −→ 1

n+ 1∆∗(1)ν(F)(A)

for A = Π(GLn(F∞)1) for n 6 2 and for A = Π0(GLn(F∞)1) for every n ∈ N.We conjecture that, in fact, the entire limiting behavior of both measures νcount

F(Q) and µcountF(Q) as

Q→∞ is governed by two measures ν and µ on Π(GLn(A)1) that are, in contrast to the countingmeasures (2.1), products of local measures. Specifically, we define a probability measure µv(F) onΠ(GLn(Fv)) by

µv(F)(Av) = νv(F)/‖νv(F)‖.Like µpl

v and νv(F), the measures µv(F) are supported on tempered spectrum. Moreover, we put

µ(F) = ν(F)/‖ν(F)‖ =∏v

µv(F);

this is a well-defined probability measure on Π(GLn(A)1).

Conjecture 2 (Equidistribution to µ(F)). As Q→∞,

ν countF(Q) −→

1

n+ 1·∆∗(1) · ν(F)

In particular, this implies the Weyl–Schanuel law (Conjecture 1) as well as

µ countF(Q) −→ µ(F).

The convergence of the above measures is taken in the sense of Sauvageot.

2.1. Sato-Tate measure. Conjecture 2 implies that the universal family F(Q) equidistributes inΠ(GLn(A)1) with respect to the measure µ(F). As a consequence, we may identify the Sato-Tatemeasure µST(F) of the universal family F.

We recall the definition of µST(F), introduced in [39]. Let T denote the diagonal torus inside theLanglands dual group GLn(C) of GLn, and let W be the associated Weyl group. For finite placesv, the Satake isomorphism identifies the unramified admissible dual with the quotient T/W . Itthen makes sense to speak of the restriction of µv(F) to T/W , which we write (abusing notation)as µv(F)|T . One then defines

µST(F) = limx→∞

1

x

∑qv<x

(log qv) · µv(F)|T ;

thus µST is a measure on T/W .

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Note that the Plancherel measure µplv is supported on the tempered unitary dual, so the same

is true for µv(F). Moreover, under the above identification, the tempered unramified unitary dualcorresponds with Tc/W where Tc is the compact torus U(n) ∩ T . Thus the restriction µp(F)|T issupported on Tc/W and we may think of the Sato-Tate measure as being defined on Tc/W . Now,as already mentioned, we show in Lemma 7.1 that for finite places v we have∫

Π(GLn(Fv))q(πv)

−n−1dµplv (πv) =

ζv(1)

ζv(n+ 1)n+1.

Thus, letting µplv |Tc denote the restriction of µpl

v to Tc/W , we have

µv(F)|Tc =ζv(n+ 1)n+1

ζv(1)µplv |Tc = (1 + O(q−1

v ))µplv |Tc .

We deduce that

µST(F) = limqv→∞

µplv |Tc .

The latter limit is well known to have the following description.

Corollary 2.1. Assume Conjecture 2. Then the Sato-Tate measure µST(F) of the universal familyis the push-forward of the probability Haar measure on U(n) to Tc/W .

Using the Weyl integration formula, we have

µST(et1 , . . . , etn) =1

n!

∫[0,2π)n

∏j<k

∣∣eitj − eitk ∣∣2dt12π· · · dtn

2π.

In particular, it follows from Corollary 2.1 that the indicators

i1(F) =

∫T|χ(t)|2 dµST(F)(t), i2(F) =

∫Tχ(t)2 dµST(F)(t), i3(F) =

∫Tχ(t2) dµST(F)(t)

introduced in [39], where χ(t) = tr(t), take values

i1(F) = 1, i2(F) = 0, i3(F) = 0

on the universal family. This is consistent with the expectation that the universal family F is ofunitary symmetry type.

3. Discussion in the classical case of GL2 over Q

In this section, we consider several concrete cases to exhibit essential features of our countingproblem, keeping the classical notation.

3.1. Connection to Weyl’s law and limit multiplicity theorems. Consider the countingproblem for the spherical spectrum for GL2 over Q, consisting of even Maaß cusp forms. In classicallanguage, we seek an asymptotic for |F(Q)|, the number of Hecke–Maaß cuspidal newforms φ oncongruence quotients Y1(q) = Γ1(q) \H such that their level q and their spectral parameter r satisfyq(1 + |r|)2 6 Q. Here, our convention is that the spectral parameter r is such that the Laplacianeigenvalue is given by λ = 1/4 + r2.

A familiar environment for such a counting problem is that of Weyl’s law. Indeed, for a fixedcongruence subgroup Γ 6 SL(2,Z), Weyl’s law provides the following asymptotic for the full countof all (not necessarily new) Maaß cusp forms ordered by their spectral parameters rj as T →∞:

(3.1) NΓ(T ) :=∑

rj on Γ|rj |6T

1 =Vol(Γ\H)

4πT 2 + TP1,Γ(log T ) + OΓ

(T

log T

).

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In the conductor count problem, the group Γ = Γ1(q) varies over q, and Vol(Y1(q)) q2. Consid-

ering only the main terms in this asymptotic with T =√Q/q − 1 and temporarily putting aside

the issue of sieving for newforms (which, as it turns out, only impacts the leading constant, notthe shape of the asymptotics), one might heuristically expect the main term in the asymptotic for

|F(Q)| of size∑

q6Q q2(√

Q/q − 1)2 Q3. In fact this expectation is confirmed by the asymptotic

|F(Q)| ∼ cQ3 provided by Theorem 1.1.For the above heuristic to go through, one must have control over the q-dependence in the implied

constants in (3.1). Several recent results [34, 24] explicate this dependence, yet not in a satisfactoryway to yield information on |F(Q)|. For example, the best available uniform GL2 Weyl Law (provedin [34, Corollary 3.2.3] with ample flexibility in specifying local conditions and with variations validover an arbitrary number field), yields

(3.2) NΓ1(q)(T ) =Vol(Y1(q))

4πT 2 − c(q)T log T + O(Vol(Γ1(q)\H)T ),

for a constant c(q) > 0 depending on q. Upon summing∑

q6QNΓ1(q)(√Q/q− 1), we find the total

error term is of size Q3, the same as the main term.To see, in a more fundamental way, why (3.2) does not suffice to count the universal family,

note that when q Q, contributions of NΓ1(q)(T ) to |F(Q)| occur for T =√Q/q − 1 1. In this

range, the estimate (3.2) is of no use, as the error term is of the same size as the main term. Now,the multiplicity of the 1/4-eigenspace (corresponding to r = 0) as q varies is a delicate arithmeticquestion, related to Artin’s conjecture, and one knows by the groundbreaking results of Duke [?]and Ellenberg [?] that this multiplicity is O(q2−δ) for some δ > 0. Thus, the contribution of the1/4-eigenspace to the universal count is provably negligible. Nevertheless, in a bounded intervalabout r = 0, limit multiplicity theorems [?] suggest a regularization of this behavior, to the extentthat NΓ1(q)(1) q2. Such a result cannot be deduced from (3.2). If true, this regularization wouldthen in turn show that the bounded eigenvalue range contributes to |F(Q)| with positive proportion,which is consistent with Conjecture 2.

The important point here is that we cannot assume even a condition of the form T > 1100 if

we wish to recover the correct leading constant in Theorem 1.1, since the complementary rangecontributes with positive proportion to the universal count. In any uniform Weyl law for NΓ1(q)(T )we thus require an error term that is not only uniform in q but in fact gives explicit savings in q inthe range T 1. Indeed, even a modest improvement in the q-dependence in the error term witha complete loss of savings in the eigenvalue aspect – something of the form

(3.3) NΓ1(q)(T ) =Vol(Y1(q))

4πT 2

(1 + O

(1

log q

))– is sufficient and yields an asymptotic (for the spherical spectrum count) of c0Q

3 + O(Q3/ logQ),with an absolute c0 > 0. For GL2 over Q we can in fact improve the above error term to

O(

Vol(Y1(q))log q T

).

We show results of the form (3.3) in the cases enumerated in Theorem 1.2, in part by relyingon recent breakthroughs by Finis-Lapid [8, 11] on limit multiplicity theorems. Our methods cansuccessfully treat ranges corresponding to those as short as T 1/ log q. Recents advances forSato-Tate laws for Maass forms [25] would actually allow us to retain a power savings for largeeigenvalues in (3.3), similarly to what can be proved for GL2 over Q, but we have not implementedthis.

3.2. Trace formula, low-lying length spectrum, and strength of savings. An essential toolto proving Weyl’s law and limit multiplicity theorems is the Arthur-Selberg trace formula. Forconcreteness in this introduction, we use Selberg’s trace formula for GL2 over Q to explain thesalient features. For an even function h in the Paley–Wiener space PW(C) and a congruence

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subgroup Γ of SL2(Z), this formula states that the spectral parameters rj of Maass cusp forms onΓ \H satisfy

(3.4)∑

rj on Γ

h(rj) =vol(Γ \H)

4π

∫Rh(r)r tanhπr dr +

∑[γ] on Γ

f(logNγ)log N γ

N γ1/2 −N γ−1/2+ . . . ,

where f denotes the inverse Fourier transform of h (which is of compact support by our assump-tions), [γ] runs through hyperbolic conjugacy classes in Γ, and the remaining terms arise fromnon-hyperbolic conjugacy classes on the geometric side and the Eisenstein spectrum on the spec-tral side.

After estimating the contributions from the Eisenstein series and the non-identity terms in thegeometric side, Weyl’s law follows by converting the smooth count of (3.4) to a sharp-cutoff. Thisconversion requires local bounds on the discrete spectrum, which itself involves another applicationof the trace formula. See [23, Section 2] for a nice overview. Throughout, one generally works withPaley-Wiener functions which approximate the characteristic function χI of a spectral interval I(or ball, in higher rank). One way of constructing such h is to convolve χI with a suitably niceh0 ∈ PW(C), centered at the origin. In fact, for a parameter R > 0, by convolving rather withthe rescaled function ξ 7→ h0(Rξ) one improves the approximation by a factor of 1/R, since thewalls around the interval I are then of length 1/R. For example, if I = [−T, T ], we may localizer to [−T + O(T/R), T + O(T/R)] and the de-smoothing process in Weyl’s Law incurs an errorof size T/R. Note, however, that the spectral test functions h obtained in this way have Fouriertransforms f supported on a ball of radius R about the origin.

In the case of a fixed level and large eigenvalue, it is possible to localize r within OΓ(1) (andobtain the error term OΓ(T ) in (3.1)) without seeing any of the hyperbolic spectrum. In light ofthe Prime Geodesic Theorem, which states that

(3.5) #primitive γ : logNγ 6 T ∼Γ eT /T,

this approach can be pushed to the limit by entering up to OΓ(log T ) of the hyperbolic spectrum,which leads to the familiar (and currently best available) error term OΓ(T/ log T ) in (3.1).

We have seen in § 3.1 that the analytic conductor count requires, among other things, estimatingNΓ1(q)(T ) in bounded ranges of T with error terms that feature explicit savings in q. Such savingscorrespond to instances of (3.4) for Γ = Γ1(q) such that the support of f is expanding for large q;thus, controlling the number and magnitude of conjugacy classes of γ ∈ Γ1(q) in (3.4) is an essentialingredient in any limit multiplicity-type statement. One can use effective Benjamini-Schramm typestatements [?], adapted to this non-compact setting [?], to show that the number of closed geodesicsof length at most R in Y1(q) is at most O(eCR), for some constant C > 0. (In fact, for the examplecase of this section, namely Y1(q), an elementary argument with congruences shows that there areno closed geodesics of length log q. However, this fact is not robust: it already disappears forΓ0(q) or for the analog of Γ1(q) over number fields.) This control allows us to use, in the traceformula for Γ = Γ1(q), spectral test functions h arising as Fourier transforms of functions supportedup to log q. After some work to estimate all other contributions to (3.4), this indeed allows one toobtain Weyl’s Law for Γ1(q) with an explicit O(T/ log q) error term.

More generally, the proof of our Theorem 1.2 shows that the non-central, non-cuspidal contri-butions may be estimated as O

(eCRNqn−δ

)when the support of f extends as far as R, for some

constant C > 0. (Recall that [GL2(ov) : K1(q)] Nqn.) The exponential factor eCR should becompared to (3.5). The latter shows that R log Nq is an allowable range in which the mainterm dominates, that we may reasonably aim for log Nq-savings in any analogue of Weyl’s Law,and finally that log Nq is a serious barrier of the same fundamental importance as the log Y -barrierin the familiar large-eigenvalue Weyl’s Law.

10

4. Outline of the proof

For any integral ideal q and any subset Ξ of the generic unitary dual of GLn(F∞)1 we put

H(q,Ξ) = π ∈ F : q(π) = q, π∞ ∈ Ξ.For example, if

(4.1) ΩX = π∞ ∈ Πgen(GLn(F∞)1) : q(π∞) 6 X,then

(4.2) |F(Q)| =∑

16Nq6Q

|H(q,ΩQ/Nq)|.

In the parlance of [39, 45], the set H(q,Ξ) is what is called a harmonic family. One of the hallmarksof a harmonic family is that it can be studied by means of the trace formula. It shall be moreconvenient to work with the weighted sum

(4.3) N(q,Ξ) =∑

π∈H(q,Ξ)

dimπK1(q)f ,

which counts each π ∈ H(q,Ω) with a weight corresponding to the dimension of the space of oldforms for πf . The quantities |H(q,Ξ)| and N(q,Ξ) can be related via newform theory, namely,

(4.4) |F(Q)| =∑

16Nq6Q

∑d|q

λn(q/d)N(d,ΩQ/Nq),

where λn = µF ? · · · ? µF is the n-fold Dirichlet convolution of the Mobius function on F . Thedetails of this passage may be found in Section 12.

Having reduced the study of |F(Q)| to that of the above harmonic families, we now outlineour arguments for Theorems 1.1 and 1.2 and highlight several results that may be of independentinterest.

Very roughly, the idea of the proof of Theorem 1.1 is as follows. For a given integral idealq and suitable subset Ξ, we approximate each term N(q,Ξ) by the discrete spectral distributionJdisc(εK1(q) ⊗ fΞ

R) of the trace formula. The quality of this approximation is estimated in Part 2,where we execute the passage from smooth to sharp count of the tempered spectrum in harmonicfamilies. We obtain asymptotic results on the size of spectrum and strong upper bounds on thesize of the complementary spectrum for individual large levels q, which are of independent interest.The successful execution of these steps of course depends on the trace formula input, which entersour argument through suitable applications of Property (ELM). Summing over q and appropriatespectral data as in (4.4) then proves Theorem 1.1.

4.1. Overview of Part 2. In the decomposition (4.4), we broke up the universal count |F(Q)| intoa sum over all non-archimedean discrete data, indexed by integral ideals q of norm up to Q. Wemay further isolate discrete data from continuous data at the archimedean places, by recalling thestructure of Πgen(GLn(F∞)1). We refer the reader to Sections 5.2 and 5.3 for some of the notationalconventions we shall use below.

Let π∞ ∈ Πgen(GLn(F∞)1). Then there exists a standard parabolic subgroup P of GLn(F∞)1

with cuspidal Levi subgroup M , a square-integrable representation δ of M(F∞)1, and ν ∈ h∗M,Csuch that π∞ is isomorphic to the fully induced representation

πδ,ν = IndGLn(F∞)1

P (δ ⊗ eν).

The pair (δ, ν) is uniquely defined up to conjugation by the Weyl group WM . Furthermore, notall ν ∈ h∗M,C arise as parameters; both the hermitian and genericity conditions impose restraints.

Letting h∗M,gen denote the subset of generic parameters, every π∞ ∈ Πgen(GLn(F∞)1) can be rec-ognized as π∞ ' πδ,ν for exactly one choice of standard Levi M , and a Weyl group orbit of δ and

11

ν ∈ h∗M,gen. For example, a nice subset Ξ = Ξ(δ, P ) of the generic unitary dual would be a Weyl

group orbit of a pair (δ, P ), where P is some bounded Borel set in h∗M,gen,Using this separation into discrete and continuous data at the archimedean place, the decompo-

sition (4.4) can then be further refined by summing upon M and δ. In general, for triplets (q, δ, P ),where P is some bounded Borel set in h∗M,gen, we may write refine the notation (4.3) by writing

(4.5) N(q, δ, P ) = N(q,Ξ(δ, P )).

This is still a sharp-cutoff; the passage from smooth to sharp then concerns the smooth approxi-mation of spectral parameter within P .

Here it should be noted that we require uniformity in q, δ, and the domain P , as all of them varyin our average (4.4). As discussed in Section 3.1, this crucially includes the range in which both δand P are bounded, in which we require asymptotics with savings in Nq in the error term. As wasdiscussed in Section 3.2, this requires the use of spectral localizing functions with thin walls, whichin turn requires the use of Arthur Trace Formula (via an application of Property (ELM)) with test

functions f δ,RR on the geometric side of expanding compact support of size R.For each individual q and δ, we analyze the quantity (4.5) for P in the tempered subspace

ih∗M ⊂ h∗M,gen by applying Property (ELM) with an appropriate test function f δ,PR from Section 9.4.

Letting hδ,PR (ν) = trπδ,ν(f δ,PR ), the basic idea is that the characteristic function χP (ν) is very well

approximated by hδ,PR (ν) on points ν that are firmly inside or outside P (in the sense that the entireball B(ν, 1/R) is either contained in or disjoint from P ). Referring to Lemma 9.1 for the precisestatement, this leads in §11.3

N(q, δ, P ) = Jtemp(εK1(q) ⊗ fδ,PR ) + O

( ∞∑`=1

`−NN(q, δ, ∂P (`/R))

),

where ∂P (ρ) denotes the “ρ-boundary of P” (see Section 8.1). The idea is that the first summandis amenable to the application of Property (ELM). We show that the error terms can also beestimated from above by smooth sums that can in turn be estimated by further applications ofProperty (ELM).

The passage from smooth to sharp also incurs a global error, coming from the difference

(4.6) Jdisc(εK1(q) ⊗ fδ,PR )− Jtemp(εK1(q) ⊗ f

δ,PR ).

Note that the spectral sampling functions hδ,PR (ν), being of Paley–Wiener type, act differently onspectral parameters νπ outside the tempered spectrum ih∗M : they exhibit exponential growth inRe νπ. In fact, the rate of exponential growth is directly related to the size R of the supportof the test functions used on the geometric side; see Section 9.4 for details. For this reason, thecontributions from π ∈ Πdisc(G(A)1)δ for which π∞ is not tempered must be estimated separately;specifically, for a suitable parameter R > 0 we require an upper bound for the exponentiallyweighted sum ∑

π∈Πdisc(G(A)1)δIm νπ∈P

dimV K1(q)πf

eR‖Re νπ‖.

This is majorized, using an application of Property (ELM) and a delicate construction of archimedeantest functions, in Section 10.

The passage from smooth to sharp count for N(q, δ, P ) for P in the tempered subspace ih∗M isproperly executed along the above lines in Section 11. Assuming Property (ELM), this leads anestimate of the form

N(q, δ, P ) = ∆∗(1)ϕn(q)

∫P

dµpl∞ + O

(ϕn(q)volR(δ, P )

).

12

where 0 < R 6 c log(2 + Nq) (see Proposition 8.3). We sum this over all discrete parameters δ andlevels q. Bounding the resulting averages of errors terms proves Theorem 1.1.

4.2. Overview of Part 3. In Part 3, we establish Theorem 1.2. The proof naturally divides intotwo parts, corresponding to bounding Jgeom − Jcent on the geometric side and Jspec − JEis on thespectral side. The estimations are not symmetric in the way they are proved, nor in the degreeof generality in which they are stated. We would like to briefly describe these results here, and inparticular explain why we are at present unable to establish Property (ELM) in all cases.

The main result on the geometric side is Theorem 14.1 in which we show the existence of constantsC, θ > 0 such that for any R > 0, integral ideal q, and test function f ∈ H(GLn(F∞)1)R, we have

(4.7) Jgeom(εK1(q) ⊗ f)− Jcent(εK1(q) ⊗ f) eCRNqn−θ‖f‖∞.This can be thought of as a sort of geometric limit multiplicity theorem, although it is only non-trivial in the q aspect. The proof of this occupies most of Sections 14 and 15. Indeed, in §14 wereduce the problem to a local one, and in §15 we bound the relevant local weighted orbital integrals.

The proof of our local estimates relies crucially on several recent developments, due to Finis-Lapid, Matz, Matz-Templier, and Shin-Templier. In particular, a central ingredient in the powersavings in Nq comes from the work of Finis-Lapid [8] on the intersection volumes of conjugacyclasses with open compact subgroups. On the other hand, the source of the factor ‖f‖∞ comesfrom estimating archimedean weighted orbital integrals trivially, by replacing f by the product of‖f‖∞ with the characteristic function of its support. As the latter is, by hypothesis, containedin K∞ exp(B(0, R))K∞, it is enough then to have polynomial control in the support of the testfunction on these weighted orbital integrals. This can be extracted from the papers of Matz [24]and Matz-Templier [25].

Comparing the bound (4.7) to the statement of Property (ELM), it is clear that if one takesf = fΞ

R , then one wants to understand ‖fΞR‖∞ in terms the Plancherel volume of Ξ. It is at this point

that, for technical reasons, we impose the condition that for n > 2 the subset Ξ ⊂ Πgen(GLn(F∞)1)be contained in the spherical unitary dual. In this case, the Paley-Wiener functions we use toapproximate Ξ can be inverted by integration against the spherical function ϕλ. Since ‖ϕλ‖∞ 6 1

for tempered parameters λ, we obtain ‖fΞR‖∞ 6

∫Ξ dµpl

∞, as desired. For GL2, we use a slightlymore general inversion formula, valid for τ -spherical functions, where τ is an arbitrary K∞-type,due to Camporesi [5]; see §16.2. In any case, the test functions fΞ which we use in Property (ELM)are all defined in Section 16, and their main properties are summarized in Proposition 16.1.

On the spectral side, our main result is Theorem 17.1, which roughly states that for R log Nqwe have

Jspec(εK1(q) ⊗ fΞR)− Jdisc(εK1(q) ⊗ fΞ

R)ε Nqn−θ+ε∫

Ξdµpl∞.

The argument uses an induction on n. Indeed the above difference can be written as a sum overproper standard Levi subgroups M 6= G of Jspec,M (εK1(q)⊗ fΞ

R), and each M is a product of GLm’sfor m < n. The induction step itself relies critically on several ingredients. Besides the boundson the geometric side of the trace formula of Sections 14 and 15, and the properties of the testfunctions of Section 16, the proof uses in an essential way the Tempered Winding Number propertyof [11] and the Bounded Degree Property of [10]. Our presentation follows that of several recentworks, such as [24, §15], but differs in the following aspects: we explicate the dependence in theparameter R and in the level q.

4.3. On power savings. All our trace formula estimates, including the Effective Limit Multiplicity(ELM) property, include power savings in Nq, as long as the parameter R (which controls inverselyproportionally the width of the walls in the spectral count) satisfies R log Nq. To see whatkind of an analytic conductor result can be obtained directly with this input, as a model fix a

function hR ∈ C∞c (ih∗M ) with R 6 c logQ with a sufficiently small c > 0, and consider fΞR = f δ,PR

13

with P = ΩQ/Nq ∩ ih∗M such that hδ,PR (ν) = trπδ,ν(f δ,PR ) = 1P ? hR. Assuming property (ELM), bysumming over all Nq 6 Q we can prove that

(4.8)

∑16Nq6Q

∑π∈F

q(π)=q

(hR ? 1ΩQ/Nq)(π∞) = ChRQ

n+1(1 + O(Q−δ)

),

ChR =1

n+ 1∆∗(1) · ζ∗F (1)

∏v<∞

ζv(1)−1νv(F) ·∫

Π(GLn(F∞))1

(hR ? q(·)−n−1)(π∞) dµpl∞(π∞).

This equality may be considered a smooth form of Theorem 1.1, with a power saving error term.It features the expected main term, resolves all difficulties related to the trace formula except forthe restriction on the support of the test function, and involves no passage from smooth to sharpcount.

The equation (4.8) lends some support to the possibility that Theorem 1.1 holds with a power-saving error term. The function hR may be considered as an approximate identity in the convolutionalgebra L1(ih∗M ), and the distinction between (4.8) and Theorem 1.1 boils down to the execution ofthe passage to the limit of hR to the Dirac delta distribution. Of course, the equation (4.8) featuresan infinite sum over π which is only essentially localized on the scale 1/R 1/ logQ. Quantifyingthis localization by passing to the sharp count, we obtain Theorem 1.1 in its present form, and itis exactly at this step that we do not know how to use (4.8) more efficiently in order to preservethe power savings.

Part 1. Preliminaries

5. General notation

The goal of this section is to put in place the basic notation associated with number fields andwith the algebraic group G = GLn.

5.1. Field notation. We recall some standard notation relative to the number field F .Let d = [F : Q] be the degree of F over Q. Let O be the ring of integers of F . For an ideal n

of O let N(n) = |O/n| be its norm. Let r1 and r2 be the number of real and inequivalent complexembeddings of F .

For a normalized valuation v of F , inducing a norm | · |v, we write Fv for the completion of Frelative to v. For v < ∞ let Ov be the ring of integers of Fv, pv the maximal ideal of Ov, $v anychoice of uniformizer, and qv the cardinality of the residue field.

Let ζF (s) =∏v<∞ ζv(s) for Re(s) > 1 be the Dedekind zeta function of F . Write ζ∗F (1) for the

residue of ζF (s) at s = 1. We let A denote the ring of adele ring of F and Af the ring of finiteadeles.

5.2. Subgroup notation. We let G = GLn, viewed as an algebraic group defined over F . Let Zdenote the center of G. Let T0 denote the diagonal torus of G.

A Levi subgroup of G is called semistandard if it contains T0; it is automatically defined over F .Let L denote the finite set of all semistandard Levi subgroups of G. If M ∈ L let L(M) = L ∈ L :M ⊂ L. An F -parabolic subgroup P of G is called semistandard if it contains T0. Let F denotethe finite set of all semistandard F -parabolic subgroups. For P ∈ F , let UP denote the unipotentradical of P and MP the unique semistandard Levi subgroup such that P = MPUP . For M ∈ L letF(M) = P ∈ F : M ⊂ P. Denote by P(M) the subset of F(M) consisting of those F -parabolicsubgroups having Levi component M . Thus P(M) = F(M)−

⋃L)M F(L).

Let P0 denote the Borel subgroup of upper triangular matrices. We call an F -parabolic subgroupP of G standard if it contains P0. Similarly, a semistandard Levi subgroup M of G is standardif it is containted in a standard parabolic subgroup. Write Fst and Lst for the respective subsetsof standard elements. Then Fst and Lst are both in bijection with the set of unordered partitions

14

of n, the correspondence sending a partition (n1, . . . , nr) to the Levi subgroup of block diagonalmatrices in G of the shape GLn1 × · · · ×GLnr .

When P = P0 we write U0 for UP0 and of course MP0 is simply T0.For every M ∈ L let X∗(M)Q be the group of Q-rational characters of the center of M (as a

group over F ). Furthermore, we put

M(A)1 =⋂

χ∈X∗(M)Q

ker(|χ|).

Then M(F ) is a discrete subgroup of finite covolume in M(A)1. More concretely, M(A)1 is theclosed subgroup of M(A) given by those elements such that each component in the block decom-position has determinant whose idelic norm is 1.

Let AM be identity component of the real points of the Q-split part of the center of ResF/QM .

Then AG ⊂ Z∞ is a Lie subgroup. One has a direct product decomposition M(A) = M(A)1×AM .Setting M(F∞)1 = M(A)1 ∩M(F∞) gives M(F∞) = M(F∞)1 ×AM .

5.3. Lie algebras. For M ∈ L we set a∗M = X∗(ResF/QM)Q ⊗Z R and aM = HomR(a∗,R). Wemay identify aM with a∗M via the Killing form whenever convenient. Write a∗M,C and a∗M,C for

their complexifications. The map AM → aM sending a to χ 7→ log |χ(a)| is an isomorphism. IfM ' GLn1 × · · · × GLnm then we can make canonical identifications aM = RrM , a∗M = (RrM )∗,a∗M,C = (CrM )∗, where rM = m(r1 + r2).

For each v | ∞ we now set aMv = X∗(M/Q)Q ⊗Z R and a∗Mv= HomR(aMv ,R), where now

we view M as a group over Q. Their complexifications are denoted aMv ,C and a∗Mv ,C. If M =

GLn1 × · · · × GLnm then we can again make canonical identifications aMv = Rm, a∗Mv= (Rm)∗,

a∗Mv ,C = (Cm)∗. Note that we have the following place by place decompositions

aM =⊕v|∞

aMv , a∗M =⊕v|∞

a∗Mv, a∗M,C =

⊕v|∞

a∗Mv ,C.

It will be important to further decompose these spaces according to various trace-zero conditions.We first do so at each archimedean place, then across all archimedean places. For every v | ∞ lethMv = Xv ∈ aMv : trXv = 0 and h∗Mv

=λv ∈ a∗Mv

: trλv = 0

be the trace-zero hyperplanes,and similarly for h∗Mv ,C. Note that aGv sits inside every aMv as the diagonally embedded copy ofR, and aMv = aGv ⊕ hMv and a∗Mv

= a∗Gv ⊕ h∗Mv. In this way, we get

aM = aG ⊕⊕v|∞

hMv , a∗M = a∗G ⊕⊕v|∞

h∗Mv, a∗M,C = a∗G,C ⊕

⊕v|∞

h∗Mv ,C.

The trace-zero hyperplane which cut across all archimedean places are defined as

hM =

X = (Xv)v|∞ ∈ aM :

∑v

trXv = 0

, h∗M =

λ = (λv)v|∞ ∈ a∗M :

∑v

trλv = 0

.

Denoting a0G = aG ∩ hM and a0,∗

G = a∗G ∩ h∗M , we obtain decompositions

(5.1) hM = a0G ⊕

⊕v|∞

hMv h∗M = a0,∗G ⊕

⊕v|∞

h∗Mv, h∗M,C = a0,∗

G,C ⊕⊕v|∞

hMv ,C.

We shall use (5.1) when constructing test functions in §9.2.

5.4. Roots and weights. For M ∈ L, we let WM = NG(M)/M denote the Weyl group of M .When M = G we simplify WG to W . Denote by 〈 , 〉 the inner product on aM and a∗M induced fromthe Killing form. Extend this to the complexifications aM,C and a∗M,C in the natural way. ThenWM acts by orthogonal transformations on each of these spaces.

15

If µ ∈ a∗M and r > 0 then BM (µ, r) will denote the ball of radius r in a∗M about µ. Finally, letA1R be the exponential of the ball of radius R in aG.Let X∗(T0) denote the lattice of rational cocharacters of T0. We identify X∗(T0) with Zn in the

usual way by associating with λ = (λ1, . . . , λn) ∈ Zn the cocharacter λ 7→ diag($λ11 , . . . , $λn

n ). Let‖ · ‖ denote the Weyl group invariant norm on X∗(T0)⊗R given by ‖λ‖ = maxi maxw∈W |wλi|. LetX+∗ (T0) = λ ∈ X∗(T0) : λ1 > · · · > λn,

∑i λi > 0 denote the set of positive cocharacters.

5.5. Twisted Levi subgroups. Although our interest in this paper is solely in G = GLn, inapplications of the trace formula one encounters more general connected reductive groups, throughthe centralizers of semisimple elements in G(F ).

If γ ∈ G(F ), we write Gγ for the centralizer of γ. If γ = σ is semisimple, then Gσ is connectedand reductive. Moreover, one knows that in this case Gσ is a twisted Levi subgroup, meaningthat there are field extensions E1, . . . , Er of F and non-negative integers n1, . . . , nr such that, ifE = E1 × · · · × Er, then Gσ = ResE/F (GLn1 × · · · ×GLnr).

If H is a twisted Levi subgroup of G containing some M ∈ L, then an F -Levi subgroup of Hwill be called semistandard (resp., standard) if it is the restriction of scalars of a semistandard(resp., standard) Levi subgroup. We similarly extend the notions of semistandard and standardto F -parabolic subgroups of H. We let LH (resp., FH) denote the set of semistandard F -Levisubgroups (resp., F -parabolic subgroups) of H. If M ∈ LH we write LH(M) = L ∈ LH : M ⊂ Land FH(M) = P ∈ FH : M ⊂ P. Finally, P(M) will denote the subset of FH consisting ofparabolics having Levi component M .

If H is a twisted Levi subgroup of G, and v is a finite place, we write KHv = Kv∩H(Fv). Similarly,

let KH∞ = K∞∩H∞. Then KH

v (resp., KH∞) is a maximal compact subgroup of H(Fv) (resp., H∞).

More generally, for r > 0 and a finite place v we put KH1,v(p

rv) = K1,v(p

rv)∩H(Fv). If integral ideal

q =∏

prv ||q let KH1 (q) = K1(q) ∩H(Af ). Then if q =

∏prv ||q we have KH

1 (q) =∏

prv ||qKH1,v(p

rv).

5.6. Open compact subgroups. At every place v of F let Kv = G(Ov),O(n), or U(n) accord-ing to whether v < ∞, v = R or v = C. One then has the Iwasawa decomposition G(Fv) =U0(Fv)T0(Fv)Kv. Let Kf =

∏v<∞Kv be the standard maximal compact open subgroup of G(Af )

and let K∞ =∏v|∞Kv. If we put K =

∏v Kv = KfK∞, then K is a maximal compact subgroup

of G(A), and we have G(A) = U0(A)T0(A)K.At a finite v and an integer r > 0 write K1,v(p

rv) for the subgroup of Kv consisting of matrices

whose last row is congruent to (0, 0, . . . , 1) mod prv. In particular, when r = 0 we obtain themaximal compact Kv. Then for an integral ideal q, whose completion in Af factorizes as

∏prvv , we

define an open compact subgroup of G(Af ) by K1(q) =∏v<∞K1,v(p

rvv ). We set

(5.2) ϕn(q) = |Kf/K1(q)| = N(q)n∏p|q

(1− 1

N(p)n

)= (µ ? pn)(q),

where pn(n) = N(n)n is the power function. When n = 1 and F = Q this recovers the Eulerϕ-function.

5.7. Automorphic notation. We write Πdisc(M(A)1) for the set of equivalence classes of ir-reducible unitary representations of M(A)1 which are equivalent to a subrepresentation of theright-regular representation on L2

disc(M(F )\M(A)1). Let Πdisc(M(A)1)δ denote the subset of thoseπ ∈ Πdisc(M(A)1) such that δπ∞ ' δ.

5.8. Canonical measures. We shall need a uniform way of fixing measures on the centralizersthat arise in the trace formula, so that we can speak of the associated adelic volumes and orbitalintegrals. The theory of canonical measures, developed by Gross in [13], will be useful for this.

Let H be a connected reductive group over F . Let MotH be the motive attached by Gross toH. For any place v of F one has the associated local factor L(Mot∨H,v(1)). Let ωH,v be the Gross

16

canonical measure of H [13, §4] and write µH,v = L(Mot∨H,v(1))|ωH,v|. For any finite set of placesS we write µH,S =

∏v∈S µH,v. Since for almost all finite places v, the measure µH,v assigns a

hyperspecial subgroup of H(Fv) measure 1, we may define µH =∏v µH,v.

Then the global canonical measure is the product measure µH =∏v µH,v. Give the automorphic

space H(F )\H(A) the quotient measure by the counting measure on H(F ). In [13], it is shownthat µH(H(F )\H(A)) = L(MotH)τ(H), where τ(H) is the Tamagawa number of H.

For example, for G = GLn we have MotG = Q+Q(−1)+· · ·+Q(1−n) and L(Mot∨G,v(1)) = ∆v(1),where ∆v is defined in (1.2). The canonical measure in this case is

dµG,v(g) = ∆v(1)| det(g)|−n∏i,j

dgij ,

where g = (gij). Since τ(G) = 1, we deduce that µG(G(F )\G(A)1) = ∆∗(1).

5.9. Plancherel measure. For any place v we write Π(G(Fv)) for the unitary dual of G(Fv),

endowed with the Fell topology. Let µplv denote the Plancherel for Π(G(Fv)). Having fixed the

measure µG,v to define the trace, we may normalize µplv so as to satisfy

φv(e) =

∫Π(G(Fv))

tr (πv)(φ∨v ) dµpl

v (πv)

for any φv ∈ C∞c (G(Fv)), where φ∨v (g) = φv(g−1).

5.10. Weyl discriminant. Let v be a place of F and σ a semisimple element in G(Fv). Let Gσ(Fv)be the centralizer of σ and gσ its Lie algebra inside g, the Lie algebra of G(Fv). Then the Weyldiscriminant of σ in G is defined to be

DG(x) = det(1−Ad(σ)|g/gσ).

This is an element in F×v . If an arbitrary γ ∈ G(Fv) has Jordan decomposition γ = σν, whereσ is semisimple and ν ∈ Gσ(Fv) is nilpotent, then we extend the above notation to γ by settingDG(γ) = DG(σ). The function γ 7→ |DG(γ)|v on G(Fv) is locally bounded and discontinuous atirregular elements. In our estimates on orbital integrals in the latter sections, it is this functionwhich will measure their dependency on γ.

5.11. Hecke algebras. The following discussion will be of use in formal calculations involving thetrace formula.

5.11.1. Local case. At any place v we define C∞c (G(Fv)) to be the space of functions on G(Fv) whichare locally constant and of compact support, for v finite, and smooth and of compact support for vinfinite. We then let H(G(Fv)) denote C∞c (G(Fv)), when considered as a convolution algebra withrespect to the measure µG,v. For non-archimedean v, and an open compact subgroup Kv of G(Fv),let

(5.3) εKv =1

µG,v(Kv)1Kv

denote the corresponding idempotent in H(G(Fv)).Given an admissible representation πv of G(Fv) any φv ∈ H(G(Fv)) define an operator on the

space of πv via the averaging

πv(φv) =

∫G(Fv)

φv(g)πv(g) dµG,v(g).

This is a trace class operator; we write trπv(φv) for its trace. If, for a finite place v, Kv is an opencompact subgroup of G(Fv), it is straightforward to see that trπv(εKv) = dimπKvv .

Similarly, for a finite set of places S containing all archimedean places, we denote by H(G(FS))the space of finite linear combinations of factorizable functions ⊗v∈S φv, where each φv lies in

17

H(G(Fv)). Convolution is taken with respect to the measure µG,S . Let H(G(FS)1) denote thespace of functions on G(FS)1 obtained by restricting those in H(G(FS)).

Finally, denote by H(G(F∞)1)R the space of all smooth functions on G(F∞)1 supported inK∞A

1RK∞.

5.11.2. Global case. The global Hecke algebra H(G(A)) is defined as the space of finite linearcombinations of factorizable functions ⊗v φv, where each φv lies in H(G(Fv)) and φv = 1Kv foralmost all finite v. Convolution is taken with respect to the canonical measure µG. We defineH(G(A)1) by restricting functions from H(G(A)). For admissible π = ⊗vπv and φ ∈ H(G(A)1) wedefine the trace-class operator π(φ) with respect to µG. Moreover, for admissible π = ⊗vπv andfactorizable φ = ⊗v φv ∈ H(G(A)1) the global trace trπ(φ) factorizes as

∏v trπv(φv).

Similarly, if S is any finite set of places of F containing all archimedean places, we let H(G(AS))denote the analogous space, with convolution taken with respect to the measure µSG.

6. Local and global conductors

In this section we review the representation theory of archimedean GLn and the Plancherelmeasure for the unitary dual. In particular, we define the archimedean conductor.

For a place v of F , let πv be an irreducible admissible representation of GLn(Fv). Let L(s, πv)denote the standard L-function of πv, as defined by Tate [46] (for n = 1) and Godement-Jacquet[12] (for n > 1).

6.1. Non-archimedean case. Let v be a non-archimedean place. For an additive character of levelzero ψ, let ε(s, πv, ψ) be the local espilon factor of πv. Then there is an integer f(πv), independent

of ψ, and a complex number ε(0, πv, ψ) of absolute value 1 such that ε(s, πv, ψ) = ε(0, πv, ψ)q−f(πv)sv .

Moreover, f(πv) = 0 whenever πv is unramified.Under the additional assumption that πv is generic, Jacquet, Piatetski-Shapiro, and Shalika [19],

show that the integer f(πv) is in fact always non-negative. One then calls f(πv) the conductor

exponent of πv. The conductor q(πv) of a generic irreducible πv is then defined to be q(πv) = qf(πv)v .

In particular, q(πv) = 1 whenever πv is unramified.For v finite, Jacquet, Piatetski-Shapiro, and Shalika [19], building on work of Casselman [6],

show that for any irreducible generic representation πv of GLn(Fv) the conductor exponent f(πv)is equal to the smallest non-negative integer r such that πv admits a non-zero fixed vector underK1,v(p

rv). Moreover, the space of all such fixed vectors is of dimension 1. By the subsequent work

of Reeder [37] it follows that for irreducible generic πv with q(πv)|q one has

(6.1) dimπK1,v(q)v = dn(q/q(πv)),

where dn = 1 ? · · · ? 1 is the n-fold convolution of 1 with itself. In particular, if πv is an irreduciblegeneric representation of GLn(Fv), one has

(6.2) tr(πv(εK1,v(q(πv)prv))

)= dn(prv).

6.2. Archimedean conductor. For v archimedean the local L-factor of πv is a product of shiftedGamma factors. To describe them, we first recall the parametrization of the unitary dual ofarchimedean GLn.

Assume that M(Fv) is a product of blocks of size n1, . . . , nr, where n1 + · · ·+ nr = n, and writeδ = δ1 ⊗ · · · ⊗ δnr . For M(Fv) to admit discrete series, we must have 1 6 ni 6 2 for v real andni = 1 for v complex, which we will henceforth assume. Let σi = δie

νi ∈ E2(M(Fv)) and writeλσi = λδi + νi denote its infinitesimal character.

Note that E2(GL1(C)) = χk : k ∈ Z, where χk is the unitary character z 7→ (z/|z|)k. Fur-thermore, E2(GL1(R)) = sgnε : ε = 0, 1 and E2(GL2(R)) = Dk : k > 2, where Dk denotes theweight k discrete series representation.

18

With the above parametrization, we have

L(s, πv) =

r1∏j=1

Γv(s− λσi)r2∏j=1

Γv(s− λσi)Γv(s+ 1− λσi).

As is customary we have put ΓR(s) = π−s/2Γ(s/2) and ΓC(s) = 2(2π)−sΓ(s); thus ΓC(s) =ΓR(s)ΓR(s + 1). Given this expression for the local L-factor, Iwaniec and Sarnak [17] define theconductor q(πv) of πv as

q(πv) =r∏j=1

(1 + |λσj |)dvnvj .

6.3. Global (analytic) conductor. Let π = ⊗vπv be a unitary cuspidal automorphic representa-tion of GLn(A). From the work of Jacquet-Piatetski-Shapiro-Shalika [20] one knows that the localcomponents of cusp forms on GLn are generic. The analytic conductor of π, is defined as

Q(π) =∏v

q(πv).

As almost all local components of the global cusp form π are unramified, the product over all vmakes sense as a finite product.

7. Local conductor zeta functions

In this section we examine the local conductor zeta functions, defined at each place v by theintegral

Zv(s) =

∫Π(G(Fv))

q(πv)−s dµpl

v (πv),

where the complex parameter s has large enough real part to ensure absolute convergence. Theseintegrals appear in the expression for the leading term constant in our main theorems.

7.1. Non-archimedean local integrals. For a finite place v and an ideal q = pdv let Mv(q) =∫q(πv)=q dµpl

v (πv) be the Plancherel measure of those tempered πv with q(πv) = q. For Re(s) large

enough, we have

Zv(s) =∑r>0

M(prv)q−rsv .

Lemma 7.1. We have

Mv(q) =∑d|q

λn+1(d)N(q/d)n and Zv(s) =ζv(s− n)

ζv(s)n+1.

In particular, Zv(n+ 1) = ζv(1)/ζv(n+ 1)n+1.

Proof. Applying Plancherel inversion to the idempotent εK1,v(q) we obtain

1

µG,v(K1,v(q))=

∫Π(G(Fv))

dimVK1,v(q)πv dµpl

v (πv).

From (5.2), the left-hand side is [K1,v(q) : Kv]/µG,v(Kv) = [K1,v(q) : Kv] = ϕn(q). Thus, from(6.2), we get

ϕn(q) =

∫q(πv)|q

dn(q/q(πv)) dµplv (πv) =

∑d|q

dn(q/d)Mv(d) = (dn ?Mv)(q).

19

By Mobius inversion (and associativity of Dirichlet convolution) this gives Mv(q) = (λn ? ϕn)(q) =(λn+1 ? pn)(q) =

∑d|q λn+1(d)N(q/d)n. From∑r>0

λn+1(prv)q−rsv =

1

ζv(s)n+1,

∑r>0

pn(prv)q−rsv = ζv(s− n),

we obtain the value of Zv(s).

7.2. Plancherel majorizer for archimedean GLn. The work of Harish-Chandra allows us to

explicitly describe the Plancherel measure µplv . Namely, for every measurable h on Π(G(Fv)) we

have

(7.1)

∫Π(G(Fv))

h(πv)µplv (πv) =

∑M∈Lv

cM∑

δ∈E2(M(Fv)1)

deg(δ)

∫ia∗M

h(πδ,ν)µGM (δ, ν)dν,

where deg(δ) is the formal degree of δ, cM are constants depending only on M , and µGM (δ, ν) isHarish-Chandra’s µ-function.

We would now like to describe an explicit majorizer for the Plancherel density function µGM (δ, ν),when v is archimedean. Using the notation of §6.2, we put

βv(δ, ν) =∏

16i<j6r

(1 + |λσi + λσj |)dvnvinvj .

The quantity βv(δ, ν) is closely related to conductor of the associated local Rankin-Selberg L-function, as will become clear in the proof of the following result.

Lemma 7.2. Let v be an archimedean place. Let M ∈ Lv be cuspidal. For δ ∈ E2(M(Fv)1) and

ν ∈ a∗M,C, we have µGM (δ, ν) βv(δ, ν).

Proof. Then the Harish-Chandra product formula expresses µGM (δ, ν) as

µGM (δ, ν) =∏

16i<j6r

µGLni+njGLni×GLnj

(δi ⊗ δj , νi ⊗ νj),

It thus suffices to describe the Plancherel density function for Levi factors GLn × GLn′ of max-imal parabolic subgroups of GLn+n′ . The latter are given as quotients of local Rankin-SelbergL-functions, namely

µGLn+n′GLn×GLn′

(δ ⊗ δ′, ν ⊗ ν ′) = deg(δ ⊗ δ′)−1Lv(1 + ν − ν ′, δ × δ′)Lv(ν − ν ′, δ × δ′)

Lv(1− ν + ν ′, δ × δ′)Lv(−ν + ν ′, δ × δ′)

.

The archimedean local Rankin-Selberg factor (over R) is explicated, for example, in [38, Appendix].

Let φv(s) = Γv(1 + s)/Γv(s). Now by Stirling’s formula we have φv(s) (1 + |s|)dv/2. Applyingthis to the above factors, and taking the product over pairs of Levi subgroups, we deduce the statedbounds.

We define a measure βv(πv)dπv on Π(G(Fv)) by putting

(7.2)

∫Π(G(Fv))

h(πv)βv(πv)dπv =∑M∈Lv

∑δ∈E2(M(Fv)1)

deg(δ)

∫ia∗M

h(πδ,ν)βGM (δ, ν)dν

for any measurable h on Π(G(Fv)). In other words, we have replaced the Plancherel density functionµGM (δ, ν) of (7.1) by the above majorizer.

20

7.3. Plancherel measure of conductor intervals for archimedean GLn. Let

Z∞(s) =

∫Π(G(F∞)1)

q(π∞)−s dµpl∞(π∞).

Lemma 7.3. For Q > 1 we have∫Q6q(π∞)62Q

β(π∞)dπ∞ Qn−1/d.

In particular Z∞(s) converges absolutely for s ∈ C with Re s > n− 1/d.

Proof. Using the decomposition (7.1) the above volume is equal to∑M∞ cuspidal

cM∑

δ∈E 2(M1∞)

‖δ‖62Q

deg(δ)

∫q(πδ,ν)∼Q

µGM (δ, ν) dν,

where the integral runs over ν ∈ ih∗M such that Q 6 q(πδ,ν) 6 2Q. Let X = (Xvj) denote acollection of dyadic integers Xvj > 1 indexed by j = 1, . . . , n and v | ∞. For M ,δ, and X let

Eδ,X = ν ∈ ih∗M : (1 + |λvj |)nvj ∼ Xvj ∀ v, j.

Here nvj = dimMi(Fv). If S(Q) denotes the set of triplets (M, δ,X ) for which∏v

∏j X

dvvj ∼ Q,

then the Plancherel volume appearing in the lemma is bounded by

(7.3)∑S(Q)

vol(Eδ,X ) maxν∈Eδ,X

β∞(δ, ν).

By vol(Eδ,X ) we of course mean with respect to Lebesgue measure in ih∗M .We refine this decomposition according to how deep into a spike the dyadic region Eδ,X lies.

For any given (M, δ,X ) ∈ S(Q) let X = maxXvj . Let mv be the number of indices j for whichXvj ∼ X and put m(X ) =

∑v dvmv. Then any (M, δ,X ) for which Boxδ,X is non-empty satisfies

m(X ) > 2. We then have S(Q) = ∪m>2Sm(Q), where (M, δ,X ) ∈ Sm(Q) satisfies m(X ) = m.For (M, δ,X ) ∈ Sm(Q) we have vol(Eδ,X ) = O(Xm−1

∏Xvj<X

Xvj). Next, applying the inequal-

ity 1 + |a− b| 6 (1 + |a|) + (1 + |b|) to each factor in the definition of βv(δ, ν) and then multiplyingout, we get

βv(δ, ν) maxw∈Sr1

r1∏j=1

(1 + |λσwj |)dv(r1−j) maxw∈Sr

r∏j=1

(1 + |λσwj |)2(r−j) maxw∈Sr2

r2∏j=1

(1 + |λσwj |)4(r2−j).

We deduce that maxν∈Eδ,X β∞(δ, ν) XA∏Xvj<X

Xn−jvj , where

A =∑v

dv(mvn−mv(mv + 1)/2) = mn− 1

2

∑v

dvmv(mv + 1).

Terms in (7.3) indexed by (M, δ,X ) ∈ Sm(Q) are therefore at most Xm−1+A∏Xvj<X

Xn−j+1vj .

As (M, δ,X ) runs over S(Q) the last expression is largest whenever X takes the largest allowed

value, namely X = Q1/m. This forces Xvj = 1 whenever Xvj < X, yielding the upper bound

Qn+1/2−(1/m)(1+∑v dvm

2v/2). This in turn is largest whenever mv is as smallest as allowed by the

constraint m > 2. When there is just one place v (so that dv = d), this means m = mv = 2, so

that the total volume is majorized by Qn−1/d. For d > 2 this means that mv 6 1 for all v, withequality holding for at least two places v. In this case one sees that the total volume is majorizedby Qn−1/m. But since mv 6 1 we have m 6 d. Thus producing an upper bound of Qn−1/d.

21

Remark 7.1. A more elegant proof of Lemma 7.3 would proceed analogously to that of Lemma 7.1.Although we have not worked out the details, we sketch here the main idea, based on some remarksof Paul Nelson. Let K1,∞(Q) be the subset of matrices in GLn(F∞)1 whose last row is within 1/Qof (0, . . . , 0, 1). Let εK1,∞(Q) denote a smooth approximation of K1,∞(X), divided by the volume

of K1,∞(Q) (roughly Qn−1/d). One would then want to show that trπ∞(εK1,∞(Q)) roughly picks

out those generic π∞ ∈ Π(G(F∞)1) with q(π∞) 6 Q (presumably with some logarithmic weight).Applying Plancherel inversion would then yield a result similar to Lemma 7.3.

Remark 7.2. Despite the “spikes” introduced by the product condition q(πδ,ν) 6 X, the asymp-totics of the Plancherel measure of the sets ν ∈ ih∗M : q(πδ,ν) 6 X as X →∞ feature pure powergrowth without logarithmic factors. This is due to the following two facts: first, the Plancherel den-sity increases into the spikes; and, second, these spikes are somewhat moderated by the trace-zerocondition. To visualize this latter feature, we include the following graphics.

Figure 1 Figure 2Figure 3

In Figure 1, the hyperboloid (1 + |x|)(1 + |y|)(1 + |z|) 6 X is drawn in R3. The spikes extend asfar as X. The intersection with x + y + z = 0 is indicated in bold and reproduced in the planein Figure 2. The spikes extend as far as X1/2.

In Figure 3, the set (1 + |x|)(1 + |y|) 6 X is drawn in R2 with spikes as far as X and volume

X logX. The intersection with x+ y = 0 is in bold. This produces a segment of length X1/2.

Part 2. Tempered count

8. Preparation for the proof of Theorem 1.1

Our principal aim in Part 2 is to establish the following result.

Theorem 8.1. Assume Property (ELM). Then for all Q > 3 we have

|F(Q)| = 1

n+ 1∆∗(1)‖ν(F)‖Qn+1 +O(Qn+1/ logQ).

The proof of Theorem 8.1 will pass through a uniform asymptotic expression for the countingfunctions N(q, δ, P ) of (4.5). To formulate this precisely, we will first need to define what class ofsubsets P we consider and associate with them an (1/R)-thickened volume volR(δ, P ). Once theseconcepts are in place we state, at the end of this short section, the desired asymptotic expressionfor N(q, δ, P ) in Proposition 8.3.

22

8.1. Nice sets and their boundaries. For M ∈ L we let BM be the σ-algebra of all Borel-measurable subsets of ih∗M , and let P ∈ BM . For every ρ > 0, let

P (ρ) = µ ∈ ih∗M : B(µ, ρ) ⊆ P, P •(ρ) = µ ∈ ih∗M : B(µ, ρ)∩P 6= ∅, ∂P (ρ) = P •(ρ)\P (ρ),

where B(µ, ρ) denotes the open r-ball of radius ρ centered at µ. Then, for every point µ ∈ ∂P (ρ),there are points ν1, ν2 ∈ B(µ, ρ) with ν1 ∈ P , ν2 6∈ P , and hence by a continuity argument there isa point ν on the boundary ∂P such that |µ− ν| < ρ; in other words,

(8.1) ∂P (ρ) ⊂⋃ν∈∂P

B(ν, ρ).

We will only use (8.1) for compact regions P with a piecewise smooth boundary (although it isvalid for every P ∈ BM ).

We record a few simple facts. For any bounded Borel set P ∈ B(ih∗) and ρ2 > ρ1 > 0, let

P •(ρ1, ρ2) = P •(ρ2) \ P •(ρ1), P (ρ1, ρ2) = P (ρ1) \ P (ρ2).

Definition 8.1. Let X,Y ⊆ ih∗M and r > 0. We say that X is r-contained in Y if, for every µ ∈ X,B(µ, r) ⊆ Y .

With these notions, we are ready for the following simple lemma.

Lemma 8.2. Let P ∈ BM be a bounded Borel set. Then:

(1) For every ρ, r > 0, the set ∂P (ρ) is r-contained in ∂P (ρ+ r).(2) For every ρ2 > ρ1 > r > 0, the set P •(ρ1, ρ2) is r-contained in P •(ρ1 − r, ρ2 + r), and the

set P (ρ1, ρ2) is r-contained in P (ρ1 − r, ρ2 + r).

Proof. The initial claims follow essentially by the triangle inequality as follows.For the first claim of (2), we need to prove that, if ν ∈ P •(ρ1, ρ2), then B(ν, r) ⊂ P •(ρ1−r, ρ2+r).

Indeed, there is a ν2 ∈ B(ν, ρ2) ∩ P while B(ν, ρ1) ∩ P = ∅. Therefore, if ν1 ∈ B(ν, r), thenν2 ∈ B(ν1, ρ2+r)∩P and so ν1 ∈ P •(ρ2+r). On the other hand, we must have B(ν1, ρ1−r)∩P = ∅,for if ν3 ∈ B(ν1, ρ1− r)∩P , then ν3 ∈ B(ν, ρ1)∩P , a contradiction; and so ν1 6∈ P •(ρ1− r), as wasto be shown.

It is shown analogously that, if ν ∈ P (ρ1, ρ2), then B(ν, r) ⊂ P (ρ1 − r, ρ2 + r). Indeed, wehave that B(ν, ρ1) ⊆ P , while there is a point ν2 ∈ B(ν, ρ2) \ P . Therefore, if ν1 ∈ B(ν, r), then,if ν3 ∈ B(ν1, ρ1 − r), then ν3 ∈ B(ν, ρ1) and hence ν3 ∈ P , so that B(ν1, ρ1 − r) ⊆ P and soν1 ∈ P (ρ1 − r). On the other hanse, ν2 ∈ B(ν1, ρ2 + r) \ P , so that ν1 6∈ P (ρ2 + r), completingthe proof of the second claim of (2).

The claim (1) is proved analogously.

8.2. Tempered count for fixed discrete data. It will be convenient to consider the followingfamily

(8.2) BM =P ∈ BM : P is bounded and ∀ρ > 0, P (ρ), P •(ρ), ∂P (ρ) ∈ BM

.

For example, every compact region with a piecewise smooth boundary clearly belongs to BM .Let P ∈ BM , R > 0, and N ∈ N. Then we define

(8.3) ∂ volR(δ, P ) =∞∑`=1

`−N∫∂P (`/R)

βGM (δ, ν)dν, volR(δ, P ) =

∫PβGM (δ, ν)dν + ∂ volR(δ, P ),

suppressing the dependence on N in the notation. If we want to emphasize this dependence, weshall write ∂ volN,R(δ, P ) or volN,R(δ, P ). Note that

volR(δ, P ) ∫ih∗M

(1 + ‖ν − Y ‖

)−NβGM (δ, ν)dν

and that volR1(δ, P ) volR2(δ, P ) if R1 R2.

23

The central ingredient to the proof of Theorem 8.1 is then the following result.

Proposition 8.3. Let M ∈ L∞ and δ ∈ E2(M1∞). Assume Property (ELM) for (M, δ). There

are constants c, C > 0 such that for any P ∈ BM , integral ideal q with Nq > C, and 0 < R 6c log(2 + Nq), we have

N(q, δ, P ) = ∆∗(1)ϕn(q)

∫P

dµpl∞ + O

(ϕn(q)volR(δ, P )

).

If Nq 6 C, then the above holds with the first term replaced by O(ϕn(q)

∫P dµpl

∞).

Proposition 8.3 will be proved in Section 11, after having introduced an appropriate class of testfunctions in Section 9 and estimated the (exponentially weighted) discrete spectrum in Section 10.Then, in Sections 12 and 13, we make the deduction from Proposition 8.3 to Theorem 8.1.

9. Spectral localizing functions

Let v be an archimedean place. In this section, given a cuspidal Levi subgroup M ∈ L∞,spectral parameters µ ∈ a∗M,C and δ ∈ E2(M1

∞), and a real number R > 0, we define a function

f δ,µR ∈ H(G1∞)R such that

hδ,µR : (τ, λ) 7−→ trπτ,λ(f δ,µR )

localizes about δ, µ. We shall do this first for each archimedean place individually, then glue themtogether, using the decomposition (5.1). Our presentation is based on [27, §4.2], but is slightlymore general in that we work with arbitrary Levi subgroups M .

9.1. Localization around µv. Let v | ∞. Fix a Levi subgroup Mv ∈ Lv. For a function g ∈C∞c (hMv) and λ ∈ h∗Mv ,C let

g(λ) =

∫hMv

g(X)e〈λ,X〉dX

denote the Fourier transform of g at λ. The image of C∞c (hMv) under this map is the Paley–Wienerspace P(h∗Mv ,C). Recall that

P(h∗Mv ,C) =⋃R>0

P(h∗Mv ,C)R,

where P(h∗Mv ,C)R consists of those entire functions h on h∗Mv ,C such that for all k > 0 we have

supλ∈h∗Mv,C

|h(λ)|e−R‖Reλ‖(1 + ‖λ‖)k

<∞.

In fact the Fourier transform C∞c (hMv) → P(h∗Mv ,C) is an isomorphism of topological algebras.

Moreover, for R > 0, if C∞c (hMv)R denotes the subspace of g ∈ C∞c (hMv) having support in theball B(0, R) = X ∈ hMv : ‖X‖ 6 R, then the Fourier transform maps C∞c (hMv)R to P(h∗Mv ,C)R.

Let g1 ∈ C∞c (hMv) be non-negative function satisfying g1(0) = 1. Let h1 = g1 be its Fouriertransform, which we assume to be non-negative on ih∗Mv

. We may assume that

(9.1)

∫B(0,1)

h1(λ) dλ >1

2.

Let gR(λ) = g1(R−1λ). For µv ∈ h∗Mv ,C we then put gµvR (X) = gR(X)e〈µv ,X〉. Then if hR = gR and

hµvR = gµvR we have hµvR (λ) = hR(λ− µv). Note that gµvR ∈ C∞c (hMv)R and hµvR ∈P(h∗Mv ,C)R.

24

9.2. Localization around (δ, µ). For a cuspidal M ∈ L∞ we fix δ ∈ E2(M1∞) and µ ∈ h∗M . We fix

additionally a real parameter R > 0. Our aim is to construct a function hδ,µR which concentratesaround the pair (δ, µ) and lies in a generalized Paley–Wiener class, to be described in the nextparagraph.

The first step is to define a function hµR ∈ P(h∗M,C)R which concentrates around µ. Afterintroducing a spectral localizer on the center, we will essentially reduce this construction to thatof the previous section.

We begin by introducing a spectral localizer on the abelian Lie algebra a0,∗G,C. We let : C∞c (a0

G)→P(a0,∗

G,C) be the Fourier transform. Let g0 ∈ C∞c (a0G) be such that g0(0) = 1. For a real parameter

R > 0 we write

gR(X) = g0(R−1X) and hR = gR.

If µZ ∈ a0,∗G we let

gµZR (X) = gR(X)e〈µZ ,X〉 and hµZR = gµZR .

Then we have hµZR (λ) = hR(λ− µZ).Let µ ∈ h∗M . Recall the decomposition (5.1), which we use to write µ = µZ +

∑v|∞ µ

0v. Define

hµR(λ) = hµZR (λZ)∏v|∞

hµ0vR (λ0

v).

Then hµR ∈P(h∗M,C)R. Finally we simply define

hδ,µR (τ, λ) =1

|WM |∑

w∈WM

1δ(wτ)hµR(wλ).

9.3. Approximating functions. For a set P ∈ BM and R > 0, we put

hδ,PR (τ, ν) =

∫Phδ,µR (τ, ν) dµ.

Note that if Property (ELM) holds for (δ,M) then it holds using hδ,PR for any P ∈ BM with errorterm

(9.2) Jerror(εK1(q) ⊗ fδ,PR ) Nqn−θ

∫PβGM (δ, ν)dν.

In the following lemma, we quantify the extent to which hδ,PR approximates the characteristicfunction of the union of all Weyl group translates of δ × P .

Lemma 9.1. Let P ∈ BM and R > 0. Then:

(1) For every N ∈ N,

hδ,PR (τ, ν)N

eR‖Re ν‖, ν ∈ h∗M,C;

eR‖Re ν‖(Rρ)−N , ν ∈ h∗M,C, Im ν 6∈ P •(ρ).

(2) For every ν ∈ ih∗M we have 0 6 hδ,PR (τ, ν) 6 1 and, for every ρ > 0, N ∈ N,

hδ,PR (τ, ν) =

1 + ON

((Rρ)−N

), ν ∈ P (ρ),

ON

((Rρ)−N

), ν 6∈ P •(ρ).

Proof. The estimates in (1) follow from Fourier inversion and an application of the trivial bound(in the first case) and a standard application of integration by parts (in the second case).

25

Next, let ν ∈ ih∗M . The inequality 0 6 hδ,PR (ν) 6 1 follows immediately from

hδ,PR (τ, ν) =1

|WM |∑

w∈WM

1δ(wτ)

∫R(wν−P )

h0(µ) dµ,

the non-negativity of h0, and the normalization g0(0) = 1.If ν ∈ P (ρ), then

hδ,PR (τ, ν) =1

|WM |∑

w∈WM

1δ(wτ)

(∫ih∗M

h0(µ) dµ+ O

(∫B(0,Rρ)c

h0(µ) dµ

))

=1

|WM |∑

w∈WM

1δ(wτ) + O

(∫ ∞Rρ

(1 + t

)−N−rtr−1 dt

)= 1 + O

((Rρ)−N

),

where B(0, Rρ) denotes the ball of radius Rρ in ih∗M . If ν 6∈ P •(ρ), then, analogously,

hδ,PR (τ, ν) = O

(∫B(0,Rρ)c

h0(µ) dµ

)= O

(∫ ∞Rρ

(1 + t

)−N−rtr−1 dt

)= O

((Rρ)−N

).

This establishes the estimates in (2).

9.4. Paley–Wiener theorem of Clozel–Delorme. The Paley–Wiener theorem of Clozel–Delorme[7] states that given a collection of functions (HM )M∈L∞ , such that each

HM : E2(M1∞)× h∗M,C → C

satisfies the conditions that

(1) HM is of finite support in δ ∈ E2(M1∞);

(2) for every δ ∈ E2(M1∞) the function

hM,δ : ν 7→ HM (δ, ν)

lies in the Paley–Wiener space P(h∗M,C)R;

(3) HM (δ, ν) = HM (wδ,wν) for every w ∈WM ,

there exists a Φ ∈ H(G1∞)R such that trπδ,ν(Φ) = HM (δ, ν).

It follows directly from their definition that spectral localizing functions hδ,µR satisfy the above

conditions. We deduce then that there is f δ,µR ∈ C∞c (G(F∞)1)R such that

hδ,µR (τ, λ) = trπτ,λ(f δ,µR ).

10. Bounding the discrete spectrum

Let M ∈ L, δ ∈ E 2(M∞), and P ⊂ h∗M . Let q be an integral ideal. Then for a real parameterR > 0 we introduce the exponentially weighted sum of the discrete spectrum

DR(q, δ, P ) =∑

π∈Πdisc(G(A)1)δIm ξπ∈P

dimV K1(q)πf

eR‖Re ξπ‖.

In the special case when P = BM (µ, 1/R), the ball of radius 1/R centered about µ in h∗M , we writeDR(q, δ, µ) in place of DR(q, δ, BM (µ, 1/R)).

The goal of this section is to prove the following result.

Proposition 10.1. Assume Property (ELM). Let M ∈ L and δ ∈ E 2(M1∞). Then there is c > 0

such that for 1 6 R 6 c log(2 + Nq) and µ ∈ h∗M we have

DR(q, δ, µ) ϕn(q)

∫BM (µ;1/R)

βGM (δ, ν)dν.

26

As will become clear shortly, the bulk of the work necessary to prove Proposition 10.1 is theconstruction of appropriate test functions. This turns out to be a highly non-trivial analyticproblem.

10.1. Existence of test functions. Granting ourselves the function hµR ∈ PW(h∗M,C)R in Lemma10.2 below, we have

DR(q, δ, µ) 6 Jdisc(εK1(q) ⊗ fδ,µR ).

Proposition 10.1 then follows from Property (ELM).

Lemma 10.2. Let µ ∈ h∗ and R > 1. There is a hµR ∈ PW(h∗C)R verifying the following properties

(1) hµR is W -invariant;(2) hµR > 0 on h∗hm;(3) there are constants η,A,C > 0 such that for all ξ ∈ h∗hm satisfying

(10.1) ‖Im ξ − µ‖ 6 R−1 and ‖Re ξ‖ > AR−1

we have hµR(ξ) > CeηR‖Re ξ‖.

Proof. We begin by noting that if µ is such that no Hermitian ξ satisfies (10.1), then condition(3) is vacuous and the function hµR identically equal to zero satisfies the remaining conditions.Otherwise, µ should be of distance at most R−1 from the singular subset

⋃M∈L,M 6=M0

h∗M of h∗.Let Mµ ∈ L be the smallest semistandard Levi subgroup, distinct from the minimal Levi M0, suchthat ‖µMµ‖ 6 R−1. Now if the lemma is true for µMµ then it is true for µ (by taking for hµR the

function hµMµR ). We may therefore assume that µ ∈ h∗Mµ

.

For M ∈ L and δ > 0 let TM (δ) denote the tube of radius δ about h∗M inside h∗. Define constants0 < δ1 < δ2 < · · · < δn−1 such that for any M,M ′ ∈ L one has TM (δrM ) ∩ TM ′(δrM′ ) ⊂ TL(δrL),where L = 〈M,M ′〉 is the semistandard Levi subgroup generated by M and M ′ and rM is the semi-simple rank of M . Note that this property is conserved under simultaneous rescaling of all δi. LetM(µ) be the largest semistandard Levi subgroup M containing Mµ for which ‖µM‖ 6 δrM ‖Re ξ‖.This is well-defined, since if M1 and M2 satisfy this bound, then so does the Levi subgroup theygenerate. Moreover, M(µ) = M(µM ). For notational convenience, we often simply write M =M(µ).

With h0 as in Lemma 10.3 we put

h0R,µ(ξ) =

∑w∈W

h0(R(wξ − iµM ))

and hµR = (h0R,µ)2. Then hµR ∈ P(h∗C)R is W -invariant by construction. Moreover, since Wξ =

Wξconj for all ξ ∈ h∗hm, we have hµR > 0 on h∗hm. This establishes the first two properties.For the third property, we set WM = w ∈W : wλ = λ ∀λ ∈ h∗M. Then

(10.2) |h0R,µ(ξ)| >

∣∣∣∣ ∑w∈WM

h0(R(wξ − iµM ))

∣∣∣∣− |W \WM | maxw∈W\WM

|h0(R(ξ − iµM ))|.

Note that W ⊂ O(h, 〈, 〉). By property (1) of Lemma 10.3, and the definition of WM , we have

(10.3)

∣∣∣∣ ∑w∈WM

h0(R(wξ − iµM ))

∣∣∣∣ = |WM ||h0(R(ξ − iµM ))| > |h0(R(ξ − iµM ))|.

In remains to bound the subtracted term in (10.2).First we record an upper bound on ‖Im ξ − µM‖. For ν ∈ h∗ we have ‖ν − µM‖ 6 ‖νM − µM‖+

‖νM‖. Now if ξ = ρ+ iν ∈ h∗C satisfies (10.1) then ‖νM − µM‖ 6 ‖ν − µ‖ 6 R−1. Moreover, from

27

‖µM‖ 6 δrM ‖Re ξ‖ and ‖ν − µ‖ 6 R−1 we have ‖νM‖ 6 δrM ‖Re ξ‖ + R−1. We find, using thesecond condition in (10.1), that

(10.4) ‖Im ξ − µM‖ 6 δrM ‖Re ξ‖+ 2R−1 6 (δrM + 2A−1)‖Re ξ‖.

Let us compare this to a lower bound on ‖wIm ξ − µM‖ for all w ∈ W \WM . By the definitionof M , we have ‖µL‖ > δrL‖Re ξ‖ for all L ∈ L properly containing M . In fact, that there is κ > 0such that ‖µL‖ > κ‖Re ξ‖ for all L 6⊂ M . To see this, let M ′ = 〈L,Mµ〉. Since µ ∈ h∗Mµ

it follows

from [?, p. 136] that ‖µM ′‖ 6 ‖µL−µM ′‖+ ‖µM ′‖ ‖µL−µ‖ = ‖µL‖. Thus, it will be enough to

show that, say, ‖µM ′‖ > δrM′‖Re ξ‖. But if this is not the case then ‖µM ′′‖ 6 δrM′′‖Re ξ‖, whereM ′′ = 〈M,M ′〉. The maximality of M forces M ′′ = M , which is the same as to say that M ′ ⊂ Mand thus L ⊂M , a contradiction. Moreover, since M(µ) = M(µM ), the same argument applies toµM to show that ‖µLM‖ > κ‖Re ξ‖ for all L 6⊂M .

Continuing, note that any ξ ∈ h∗hm satisfying (10.1) must in fact lie in h∗hm,⊆M =⋃M ′⊂M h∗hm,M ′ .

In other words, Im ξ ∈ h∗M ′ for some M ′ ⊂ M . Indeed if Im ξ ∈ h∗L with L 6⊂ M then we have

‖Im ξ − µ‖ > ‖µL‖ > κ‖Re ξ‖ > κAR−1. But if 1 < κA this contradicts (10.1), whence the claim.From this it follows that for any w ∈W \WM there is L 6⊂M such that w Im ξ ∈ h∗L. Thus

(10.5) ‖w Im ξ − µM‖ = ‖w Im ξ − (µM )L‖+ ‖µLM‖ > ‖µLM‖ > κ‖Re ξ‖.

In view of (10.5), we may use property (3) of Lemma 10.3, with ε = 12 |W \WM |−1, to deduce

that there is 0 < δ 6 1 such that

(10.6) |W \WM | maxw∈W\WM

|h0(R(wξ − iµM ))| 6 1

2min

‖Im ξ−µM‖6δσ‖Re ξ‖=σ

|h0(R(ξ − iµM ))|.

Rescaling the constants δi and increasing A, we may assume that the upper bound in (10.4) is lessthan δ‖Re ξ‖. Inserting (10.3) and (10.6) into (10.2) yields |h0

R,µ(ξ)| > 12 |h0(R(ξ − iµM ))|. From

this and property (2) of Lemma 10.3 it follows that hµR(ξ) = |h0R,µ(ξ)|2 eηR‖Re ξ‖, as desired.

10.2. Stationary phase estimates. We now prove the establish the following technical result,used in the proof of Lemma 10.2.

Lemma 10.3. There is f0 ∈ C∞c (h) whose Fourier transform h0(ξ) =∫h f0(H)e〈ξ,H〉dH satisfies:

(1) h0(kξ) = h0(ξ) for all k ∈ O(h, 〈, 〉);(2) There are constants A,B, η > 0 such that for all R > 1 and σ > AR−1 we have

max‖Im ξ‖6R1/2σ3/2/3

‖Re ξ‖=σ

|h0(Rξ)| > BeηRσ.

(3) for every ε, κ > 0 there is 0 < δ 6 1 (depending only on κ) and A > 1 (depending on ε, κ)such that for all R > 1 and σ > AR−1 we have

(10.7) max‖Im ξ‖>κσ‖Re ξ‖=σ

|h0(Rξ)| 6 ε min‖Im ξ‖6δσ‖Re ξ‖=σ

|h0(Rξ)|.

Proof. Let b ∈ C∞c (R) be the bump function equal to e−1/(1−x2) in [−1, 1] and vanishing outside ofthis interval. Define f0(H) = b(‖H‖). We shall show that f0 satisfies all properties of the lemma.

Let ω denote the surface measure of the unit sphere Sd−1 in (h, 〈, 〉), so that

(10.8) h0(ξ) =

∫ 1

0e−1/(1−r2)dω(rξ)rd−1dr.

Then property (1) follows from the O(h, 〈, 〉)-invariance of ω.

28

Properties (2) and (3) will follow from an asymptotic evaluation of h0. We claim that for allξ ∈ h∗C and R > 1 we have

(10.9) h0(Rξ) = (2π)(d−1)/2R−(d−1)/2(sR)−3/4eRs−√

2Rs(1 + O(R−1)

),

where s = s(ξ) is the complex number k.ξ(ed) with k ∈ SO(h, 〈, 〉) such that k.Re ξ = ‖Re ξ‖e∗d.Let k ∈ SO(h, 〈, 〉) denote the unique rotation such that k.Re ξ = ‖Re ξ‖e∗d and write ν = k.Im(ξ).

From the rotational invariance of ω we have

dω(Rξ) =

∫Sd−1

eRξ(H)dω(H) =

∫Sd−1

eR(‖Re ξ‖e∗d+iν)(H)dω(H).

Let ujNj=1 be a smooth partition of unity of Sd−1, satisfying u1(ed) = 1 and ed /∈ supp(uj) for

j > 2. Then dω =∑

j dωj with

dωj(Rξ) =

∫Sd−1

uj(H)eR(‖Re ξ‖e∗d+iν)(H)dω(H) =

∫Rd−1

aj(X)eRφk(X)dX,

where aj (resp., φj) is the composition of uj (resp., ‖Re ξ‖e∗d + iν) with a coordinate patch. For

j > 2 we have dωj(Rξ) = O(e(1−η)‖Re ξ‖) for some η > 0. For j = 1 we may use the standard chart

X 7→ (X,√

1− ‖X‖2). Writing s = φ1(0) = ‖Re ξ‖+ iν(ed) ∈ C and f(X) = φ1(X)− s, we have

dω1(Rξ) = eRs∫Rd−1

a1(X)eRf(X) 1√1− ‖X‖2

dX.

We have Re f(X) = ‖Re ξ‖√

1− ‖X‖2 − ‖Re ξ‖. Now√

1− ‖X‖2 has a unique non-degenerate

critical point at 0, with Hessian matrix −Id. Thus Re f(X) = −12‖Re ξ‖‖X‖2(1 + o(1)), so that

(shrinking the support of u1 if necessary) Re f(X) is non-positive and vanishes only at the origin.Stationary phase estimates (with a complex phase, see [?, Theorem 4.1]) give

(10.10) dω1(ξ) = eRs(2π/iR)(d−1)/2(1 + O(R−1)).

Inserting these estimates into (10.8) we find

h0(ξ) =

(2π

iR

)(d−1)/2 ∫ 1

0e− 1

1−r2+Rrs

rd−1 dr(1 + O(R−1)

).

Changing variables r 7→√R(1− r) we obtain∫ 1

0e− 1

1−r2+Rrs

rd−1 dr = eRsR−1/2

∫ √R0

(1− rR−1/2)d−1e−√R(rs+(r(2−rR−1/2))−1) dr,

The phase is rs+ 1/2r + O(R−1/2) and the amplitude is 1 + O(rR−1/2), so the above integral is

eRsR−1/2

∫ √R0

e−√R(rs+1/2r)dr + O

(R−1/2

∫ √R0

r1/2e√R(rs+1/2r)dr

).

There is just one critical point, at r = 1/√

2s, which is non-degenerate, with second derivative 2s3/2.We may therefore localize the integral to a small interval about r = 1/|

√2s|. The method of steepest

descent (see [?, Theorem 4.1] again, with d = 1) then gives√π(sR)−3/4eRs−

√2Rs(1 + O(R−1)).

Inserting this into (10.10) yields the claim (10.9).To see how property (2) follows from (10.9), it suffices to remark that there is A > 1 such that

for all R > 1 and all s = σ + it ∈ C with σ > AR−1 and |t| 6 Rσ/3 one has eRe√Rs 6 e

12Rσ.

Indeed, the latter inequality is equivalent to (Re√

1 + it/σ)2 6 14Rσ, and a short calculation

shows that (Re√

1 + ir)2 = 12(1 +

√1 + r2), a monotonically increasing function with |r|. Taking

|t|/σ 6√Rσ/3, where Rσ > 10, the inequality is satisfied, so A = 10 is admissible.

29

We now prove property (3). Using (10.9) we find that (10.7) is equivalent to

(10.11) maxs=σ+it∈C|t|>κσ

HR(s) 6 ε mins=σ+it∈C|t|6δσ

HR(s),

where HR(s) = |s|−3/4e−Re√

2Rs. For fixed σ > 0 the function |σ + it|−3/4 is monotonically de-creasing in |t|, as is −Re

√σ + it. The maximum in (10.11) is therefore attained for |t| = κσ,

yielding

maxs=σ+it|t|>κσ

HR(s) 6 σ−3/4(1 + κ2)−3/8e−√

2RσRe√

1+iκ 6 σ−3/4e−√

2RσRe√

1+iκ.

Similarly, the minimum in (10.11) is achieved when |t| = δσ, giving

mins=σ+it|t|6δσ

HR(s) > σ−3/4(1 + δ2)−3/8e−√

2RσRe√

1+iδ > 2−3/8σ−3/4e−√

2RσRe√

1+iδ.

Thus (10.7) would follow from e−√

2RσRe√

1+iκ 6 ε2−3/8e−√

2RσRe√

1+iδ, which is

Re√

1 + iδ 6 Re√

1 + iκ+ (2Rσ)−1/2(log ε− log 23/8).

We may now take A > 1 large enough with respect to ε, κ so that for any R > 1 and σ > AR−1 thesecond term on the right is greater than the negative real number (1−Re

√1 + iκ)/2. Then taking

δ small enough so that Re√

1 + iδ 6 12(1 + Re

√1 + iκ) finishes the proof of property (3).

11. Proof of Proposition 8.3

Similarly to the existing literature on uniform Weyl laws, our basic strategy in the proof ofProposition 8.3 is to relate the sharp count N(q, δ, P ) to a corresponding smooth count, which wecan control via trace formula input, as represented by Property (ELM). Note that, since P lies inih∗M , we may write

N(q, δ, P ) =∑

π∈Πdisc(G(A)1)δνπ∈P

dimV K1(q)πf

.

Indeed, by the description of the discrete spectrum by Moeglin-Waldspurger [28], any π ∈ Πdisc(G(A)1)such that π∞ is tempered is necessarily cuspidal.

We need to pass from a smooth to a sharp test function in two terms, the central contribution

Jcent(εK1(q) ⊗ fδ,PR ) and smooth count of over the discrete spectrum Jdisc(εK1(q) ⊗ f

δ,PR ). For the

central contributions, this passage will be accomplished in §11.1 below.

11.1. Central contributions. This is the density that naturally appears in the central contribu-tions after the application of the trace formula with a test function whose archimedean componentis supported inside K∞A

1RK∞. Indeed, we have that

Jcent(εK1(q) ⊗ fδ,PR ) =

∑γ∈Z(F )

J(γ, εK1(q) ⊗ fδ,PR ) = ∆∗(1)ϕn(q)

∑γ∈Z(F )∩K1(q)γ∞∈K∞A1

RK∞

f δ,PR (γ).

Now, if ωδ,ν denotes the central character of πδ,ν , Plancherel inversion gives

f δ,PR (γ) =

∫ih∗M

hδ,PR (δ, ν)ωδ,ν(γ)µGM (δ, ν)dν.

Thus

(11.1) Jcent(εK1(q) ⊗ fδ,PR ) =

∑γ∈Z(F )∩K1(q)γ∞∈K∞A1

RK∞

∫ih∗M

hδ,PR (δ, ν)ωδ,ν(γ)µGM (δ, ν)dν.

30

Note that, by compactness, the sum in (11.1) is always finite. In fact we will prove the followinglemma.

Lemma 11.1. There exist constants c2, C2 > 0 such that the following holds. Let δ be a discreteparameter, let q be an integral ideal, let R > 0. Then

(1) if 0 < R 6 c2 log(2 + Nq), the sum over γ in (11.1) consists only of γ = 1 and possibly asubset of non-identity roots of unity in O×F .

(2) if, additionally Nq > C2, then only γ = 1 contributes to the sum in (11.1).

Proof. Note that γ ∈ Z(F ) ∩ K1(q) are given by diagonal elements corresponding to a unit u inO× congruent to 1 mod q. If C is taken small enough, the image under the logarithm map of suchu has trivial intersection with B(0, R). Thus the only γ ∈ Z(F ) ∩K1(q) lying in the support of Φcorrespond to roots of unity congruent to 1 mod q. This is a finite set which for q large enough isjust 1.

Lemma 11.2. For P ∈ BM and R > 0 we have∫ih∗M

hδ,PR (δ, ν)µGM (δ, ν)dν =

∫PµGM (δ, ν)dν +ON

(∂ volR(δ, P )

).

Proof. We begin by decomposing the integral according to

(11.2) ih∗M = ∂P (1/R)⋃P c \ ∂P (1/R)

⋃P \ ∂P (1/R).

From Lemma 9.1 the integral over ih∗M in the lemma is equal to∫PµGM (δ, ν)dν + O

(∫∂P (1/R)

βGM (δ, ν)dν

)+∞∑`=1

∫P •(`/R,(`+1)/R)

hδ,PR (δ, ν)µGM (δ, ν) dν

+∞∑`=1

∫P (`/R,(`+1)/R)

(1− hδ,PR (δ, ν)

)µGM (δ, ν) dν.

For every N ∈ N the last three terms above are majorized by∫∂P (1/R)

βGM (δ, ν)dν +∞∑`=1

`−N(∫

P •((`+1)/R)\PβGM (δ, ν)dν +

∫P\P ((`+1)/R)

βGM (δ, ν)dν

).

The last error term is indeed O(∂ volR(δ, P )

), as desired.

Corollary 11.3. There exist constants c2, C2 > 0 such that the following holds. Let δ be a discreteparameter, let q be an integral ideal. Let 0 < R 6 c2 log(2 + Nq). Then

(1) we have

Jcent(εK1(q) ⊗ fδ,PR ) ϕn(q)

∫PβGM (δ, ν)dν;

(2) we have

Jcent(εK1(q) ⊗ fδ,PR ) = ∆∗(1)ϕn(q)

∫PµGM (δ, ν)dν +ON

(ϕn(q)∂ volR(δ, P )

),

whenever Nq > C2.

31

11.2. Upper bounds. Recall the notion of µ-containment from Definition 8.1 and the correspond-

ing Lemma 8.2. The main tool in estimating the total deviations of hδ,PR from the sharp-cutoff con-dition of belonging to P over the “transition zones” (where the smooth test function is transitioningbetween 0 and 1) is the following upper bound.

Proposition 11.4. Let M ∈ L∞, δ ∈ E2(M1∞), and q an integral ideal. Let 0 < R < c log(2+Nq).

Let a set S ⊆ ih∗M be 1/R-contained in a set P ∈ BM , and let N ∈ N. Then,

N(q, δ, S)N ϕn(q)

(∫PµGM (δ, ν) dν + ∂ volR(δ, P )

)+ |Jerror(εK1(q) ⊗ f

δ,PR )|+DR(q, δ, P ),

where

DR(q, δ, P ) =∑

π∈Πdisc(G(A)1)δ

dimV K1(q)πf

eR‖Re(νπ)‖(1 +R‖Im(νπ)− P‖)−N

.

Proof. Let us decompose

(11.3) Jdisc(εK1(q) ⊗ fδ,µR ) = Jtemp(εK1(q) ⊗ f

δ,µR ) + Jcomp(εK1(q) ⊗ f

δ,µR ),

according to whether or not the archimedean component of the π contributing to Jdisc(εK1(q)⊗fδ,µR )

are tempered. By definition we have

Jtemp(εK1(q) ⊗ fδ,PR ) = −Jcomp(εK1(q) ⊗ f

δ,PR ) + Jcent(εK1(q) ⊗ f

δ,PR ) + Jerror(εK1(q) ⊗ f

δ,PR ).

Using that S is 1/R-contained in P and the non-negativity of f , we obtain that, for every ν ∈ S,

hδ,PR (ν) = Rr∫Pf (R(ν − τ)) dτ > Rr

∫B(ν,1/R)

f (R(ν − τ)) dτ =

∫B(0,1)

f(τ) dτ >1

2,

so that, by non-negativity of hδ,PR on ih∗M , we have Jtemp(εK1(q) ⊗ fδ,PR ) > 1

2N(q, δ, S). Further, byLemma 9.1 we have that

hδ,PR (δ, ν)N eR‖Re νπ‖(1 +R‖Im νπ − Y ‖)−N

for every ν ∈ h∗M,C, so that

(11.4) Jcomp(εK1(q) ⊗ fδ,PR )N DR(q, δ, P ).

Finally, Lemma 11.1 implies that the number of terms contibuting to Jcent(εK1(q) ⊗ fδ,PR ) is abso-

lutely bounded. Decomposing the integral over ih∗M as in the proof of Lemma 11.2 and applyingLemma 9.1, we get

Jcent(εK1(q) ⊗ fδ,PR )N ϕn(q)

∫ih∗M

(1 +R‖ν − P‖

)−NdµGM (δ, ν)

N ϕn(q)

(∫PµGM (δ, ν) dν +

∞∑`=1

`−N∫P •(`/R)\P

µGM (δ, ν) dν

).

The statement of the proposition follows by combining the above estimates.

Combining Proposition 11.4 with the fact that ∂P (ρ) is (1/R)-contained in ∂P (ρ + 1/R) (byLemma 8.2), we obtain the following corollary.

Corollary 11.5. For every P ∈ BM and for every ρ,R > 0,

N(q, δ, ∂P (ρ)) ϕn(q)

(∫∂P (ρ+1/R)

µGM (δ, ν) dν + ∂ volR(δ, ∂P (ρ+ 1/R))

)+ |Jerror(εK1(q) ⊗ f

δ,∂P (ρ+1/R)R )|+DR(q, δ, ∂P (ρ+ 1/R)).

32

11.3. Proof of Proposition 8.3. Let c = min(c1, c2), where c1 is defined in Property (ELM) andc2 is given in Corollary 11.3. Assume that 0 < R < c log(2 + Nq).

If Nq > C2, where C2 is given in Part 2 of Corollary 11.3, then using the upper bound (11.4),we obtain

Jtemp(εK1(q) ⊗ fδ,PR ) = ∆∗(1)ϕn(q)

∫PµGM (δ, P ) dν

+ O(ϕn(q)∂ volR(δ, P ) + |Jerror(εK1(q) ⊗ f

δ,PR )|+DR(q, δ, P )

).

The case Nq 6 C2 can be treated similarly, yielding an upper bound, through the use of Part 1 ofCorollary 11.3.

We now pass from the smooth count Jtemp(εK1(q) ⊗ fδ,PR ) to the sharp count N(q, δ, P ).

We first decompose the sum Jtemp(εK1(q)⊗ fδ,PR ) according to (11.2). For every integer ` > 1 we

write J•/,`temp for the restriction of these spectral sums to νπ ∈ P •/(`/R, (`+ 1)/R). Then

Jtemp(εK1(q) ⊗ fδ,PR ) = N(q, δ, P ) + O

(N(q, δ, ∂P (1/R))

)+∞∑`=1

J•,`temp(εK1(q) ⊗ fδ,PR ) +

∞∑`=1

J,`temp(εK1(q) ⊗ (1− f δ,PR )).

Lemma 9.1 then shows that Jtemp(εK1(q) ⊗ fδ,µR ) is equal to

N(q, δ, P ) + O(N(q, δ, ∂P (1/R))

)+ O

( ∞∑`=1

`−N∑•/

N(q, δ, P •/(`/R, (`+ 1)/R)

))

= N(q, δ, P ) + O

( ∞∑`=1

`−N−1N(q, δ, ∂P (`/R)

)).

It follows from this and Corollary 11.5 that Jtemp(εK1(q) ⊗ fδ,µR )−N(q, δ, P ) is majorized by

ϕn(q)

∞∑`=1

`−N−1[( ∫

∂P (`+1)/R)µGM (δ, ν) dν + ∂ volR(δ, ∂P

((`+ 1)/R

)))+ |Jerror(εK1(q) ⊗ f

δ,∂P ((`+1)/R)R )|+DR(q, δ, ∂P

((`+ 1)/R)

].

(Here the boundary terms are taken implicitly with N + 1.) We then estimate

∞∑`=1

`−N−1(∫

∂P(

(`+1)/RµGM (δ, ν) dν + ∂ volR(δ, ∂P

((`+ 1)/R

)))

6∞∑`=1

∞∑m=0

`−N−1(m+ 1)−N−1

∫∂P(

(`+m+1)/R) µGM (δ, ν) dν

=

∞∑`=2

∫∂P (`/R)

µGM (δ, ν) dν∑

`=k1+k2

(k1k2)−N−1

∞∑`=1

`−N∫∂P (`/R)

µGM (δ, ν) dν.

33

By definition, this last sum is just ∂ volR(δ, P ). Similarly, recalling the notation DR(q, δ, P ) at thebeginning of Section 10,

∞∑`=1

`−N−1DR(q, δ, ∂P((`+ 1)/R)) 6

∞∑`=1

∞∑m=1

`−N−1m−N−1DR

(q, δ, ∂P

((`+m+ 1)/R

))=∞∑`=2

DR

(q, δ, ∂P (`/R)

) ∑`=k1+k2

(k1k2)−N−1

∞∑`=1

`−NDR

(q, δ, ∂P (`/R)

).

This last sum is bounded by DR(q, δ, P ).Let Λ be the square lattice in h∗M of size length 1 and write C for the collection of all centers of

the rescaled lattice (√

2R)−1Λ. Then h∗M =⋃µ∈C BM (µ, 1/R). We have

DR(q, δ, P ) 6∑

µ∈C∩PDR(q, δ, µ) +

∑L=2``>0

L−N∑µ∈C

R‖µ−P‖∼L

DR(q, δ, µ).

Under the assumption of Property (ELM), Proposition 10.1 implies

DR(q, δ, P ) ϕn(q)

( ∑µ∈C∩P

∫BM (µ,1/R)

βGM (δ, ν)dν +∑L=2``>0

L−N∑µ∈C

R‖µ−P‖∼L

∫BM (µ,1/R)

βGM (δ, ν)dν

).

Recallling the definition (8.3) we obtain DR(q, δ, P ) ϕn(q)volR(δ, P ).It remains to treat the contribution

|Jerror(εK1(q) ⊗ fδ,PR )|+

∞∑`=1

`−N |Jerror(εK1(q) ⊗ fδ,∂P (`/R)R )|,

which we do by invoking Property (ELM). Indeed, from (9.2) the above sum is bounded by

Nqn−θ(∫

PβGM (δ, ν)dν +

∞∑`=1

`−N∫∂P (`/R)

βGM (δ, ν)dν) Nqn−θvolR(δ, P ),

completing the proof.

12. Reduction to error estimates

To pass from Proposition 8.3 to Theorem 8.1, we need to sum over various discrete data. In thisparagraph, we package together the terms arising in Proposition 8.3 after executing this summation.

12.1. Summing over discrete data. Refining the set ΩX of (4.1), we put

Ω0X = π∞ ∈ Πtemp(GLn(F∞)1) : q(π∞) 6 X.

Then Ω0X ⊂ ΩX and we have Ω0

X =∐

(M∞,δ)/∼πδ,ν : ν ∈ Pδ,X, where

Pδ,X = ν ∈ ih∗M : q(πδ,ν) 6 X.

Using this decomposition, we may extend the definition of the boundary term volR(δ, P ) of (8.3)as follows. For X > 1 we put

(12.1) volR(ΩX) =∑

(M∞,δ)/∼

volR(δ, Pδ,X).

We begin by proving the following result.

34

Lemma 12.1. Assume Property (ELM). There is θ > 0 such that for Q > 1 we have

|F(Q)| = MT (Q) + O(Qn+1−θ + volR(Q)),

where

MT (Q) = ∆∗(1)∑

16Nq6Q

∑d|q

λn(q/d)ϕn(d)

∫ΩQ/Nq

dµpl∞

and

volR(Q) =∑

Nq6Q

∑d|q

|λn(q/d)|ϕ(d)volR(d)(Ω0Q/Nq).

Proof. Assuming Property (ELM), Proposition 8.3 can be seen to yield

N(q,ΩX) = ∆∗(1)ϕn(q)

∫ΩX

dµpl∞ + O(ϕn(q)volR(Ω0

X)),

for 0 < R < c log(2 + Nq) and Nq > C, and N(q,ΩX) ϕn(q)∫

ΩXdµpl∞ for Nq 6 C.

We first recall and justify the combinatorial decomposition (4.4) of the universal family intoweighted harmonic families. Let q be an integral ideal and Ξ ⊂ Πgen(G1

∞). From the dimensionformula (6.1) of Reeder we deduce

N(q,Ξ) =∑d|q

∑π∈H(d,Ξ)

dn(q/d) =∑d|q

dn(q/d)|H(d,Ξ)|.

This equality holds for every integral ideal q and is hence an equality of arithmetical functions.Since the inverse of dn(m) under Dirichlet convolution is λn(m), Mobius inversion yields

|H(q,Ξ)| =∑d|q

λn(q/d)N(d,Ξ).

Taking Ξ = ΩQ/Nq, the claim (4.4) then follows from (4.2).We now execute the summation over q and d | q in (4.4). First we deal with the contribution for

divisors Nd 6 C. From the trivial estimate

(12.2)∑d|q

Nd6X

|λn(q/d)|ϕn(d)ε XnNqε

and Lemma 7.3, we easily deduce∑Nq6Q

∑d|q

Nd6C

|λn(q/d)|ϕn(d)

∫ΩQ/Nq

dµpl∞ ε Q

n−1/d+ε.

The terms MT (Q) and O(volR(Q)) arise from the remaining q and d | q with Nd > C.

12.2. Main term evaluation. We now evaluate the main termMT (Q) asymptotically, asQ→∞.This will use the local conductor zeta function analysis from Section 7.

Lemma 12.2. Let 0 < θ 6 2/(d+1) when n > 2 and 0 < θ < min1, 2/(d+1) when n = 1. Thenfor every Q > 0 we have

MT (Q) =1

n+ 1∆∗(1) ‖ν(F)‖Qn+1 + Oθ

(Qn+1−θ).

35

Proof. Recall the classical estimate∑

Nd6X 1 = ζ∗F (1)X + O(X1−2/(d+1)) on the ideal-countingfunction [22, Satz 210, p. 131]. From this we deduce that given any σ > −1, 0 < θ 6 2/(d + 1),and X > 0, we have

(12.3)∑

Nq6X

Nqσ =ζ∗F (1)

σ + 1Xσ+1 + Oσ,θ

(Xσ+1−θ),

where for X > 1 we simply estimate Xσ+1−2/(d+1) = O(Xσ+1−θ), and for X < 1 the estimate(12.3) holds vacuously.

Now let wn = ϕn ? λn = (pn ? µ) ? µ?n = pn ? µ?(n+1) = pn ? λn+1 and Wn(X) =

∑Nq6X wn(q).

Using (12.3), we find that, for every X > 0,

Wn(X) =∑∑N(de)6X

λn+1(e)Ndn =∑

Ne6X

λn+1(e)

(ζ∗F (1)

n+ 1

(X

Ne

)n+1

+ O

(X

Ne

)n+1−θ)

=ζ∗F (1)

n+ 1Xn+1

∑Ne6X

λn+1(e)

Nen+1+ O

(Xn+1−θ

∑Ne6X

|λn+1(e)|Nen+1−θ

).

From |λn+1(n)| 6 dn+1(n)ε (Nn)ε and the identity∑λn+1(n)Nn−s = ζF (s)−n−1 we obtain

(12.4) Wn(X) =1

n+ 1

ζ∗F (1)

ζF (n+ 1)n+1Xn+1 + O

(Xn+1−θ).

Exchanging the order of summation and integration, we find that

MT (Q) = ∆∗(1)∑

Nq6Q

wn(q)

∫ΩQ/Nq

dµpl∞(π∞) = ∆∗(1)

∫Π(G1

∞)Wn(Q/q(π∞)) dµpl

∞(π∞).

Using (12.4) we see that the right-hand side above is equal to

1

n+ 1∆∗(1)

ζ∗F (1)

ζF (n+ 1)n+1Z∞(n+ 1)Qn+1 + O

(Z∞(n+ 1− θ)Qn+1−θ

).

In light of Lemma 7.3, both integrals converge. The zeta value (including the factor Z∞(n+ 1)) inthe main term can be expressed as ‖ν(F)‖ by Lemma 7.1.

12.3. Main corollary. Let R = (R(n))n⊆O be a sequence of real number indexed by integralideals n. We shall say that R is admissible if 0 < R(n) < c log(2 + Nn), where c is the constant inProposition 8.3. From Lemmas 12.1 and 12.2 we deduce the following result.

Corollary 12.3. Assume Property (ELM). There is θ > 0 such that, for all admissible sequencesR and real numbers Q > 3, we have

|F(Q)| = 1

n+ 1∆∗(1) ‖ν(F)‖Qn+1 + O

(Qn+1−θ + volR(Q)

).

13. Boundary error term

We now prove satisfactory bounds on the error terms appearing in Corollary 12.3. Recall thenotion of admissible sequence just before that result.

Proposition 13.1. There is θ > 0 and a choice of admissible sequence R such that

volR(Q) Qn+1/ logQ.

Putting Corollary 12.3 together with Proposition 13.1, we deduce Theorem 8.1.

36

13.1. A preparatory lemma. For real parameters R,X > 0, recall the definition of volR(Ω0X) in

(12.1). The following lemma will go a long way towards proving Proposition 13.1. It bounds a verysimilar quantity to volR(Q), but with the arithmetic weights in the average replaced by powers ofthe norm, and with the admissible sequence R taken to be constantly equal to the real number R.Dealing with these arithmetic weights and choosing an appropriate admissible sequence to proveProposition 13.1 will be done in §13.2.

Lemma 13.2. Let 0 < θ 6 2/(d+ 1) and σ > n− 1/d− 1 + θ. For R 1, Q > 1, we have∑16Nq6Q

NqσvolR(Ω0Q/Nq) = Oσ,θ(R

−1Qσ+1 +Qσ+1−θ).

We remark that the integer N implicit in the volume factors should satisfy N > 3 + d(σ + 1).

Proof. Applying the definitions, we get∑16Nq6Q

Nqσ∂ volR(Ω0Q/Nq) =

∞∑`=1

`−N∑

16Nq6Q

Nqσ∫∂Ω0

Q/Nq(`/R)

β(π)dπ,

where ∂Ω0X(r) is the union over all [M, δ] of πδ,ν : ν ∈ Pδ,X(r). Now for any r > 0 we have∑

16Nq6Q

Nqσ∫∂Ω0

Q/Nq(r)β(π)dπ =

∫∂Ω0

Q(r)

( ∑16Nq6Q

π∈∂Ω0Q/Nq

(r)

Nqσ)β(π)dπ,

upon exchanging the order of the sum and integral. From Lemma 13.3 below we deduce that theright-hand side is bounded by

r(1 + rd(σ+1)

)Qσ+1

∫Π(G1

∞)q(π)−σ−1β(π)dπ + (1 + r)d(σ+1−θ)Qσ+1−θ

∫Π(G1

∞)q(π)−σ−1+θβ(π)dπ.

In view of Lemma 7.3, both integrals converge, yielding∑16Nq6Q

Nqσ∫∂ Ω0

Q/Nq(r)β(π)dπ σ,θ r

(1 + rd(σ+1)

)Qσ+1 + (1 + r)d(σ+1−θ)Qσ+1−θ.

Applying the above estimate with r = `/R > 0 and executing the sum over ` > 1, we get∑16Nq6Q

Nqσ∂ volR(Ω0Q/Nq)

∞∑`=1

`−N(`

R

(1 +

(`

R

)r(σ+1))Qσ+1 +

(1 +

`

R

)d(σ+1−θ)Qσ+1−θ

) R−1

(1 +R−d(σ+1)

)Qσ+1 +

(1 +R−d(σ+1−θ))Qσ+1−θ.

The last expression is bounded by R−1Qσ+1 +Qσ+1−θ. It remains to observe∑16Nq6Q

Nqσ∫

Ω0Q/Nq

β(π)dπ =

∫Ω0Q

∑16Nq6Q/q(π)

Nqσ

β(π)dπ σ Qσ+1

∫Ω0Q

q(π)−σ−1β(π)dπ,

the last integral converging in view of Lemma 7.3 and our assumptions on σ.

We now establish the following result, which was used in the proof of Lemma 13.2.

Lemma 13.3. Fix an irreducible tempered representation π of G(F∞)1. Let r > 0, 0 < θ 62/(d+ 1), and σ > −1 + θ. Then∑

16Nq6Qπ∈∂Ω0

Q/Nq(r)

Nqσ σ,θ r(1 + rd(σ+1)

)( Q

q(π)

)σ+1

+ (1 + r)d(σ+1−θ)(

Q

q(π)

)σ+1−θ.

37

Proof. We first convert the condition π ∈ ∂Ω0Q/Nq(r) on the ideal q to a more amenable condition

on the norm of q. We may assume that π = πδ,ν for a class [M, δ] and ν ∈ ih∗M . For parametersr,X > 0 we have π ∈ ∂ΩX(r) precisely when there is µ ∈ ih∗M with ‖µ− ν‖ < r and q(πδ,µ) 6 X.Letting Mr(π) (resp., mr(π)) denote the maximum (resp. minimum) value of q(πδ,µ) as µ variesover ih∗M with ‖µ− ν‖ < r, we see that π ∈ ∂Ω0

Q/Nq(r) implies mr(π) 6 Q/Nq 6Mr(π), so that∑16Nq6Q

π∈∂Ω0Q/Nq

(r)

Nqσ 6∑

Q/Mr(π)6Nq6Q/mr(π)

Nqσ.

Using the asymptotic (12.3) we see that this is bounded by

(13.1)1

σ + 1Qσ+1

(1

mr(π)σ+1− 1

Mr(π)σ+1

)+ O

(Qσ+1−θ

mr(π)σ+1−θ

).

To conclude the proof of the lemma we shall need to relate mr(π) and Mr(π) to expressions involving(1 + r) and q(π). This will require some basic analytic properties of the archimedean conductorq(πδ,ν) as ν varies.

For ν ∈ ih∗M and r > 0, let ν(r) ∈ ih∗M denote the translation ν + rν0, for some fixed ν0 ∈ ih∗,+Min the positive chamber, and write π(r) = πδ,ν(r). Since q(πδ,ν) is monotonically increasing in ν,it follows that for ν0 large enough we have Mr(π) 6 q(π(r)). Similarly, q(π(−r)) 6 mr(π), forr 6 1

2‖ν‖, say. In this interval we have q(π(−r)) q(π), while, if r > 12‖ν‖, we have

q(π(−r)) > q(πδ,0) (1 + ‖ν‖)−dvq(π) (1 + r)−dq(π).

Thus, in either case, q(π(−r)) (1 + r)−dq(π), proving mr(π) (1 + r)−dq(π). When insertedinto the second term in (13.1) we obtain the second term of the lemma.

Now let s 7→ ν(s) be a unit length parametrization of the line between ν(−r) and ν(r) in ih∗M .Since s 7→ q(π(s)) is a real-valued differentiable map on the interval [0, 1], we have

q(π(−r))σ − q(π(r))σ =

∫ 1

0

d

dsq(π(s))σ ds =

∫ 1

0σq(π(s))σ−1 dq(π(s))

dπ(s)

dπ(s)

dsds.

Since dq(π(s))dπ(s) q(π(s)) and dπ(s)

ds r, the latter integral is bounded by

rσ

∫ 1

0q(π(s))σ dsσ rq(π(r))σ σ r(1 + r)−dσq(π)σ.

Since σ 6 0 we have (1 + r)−dσ σ (1 + r−dσ), proving mr(π)σ −Mr(π)σ σ r(1 + r−dσ

)q(π)σ.

Inserting this into the first term of (13.1) then completes the proof of the lemma.

13.2. End of proof. We now return to the proof of Proposition 13.1.We first choose the sequence R = (R(n))n⊆O of the form

(13.2) R(n) =

R1, if Nn 6 Q1/2

R2, if Q1/2 < Nn 6 Q,

where R1, R2 > 0 will be chosen shortly. With the above choice of R the term ∂ volR(Q) is equal to∑Nq6Q

∑d|q

Q1/2<Nd6Q

|λn(q/d)|ϕn(d)volR1(Ω0Q/Nq) +

∑Nq6Q

∑d|q

Nd6Q1/2

|λn(q/d)|ϕn(d)volR2(Ω0Q/Nq).

Bounding the first term using∑d|q

|λn(q/d)|ϕn(d) 6 Nqn∏p|q

(1−Np−n)(1 + nNp−n +

(n2

)Np−2n + · · ·

) Nqn

38

and second term using (12.2), the above expression can be bounded by∑Nq6Q

NqnvolR1(Ω0Q/Nq) +Qn/2

∑Nq6Q

NqεvolR2(Ω0Q/Nq).

Combining this with Lemma 13.2 shows that volR(Q)ε R−11 Qn+1 +R−1

2 Qn2

+1+ε.Now let c > 0 be as in Proposition 8.3. Then taking R1 = c

2 logQ and R2 = c in the definitionof R in (13.2) yields an admissible sequence according to the definition preceding Corollary 12.3.Inserting these values establishes the stated bounds of Proposition 13.1.

Part 3. Trace formula estimates

14. Bounding the non-central geometric contributions

Arthur defines a distribution Jgeom on H(G(A)1) related to geometric invariants of G. Thisdistribution Jgeom admits an expansion along semisimple conjugacy classes of G(F ), and our taskin this section is to bound all but the most singular terms (the central contributions) appearingin this expansion. We must do so uniformly with respect to the level and the support of the testfunctions at infinity.

Theorem 14.1. Let n > 1. There exists θ > 0 and c > 0 satisfying the following property. Forany integral ideal q, R > 0, and f ∈ H(G(F∞)1)R we have

Jgeom(εK1(q) ⊗ f)− ϕn(q)∑

γ∈Z(F )∩K1(q)

f(γ) ecRNqn−θ‖f‖∞.

The implied constant depends on F and n.

Our presentation is by and large based on the papers [8], [24], and [25]. Many aspects of ourargument are simplified by the absence of Hecke operators in our context. On the other hand, wehave to explicate (in various places) the dependence in R.

14.1. Fine geometric expansion. Let O denote the set of semisimple conjugacy classes of G(F ).For G, a semisimple conjugacy class consists of all those γ ∈ G(F ) sharing the same characteristicpolynomial. Then Arthur defines distributions Jo associated with each o ∈ O so that

Jgeom(φ) =∑o∈O

Jo(φ).

The fine geometric expansion expresses each Jo(φ) as a linear combination of weighted orbitalintegrals JM (γ, φ), where M ∈ L and γ ∈M(F ).

More precisely, Arthur shows that for every equivalence class o ∈ O, there is a finite set of placesSadm(o) (containing S∞) which is admissible in the following sense. For any finite set of placesS containing Sadm(o), there are real numbers aM (γ, S), indexed by M ∈ L and M(F )-conjugacyclasses of elements γ ∈M(F ) (and, in general, depending on S), such that

Jo(1KS ⊗ φS) =∑M∈L

|WM ||W |

∑γ

aM (γ, S)JM (γ,1KS ⊗ φS)

for any function φS ∈ C∞c (G(FS)1). In the inner sum, γ runs over those M(F )-conjugacy classesof elements in M(F ) meeting o. We shall describe the integrals JM (γ,1KS ⊗ φS) later, in Section14.4. For the moment, we simply record the fact that JM (γ,1KS ⊗φS) = 0 for any γ /∈ KS ∩ o andJM (γ, φ) = JM (γ, φS) otherwise, the latter being an S-adic integral.

Following [24, §6] and [25, (10.4)], for o ∈ O and γ = σν ∈ σUGσ(F ) ∩ o we let

So = Swild ∪ v <∞ : |DG(σ)|v 6= 1,

39

where Swild is a certain finite set of finite places depending only on n. Then [24, Lemma 6.2] or[25, Proposition 10.8] shows that one can take Sadm(o) = So ∪∞ in the fine geometric expansion.

We will apply the fine geometric geometric expansion in the case where

(14.1) S = So ∪ Sq ∪ S∞ and φS =∏

v∈So,v /∈Sq

1Kv ⊗∏v∈Sq

εK1,v(q) ⊗ f,

for certain f ∈ H(G(F∞)1)R (those appearing in the statement of Theorem 14.1).

14.2. Contributing classes. We now wish to bound the number of equivalence classes o ∈ Ocontributing to the course geometric expansion of Jgeom(1K1(q) ⊗ f), for f ∈ H(G(F∞)1)R. Itclearly suffices to take q trivial.

Definition 14.1. For R > 0 let OR denote the set of o ∈ O for which there is f ∈ H(G(F∞)1)Rwith Jo(1Kfin

⊗ f) 6= 0.

Our main result in this subsection is the following estimate. The argument is based largely on[24, Lemma 6.10].

Proposition 14.2. There is c > 0, depending on n and [F : Q], such that |OR| = O(ecR).

Proof. Let o be a semisimple G(F )-conjugacy class represented by some semisimple element σ ∈G(F ). Let χo denote the characteristic polynomial of σ. It is a monic polynomial of degree n withcoefficients in F , independent of the choice of σ, and the map o 7→ χo is a bijection onto suchpolynomials. We shall count the o appearing in OR by counting the corresponding χo.

Let UGσ denote the algebraic variety of unipotent elements in the centralizer Gσ. The conditiono ∈ OR is equivalent to the following collection of local conditions at every place v:

(1) for every v -∞ there is νv ∈ UGσ(Fv) such that the G(Fv)-conjugacy class of σνv meets Kv;(2) there is ν∞ ∈ UGσ(F∞) such that the G(F∞)-conjugacy class of σν∞ meets G(F∞)1

R.

Note that any element γ ∈ G(A) lying in the G(A)-conjugacy class of o has characteristic polynomialequal to χo. From the above local conditions we deduce that the coefficients of χo (for o ∈ OR) arev-integral for all finite v, and so lie in O. Moreover, their archimedean absolute value is boundedby ecR for some constant c > 0. Each coefficient of χo for contributing classes o then lies in theintersection of O ⊂ F with

∏v|∞[−X,X] ⊂ F∞. As there are at most O(X [F :Q]) such lattice points,

the proposition follows.

14.3. Bounding global coefficients. Next we bound the coefficients aM (γ, So), for γ lying in acontributing class o. Once again, we are free to assume that q is trivial, so that o ∈ OR.

For any finite set of finite places T we put

qT =∏v∈T

qv.

We begin with the following useful result.

Lemma 14.3. For o ∈ OR we have qSo ecR. In particular,

(14.2) |So| R

for o ∈ OR.

Proof. If o ∈ OR and σ ∈ G(F ) is a semisimple element representing o, then there is y ∈ G(F )such that y−1σUGσ(F )y ∩ Kf 6= ∅. In other words there are y ∈ G(F ) and u ∈ UGσ(F ) such

y−1σuy ∈ Kf . Thus, for every finite place v we have |DG(σ)|v = |DG(y−1σuy)|v 6 1. Moreover, itfollows from [25, Lemma 3.4] that (under the same assumptions on o and σ)

(14.3) |DG(σ)|∞ ecR

40

Thus for o ∈ OR we in fact have

So = Swild ∪ v <∞ : |DG(γ)|v < 1 = Swild ∪ v <∞ : |DG(γ)|v 6 q−1v .

By the product formula, we deduce that

1 =∏

v/∈So∪S∞

|DG(γ)|v∏v∈So

|DG(γ)|v∏v∈S∞

|D(γ)|v qSwildq−1SoecR n q

−1SoecR,

as desired.To deduce (14.2) from this we note

∑v∈So

1 6∑

v∈Solog qv = log qSo R.

For the next estimate, we invoke the main result of [26], a corollary of which is the following. Let σbe elliptic semisimple in M(F ). Then σ is conjugate in M(C) to a diagonal matrix diag(ζ1, . . . , ζn).Let

∆M (σ) = N

∏i<j:ζi 6=ζj

(ζi − ζj)2

,

where N is the norm from F to Q, and the product is taken over indices i < j such that the stringαi + · · ·+αj lies in the set of positive roots ΦM,+ = Φ+ ∩ΦM for M . For an equivalent expressionfor ∆M (σ) using based root data, see [25, Section 9]. Then it follows from [26] (see also [24, (22)])that there is κ > 0 such that for any finite set of places S containing S∞, if γ = σν is the Jordandecomposition of γ, we have

(14.4) aM (γ, S) |S|n∆M (σ)κ(∏v∈S

log qv

)nfor σ elliptic in Mv and aM (γ, S) = 0 otherwise.

Proposition 14.4. There is c > 0 such that for any M ∈ L, any o ∈ OR meeting M , and anyγ ∈ o, we have aM (γ, So) = O(ecR).

Proof. It follows from (14.2) that the factor of |So|n in the upper bound (14.4) is at most O(Rn).To bound the factor ∆M (σ) we follow the argument of [24, Lemma 6.10, (iv)]. The eigenvaluesζ1, . . . , ζn are the roots of the characteristic polynomial of o. As in the proof of Proposition 14.2, foro ∈ OR, these coefficients have v-adic (for archimedean v) absolute value bounded by O(ecR). Anapplication of Rouche’s theorem shows that each ζi has complex absolute value bound by O(ecR),from which it follows that ∆M (σ) ecR. To bound the last factor in (14.4) we apply the first partof Lemma 14.3.

14.4. Review of the constant term map. In this section we review the definitions of the con-stant term map and the weighted orbital integrals.

Let M ∈ L and P ∈ P (M) with P = MU . Let S be a finite set of places of F . Let φS ∈ C∞c (GS).Then the constant term of φS along P is defined by

(14.5) φ(P )S (m) = δP (m)1/2

∫U(FS)

∫KS

φS(k−1muk) dk du (m ∈M(FS)).

The φ(P )S ∈ C∞c (M(FS)).

14.5. Review of weighted orbital integrals. We come now to the definition of the weightedorbital integral. Throughout, we let γ = σν ∈ G(Fv) be the Jordan decomposition of γ.

Before stating the full formula, we begin with two special cases.

41

(1) Assume that Gγ ⊂M(Fv). Then

(14.6) JM (γ, φv) = |DG(γ)|1/2v

∫Gγ(Fv)\G(Fv)

φv(y−1γy)v′M (y) dy.

Here, the weight function v′M is the volume of a certain complex hull; as a function on G(Fv),it is left invariant under M(Fv) (so the above integral is well-defined) and constantly equalto 1 when M = G.

(2) Assume γ = ν ∈M(Fv) is unipotent. Let V0 be the M(Fv)-conjugacy class of ν and denoteby V1 the G(Fv)-conjugacy class given by inducing V0 from M to G along any parabolic of Gcontaining M as a Levi subgroup. (The induced class V1 is independent of the choice of sucha parabolic.) Then, since unipotent conjugacy classes in G are of Richardson type, thereis a unique standard parabolic subgroup P1 ∈ F , say with Levi decomposition P1 = L1U1,such that V1 has dense intersection with U1. (The Levi factor of P1 is given by the dualpartition of the Jordan form of V1. See, for example, [16, §5.5].) Then

(14.7) JM (ν, φv) =

∫Kv

∫U1(Fv)

φv(k−1uk)wM,U1(u) dudk.

Here, the weight function wM,U1 is a complex-valued function defined on U1(Fv); it is

invariant under conjugacy by KL1v (so that the above integral is well-defined) and constantly

equal to 1 when M = G. See [23, p. 143] for more details.

In fact, case (2) (indeed the general case) is obtained as a limit of linear combinations of case (1).Note that for ν unipotent Gν 6⊂ M , unless M = G. When γ = ν is unipotent and M = G

then the two formulae coincide, giving the invariant unipotent orbital integral. For example, theRichardson parabolic of the trivial class in G is of course G itself, so that both formulae collapsein this case to JG(1, φv) = φv(1).

The general formula is considerably more complicated. It will be expressed in terms of weightedorbital integrals for the Levi components of parabolic subgroups in FGσ(Mσ). Indeed, for any

R ∈ FGσ(Mσ), we will evaluate the unipotent orbital integrals JMRMσ

(ν, ·) on certain descent functionsΦ ∈ C∞c (MR(Fv)). More precisely, if γ = σν ∈ σUGσ(Fv) ∩M(Fv), then [2, Corollary 8.7] statesthat

(14.8) JM (γ, φv) = |DG(σ)|1/2v

∫Gσ(Fv)\G(Fv)

∑R∈FGσ (Mσ)

JMRMσ

(ν,ΦR,y)

dy,

where, for m ∈MR(Fv) and y ∈ G(Fv), we have put

ΦR,y(m) = δR(m)1/2

∫KGσv

∫NR(Fv)

φv(y−1σk−1mnky)v′R(ky) dn dk.

The complex-valued weight function v′R on G is set to be

(14.9) v′R(z) =∑

Q∈F(M): Qσ=RaQ=aR

v′Q(z),

where v′Q, defined in [1, §2], generalizes the weight v′M to arbitrary parabolics Q ∈ F(M).

Using the expression (14.7) for the unipotent weighted orbital integral, we may write JMRMσ

(ν,ΦR,y)more conveniently. To see this, first let V0 denote the Mσ(Fv)-conjugacy class of the unipotent el-ement ν ∈ Mσ(Fv). Next we write V1 for the induced unipotent class of V0 to MR along anyparabolic in MR containing Mσ as a Levi subgroup. (The induced class V1 is independent of thechoice of such a parabolic.) Let P1 = L1U1 ⊂MR be a Richardson parabolic for V1. Finally, let V

42

be the induced unipotent class of V1 to Gσ along R. Then the Richardson parabolic P = LV ⊂ Gσof V satisfies U = U1NR. We deduce that

(14.10) JMRMσ

(ν,ΦR,y) =

∫KGσv

∫U(Fv)

φv(y−1σk−1uky)wMR

Mσ ,U(u)v′R(ky) dudk,

where wMRMσ ,U

is the trivial extension of wMRMσ ,U1

to all of U(Fv). For more details, see [24, §10.4].We make the following observations:

(1) Assume Gγ ⊂M . This condition is clearly equivalent to Gγ = Mγ and the uniqueness of theJordan decomposition then implies Mσ = Gσ. In this case FGσ(Mσ) = Gσ and the sumover R in (14.8) reduces to the single term R = Gσ; clearly, MR = Gσ and NR = e. Thus

U = U1 and the weight function wMσMσ ,U

is constantly equal to 1 on all of U . Furthermore,

v′R = v′M in this case. From the left M(Fv)-invariance of v′M , we have v′M (ky) = v′M (y) fork ∈ KGσ

v and y ∈ G(Fv). The corresponding integral in (14.10) then reduces to

v′M (y)

∫KGσv

∫U1(Fv)

φv(y−1σk−1uky) dudk = v′M (y)JGσGσ (σν, φyv),

where φyv(x) = φv(y−1xy). Note that the latter integral is∫

Gγ(Fv)\Gσ(Fv)φv(y

−1x−1σνxy) dx.

Inserting this into the integral over y ∈ Gσ(Fv)\G(Fv) we obtain the expression (14.6).

(2) When σ = 1 so that γ = ν is unipotent, the outer integral in (14.8) is trivial. Moreover, thefunction v′R vanishes on Kv unless R = G when it is constantly equal to 1. From (14.10)

we deduce that JMRM (ν,ΦR,e) = 0 unless R = G, in which case (since U = U1) we obtain∫

Kv

∫U1(Fv)

φv(k−1uk)wGM,U1

(u) dudk,

recovering the previous expression (14.7) for JM (ν, φv).

14.6. Reduction to local estimates. We first recall that for factorizable test functions φS =⊗v∈Sφv and γ ∈ M(F ) one has a splitting formula which reduces JM (γ, φS) to a sum of productsof local distributions. More precisely (see [24, Lemma 6.11] or [25, (10.3)]), there are real numbersdGM (LS), indexed by tuples of Levi subgroups LS = (Lv)v∈S where Lv ∈ L(M), such that

JM (γ, φS) =∑LS

dM (LS)∏v∈S

JLvM (γv, φ(Qv)v ).

Here, we are using an assignment L(G) 3 Lv 7→ Qv ∈ P(Lv) which is independent of S, and forevery v ∈ S the element γv ∈ M(Fv) is taken to be M(Fv)-conjugate to γ. The properties ofinterest for us on the coefficients dM (LS) are the following, proved in [24, Lemma 6.11]:

(1) as LS varies, the coefficients dM (LS) can attain only a finite number of values; these valuesdepend only on n.

(2) the number of contributing Levi subgroups LS can be bounded as

|LS : dM (LS) 6= 0| |S|n−1.

(3) If dM (LS) 6= 0 then the natural map⊕v∈S

aLvM → aGM is an isomorphism.

In particular, from the first two of these properties, it follows immediately that for any o ∈ O,γ ∈ o, admissible S, and factorisable φS = ⊗v∈Sφv ∈ H(G(FS)) we have

JM (γ, φS) |S|n−1 maxLS

∏v∈S|JLvM (γv, φ

(Qv)v )|.

43

Thus, if o ∈ OR, f ∈ H(G(F∞)1)R, and S and φS are taken as in (14.1) we obtain

(14.11)

JM (γ, εK1(q) ⊗ f) Rn−1 maxLS

∏v|∞

|JLvM (γ, f (Qv))|

×∏v∈Sq

|JLvM (γ, ε(Qv)K1,v(q))|

∏v∈So,v /∈Sq

|JLvM (γ,1KLvv

)|,

where we have used (14.2) as well as the fact (see, for example, [24, §7.5] or [45, Lemma 6.2]) that

1(Qv)Kv

= 1KLvv

.

15. Estimates on local weighted orbital integrals

It remains to bound the local weighted orbital integrals appearing in (14.11). In this section,we provide (or recall) such bounds at every place, and show how they suffice to establish Theorem14.1.

For v dividing q, we offer the following proposition, the proof of which is based heavily on worksof Finis-Lapid [8], Matz [24], and Shin-Templier [45].

Proposition 15.1. There are constants B,C, θ > 0 such that the following holds. Let M ∈ L,L ∈ L(M), and Q ∈ P(L). Then for any v ∈ Sq and γ = σν ∈ σUGσ(Fv) ∩M(Fv) and r > 0 wehave

JLM (γ,1(Q)K1,v(prv)) qaB−θrv |DL(σ)|−Cv ,

where a = 0 whenever the residue characteristic of Fv is larger than n! and v /∈ So, and a = 1otherwise.

A great deal of work has been recently done by Matz [24] and Matz–Templier [25] in boundingarchimedean weighted orbital integal for GLn. Their bounds are almost sufficient for our purposes,except for the dependency in the support of R. By simply explicating this dependence in theirproofs, we obtain the following result.

Proposition 15.2. There are constants c, C > 0 satisfying the following property. Let M ∈ L,L ∈ L(M), and Q ∈ P(L). Let v ∈ S∞. Let R > 0. Then for γ = σν ∈ σUGσ(Fv) ∩M(Fv) andf ∈ H(G(Fv)

1)R we have

JLM (γ, f (Q)) ecR|DL(σ)|−Cv ‖f‖∞.

15.1. Deduction of Theorem 14.1. We now show how the above results imply Theorem 14.1.We will need an additional result for places v ∈ So, v /∈ Sq (as in the last factor of (14.11)). Namely,it is proved in [24, Corollary 10.13] that there are constants B,C > 0 such that for any finite placev one has

(15.1) JLM (γ,1KLv) qBv |DL(σ)|−Cv .

Returning to the global situation of Theorem 14.1, we let o ∈ O be such that o∩M is non-empty,and let σ ∈M(F ) be a semisimple element representing o. If in fact o ∈ OR, then we can argue asin the proof of Lemma 14.3 to see that |DG(σ)|v 6 1 for every v <∞. Thus, we may increase thevalue of C in Proposition 15.1 and display (15.1) at the cost of a worse bound. Let Cv denote thevalue of C at each place v ∈ So ∪ S∞ as given in Proposition 15.1, display (15.1), and Proposition15.2, and put C = maxv∈So∪S∞ Cv. An application of the product theorem yields∏

v∈So∪S∞

|DG(σ)|−Cvv 6∏v∈So

|DG(σ)|−Cv∏v∈S∞

|DG(σ)|−Cvv =∏v∈S∞

|DG(σ)|C−Cvv .

Since C − Cv > 0 we deduce from (14.3) that∏v∈So∪S∞ |D

G(σ)|−Cvv ecR.

44

We now return to (14.11). From Proposition 15.1 and the above bound on the product of Weyldiscriminants, we deduce that for γ = σν ∈ σUGσ(Fv) ∩M(Fv), we have

JM (γ, εK1(q) ⊗ f) ecRNqn−θqBSo|DG(σ)|Cvv max

L∞

∏v∈∞|JLvM (γ, f (Qv))|,

where we have incorporated the (∏

char(Fv)6n! qv)B into the implied constant (which is allowed to

depend on the number field F and n). Applying Lemma 14.3 to qBSoand Proposition 15.2 to the

last quantity completes the proof of Theorem 14.1.

15.2. Proof of Proposition 15.1. The basic idea of the proof of Proposition 15.1 is to show thatthe semisimple conjugacy class o has small intersection with K1,v(p

rv). One has to do this in the

framework of the definition of the general weighted orbital integrals (14.8), which involve variousweight functions. We shall divide the proof into three steps as follows:

Step 1. Reduce to the case that L = G. We do this by showing that whenever L is a proper Levisubgroup of G we can get savings in the level by means of the constant term alone.

Step 2. Reduce to the case of M = G and γ semisimple non-central. This involves bounding theparenthetical expression in (14.8), as a function of y ∈ Gσ(Fv)\G(Fv), γ, and the level prv.

• If γ is not semisimple, then for every R ∈ FGσ(Mσ) we get savings in the level for the weighted

unipotent integrals JMRMσ

(ν,ΦR,y) of (14.10) by bounding the intersection of unipotent conjugacyclasses (in the centralizer of σ) with congruence subgroups (which depend on y).

• If γ is semisimple non-central, but M 6= G, then the same argument as above applies to allterms except the one associated with R = Mσ, since in that case the unipotent integral collapsesand one has simply JMσ

Mσ(1,ΦMσ ,y) = 1K1,v(prv)(y

−1σy)v′Mσ(y).

Step 3. Bound the invariant orbital integrals of 1K1,v(prv) for semisimple non-central γ.

In all cases, the central ingredient to bounding intersections of conjugacy classes with open compactsubgroups is the powerful work of Finis-Lapid [8]. We shall give a brief overview of their results inSection 15.2.1 below.

It is instructive to examine the division into Steps 2 and 3 in the case where Gγ ⊂ M , as thenotation greatly simplifies under this assumption. As usual, let γ = σν ∈ σUGσ(Fv) ∩M(Fv). LetV ⊂ UGσ(Fv) be the Gσ-conjugacy class of ν in Gσ, endowed with the natural measure. Then weare to estimate the integral

JGM (γ,1K1,v(prv)) =

∫Gσ(Fv)\G(Fv)

voly−1σVy(y−1σVy ∩K1,v(p

rv))v

′M (y) dy.

We proceed differently according to whether ν is trivial or not.

• If ν is trivial, then the inner y−1σVy-volume is just 1K1,v(prv)(y−1σy). Thus Step 2 is vacuous is

this case, and Step 3 then bounds∫Gσ(Fv)\G(Fv)

1K1,v(prv)(y−1σy)v′M (y) dy

by estimating the intersection volume of the conjugacy class of σ with the congruence subgroupK1,v(p

rv).

• If ν is non-trivial, then V is of positive dimension and Step 2 bounds the inner y−1σVy-volumeby a quantity which is roughly of the form q−rv 1B(tσ)(yσy

−1). Here, for a real parameter t > 0, wehave denoted by 1B(t) the characteristic function of the ball B(t) of radius t about the origin, and

45

tσ roughly of size |DG(σ)|−Cv . One may then estimate the volume of Gσ\B(t) (a compact piece ofthe tube of radius t about Gσ) by appealing to the work of Shin-Templier [45].

In either case, the weight function v′M is easy to control, as it grows by a power of log with thenorm of y, using results from [24].

The more general case, when Gγ is not necessarily contained in M , is complicated by the presenceof the various terms parametrized by R ∈ FGσ(Mσ) in (14.8). For example, whenever R 6= Gσthese terms include the unipotent weight functions wGσV,U , which must be dealt with. Neverthelessthe case Gγ ⊂M described above already contains most of the difficulties.

15.2.1. The work of Finis-Lapid. It will be convenient to use the results of [8]. We now recall theirnotation (specialized to our setting) and describe two of their main theorems.

For r > 0 let Kv(prv) = k ∈ Kv : k ≡ Id (mod prv) be the principal congruence subgroup of

level exponent r. Let g = Mn(Fv) be the Lie algebra of G and write Λ = Mn(Ov). Following [8,Definition 5.2] for γ ∈ G(Fv) we put

λGv (γ) = maxr ∈ Z ∪∞ : (Ad(γ)− 1)Λ ⊂ $rvΛ.

In other words, if we make the identification GL(g) = GLn2 , then λGv (γ) is the maximal r ∈ Z∪∞such that Ad(γ) lies in the principal congruence subgroup of GLn2(Ov) of level exponent r, c.f. [8,Remark 5.23]. The function λGv on G(Fv) descends to one on PGLn(Fv), and one has λGv (γ) > 0whenever γ ∈ Kv.

For a twisted Levi subgroup H recall that KH = H(Fv) ∩Kv. Then KHv (prv) = Kv(p

rv) ∩KH

v

is the principal congruence subgroup KHv of level exponent r. We define the level exponent of

an arbitrary open compact subgroup K of KHv as the smallest non-negative integer f such that

KHv (pfv ) is contained in K. For example, the level exponent of K1,v(p

rv) est r.

We shall make critical use of the following result, which can be deduced from Propositions 5.10and 5.11 of [8]. See [4] for more details on this deduction.

Proposition 15.3 (Finis-Lapid [8]). For every ε > 0 small enough there is θ > 0 such that thefollowing holds. Let r be a non-negative integer, v a finite place of F , H a reductive subgroup ofG, and x ∈ KH

v . If λGv (x) < εr then for any open compact subgroup K of KHv of level exponent r

we have

µHvk ∈ KHv : k−1xk ∈ K q−θrv .

We shall also need the following result, which can be deduced from [8, Lemma 5.7]; see also theproof of Corollary 5.28 in loc. cit.

Proposition 15.4 (Finis-Lapid [8]). Let H be a reductive subgroup of G. Let P be a properparabolic subgroup of H, with unipotent radical U . Let v be a finite place of F . Then

volu ∈ U(Fv) ∩KHv : λv(u) > m q−mv ,

uniformly in v.

Finally we remark that in [4, Remark 5.5] it is shown that for semisimple σ ∈ Kv we have

(15.2) qλGv (σ)v |DG(σ)|−1

v .

This inequality will be occasionally used to convert from large values of λGv (σ) to large values of− logqv |D(σ)|v.

46

15.2.2. Reduction to L = G. We begin by reducing the proof of Proposition 15.1 to the case L = G.The first step in this reduction it to estimate the constant terms of the functions 1K1,v(prv), uniformlyin r and γ.

Proposition 15.5. There is θ > 0 such that the following holds. Let v be a finite place. LetM ∈ L, M 6= G, and P ∈ P(M). Then for γ ∈M(Fv), and r > 0 we have

1(P )K1,v(prv)(γ) q−θrv 1KM

v(γ).

Proof. It follows from [24, Lemma 7.3 (i)] or [45, Lemma 6.2] that 1(P )Kv

= 1KMv

. Now 0 6 f 6 g

implies that f (P ) 6 g(P ). From this we deduce that 1(P )K1,v(prv)(γ) = 0 unless γ ∈ KM

v . Henceforth

we may and will assume that γ ∈ KMv ; note that δP = 1 on KM

v .Recall the definition of the constant term map in (14.5). Note that if u ∈ U(Fv) is such that

k−1γuk ∈ Kv then u ∈ U(Fv) ∩Kv. Fixing u ∈ U(Fv) ∩Kv the inner integral is

µGv(k ∈ Kv : k−1γuk ∈ K1,v(prv)).

From Proposition 15.3, for every ε > 0 small enough there is θ > 0 such that

1(P )K1,v(prv)(γ) volu ∈ U(Fv) ∩Kv : λGv (γu) > εr+ q−θrv volu ∈ U(Fv) ∩Kv : λGv (γu) 6 εr.

We apply the trivial bound vol(U(Fv)∩Kv) = 1 to the latter volume. To deal with the former, wenote that λGv (γu) 6 λGv (u) for γ ∈ KM

v and u ∈ U(Fv) ∩Kv and then apply Proposition 15.4 tofinish the proof.

We now prove Proposition 15.1 in the the case that L 6= G. Let γ = σν ∈ σUGσ(Fv) ∩M(Fv).From Proposition 15.5 we deduce

JLM (γ,1(Q)K1,v(prv)) qbB−θrv JLM (γ,1KL

v),

where b = 0 or 1 according to whether the residual characteristic of Fv is > n! or not, and JLMdenotes the weighted orbital integral JLM but with absolute values around the weight functions in

(14.10). If v ∈ So we apply (15.1) to the latter integral (which is valid with JGM replaced by JGM ).Otherwise, if the finite place v is not in So, then it follows from [24, Lemma 10.12] (which, again,

is valid with JGM replaced by JGM ) and the identity JLL (σ,1KLv) = 1 (for semisimple σ) that

JLM (γ,1KLv) qbBv ,

with the same convention on b as before. This yields the desired estimate in both cases.

15.2.3. Bounding the weighted unipotent orbital integrals on descent functions. We shall now boundthe parenthetical expression in (14.8), with φv the characteristic function of K1,v(p

rv). Before doing

so, we shall need to introduce slightly more notation.If y = k1diag($m1

v , . . . , $mnv )k2 ∈ G(Fv), where k1, k2 ∈ Kv and m1 > · · · > mn are integers,

then we write

‖y‖ = qmax|m1|,|mn|v .

For t > 0 we write B(t) = g ∈ G(Fv) : ‖g‖ 6 t for the ball of radius t about the origin in G(Fv).Then 1B(t) is the characteristic function of B(t).

Lemma 15.6. There are constants B,C, θ > 0 such that the following holds. Let γ = σν ∈σUGσ(Fv) ∩M(Fv) be non-central. Let b = 0 or 1 according to whether the residual characteristic

47

of Fv is > n! or not, and put tσ = |DG(σ)|−Cv qbBv . Let r > 0 be an integer. Then there is a set ofrepresentatives y ∈ Gσ(Fv)\G(Fv) such that the expression∑

R∈FGσ (Mσ)

JMRMσ

(ν,ΦR,y),

where the descent functions ΦR,y are associated with 1K1,v(prv), is majorized by

(1 + log tσ)n−1 1K1,v(prv)(y−1σy) + qbB−θrv |DG(σ)|−Cv 1B(tσ)(y

−1σy).

Proof. Let (y, u, k) ∈ G(Fv)×UGσ(Fv)×KGσv be such that y−1σk−1uky ∈ Kv. From [24, Corollary

8.4], there are constants B,C > 0, and for a triplet as above there is g ∈ Gσ(Fv) (which can betaken to be independent of u) such that

‖gy‖ 6 |DG(σ)|−Cv qbBv ,(15.3)

‖gug−1‖ 6 |DG(σ)|−Cv qbBv ,(15.4)

where the convention on b is as in the lemma. Henceforth we take a set of representatives y ∈Gσ(Fv)\G(Fv) whose norm is bounded by the right-hand side of (15.3), in which case it can beassumed that the norm of u is bounded by the right-hand side of (15.4).

We furthermore recall the bound on the weight function |v′Q(x)| (1 + log ‖x‖)n−1 established

in [24, Corollary 10.9], valid for any parabolic Q ∈ F(M). Thus, using (14.9), we deduce that forany y as above and any k ∈ KGσ

v we have

|v′R(ky)| (

1 + log |DG(σ)|−Cv qbBv

)n−1.

In particular, if ν = 1 and R = Mσ then we may go ahead and bound the integral JMσMσ

(1,ΦR,y)appearing in (14.10). (We are of course taking the descent function ΦR,y to be associated with1K1,v(prv).) Indeed, the unipotent subgroup U of that formula is reduced to the identity in this case,

so that the U(Fv) integral collapses and one has

JMσMσ

(1,ΦMσ ,y) = 1K1,v(prv)(y−1σy)v′Mσ

(y).

Using the above bound on the weight factor, we obtain the first term of the majorization of thelemma.

Next, for ρ ∈ R let Den(ρ) denote the set of matrices in Mn(Fv) all of whose coefficients havevaluation at least −ρ. Note that if g ∈ G(Fv) is such that ‖g‖ 6 qρv then g ∈ Den(ρ); indeedit suffices to establish this for diagonal elements in the positive chamber, where it is immediate.Thus, for u as above, we have u ∈ U(Fv) ∩Den(ρσ), where ρσ = bB − C logqv |D

G(σ)|v.We return to estimation of the integral JMR

Mσ(ν,ΦR,y), this time in the case where either γ is not

semisimple or M 6= G. In either of these cases, the U appearing in (14.10) satisfies dimU ≥ 1.Again, the descent function ΦR,y is taken to be associated with 1K1,v(prv). From the above discussion

we deduce that JMRMσ

(ν,ΦR,y) is majorized by

(15.5)(

1 + log |DG(σ)|−Cv qbBv

)n−1∫U(Fv)∩Den(ρσ)

|wMRMσ ,U

(u)|∫KGσv

1K1,v(prv)(y−1σk−1uky) dk du.

For y ∈ G(Fv) and r > 0 let us put Kσv (y, r) = yK1,v(p

rv)y−1 ∩ Gσ. In the special case when

r = 0 we shall simply write Kσv (y) = Kσ

v (y, 0). With this notation, the inner integral in (15.5) is∫KGσv

1Kσv (y,r)(k

−1σvk) dk. After an application of Cauchy-Schwarz, we see that the double integral

in (15.5) is bounded by

(15.6)

(∫KGσv

∫U(Fv)

1Kσv (y,r)(k

−1σuk) du dk

)1/2(∫U(Fv)∩Den(ρσ)

|wMRMσ ,U

(u)|2 du

)1/2

.

48

Using [24, Lemma 10.5], we see that the second factor in (15.6) is O(|DG(σ)|−Cv qbBv ), with the same

convention on b. (The aforementioned result in fact bounds the integral∫U(Fv)∩Den(ρσ) |w

MRMσ ,U

(u)|du,

but the same proof applies with |wMRMσ ,U

(u)|2 as integrand, simply by replacing the polynomial q in

[24, Lemma 10.4] by its square.)Next, we treat the first factor in (15.6). We follow closely the presentation in [8, Corollary 5.28],

explicating several small differences. We first write the double integral as∫KGσv

volU (U ∩ σ−1Kσv (ky, r)) dk.

We may suppose that U∩σ−1Kσv (ky, r) is non-empty, in which case, fixing any u0 in this intersection,

we have U ∩ σ−1Kσv (ky, r) = u0(U ∩Kσ

v (ky, r)). By invariance of the Haar measure on U(Fv) weobtain in this case

volU (U ∩ σ−1Kσv (ky, r)) = volU (U ∩Kσ

v (ky, r)).

We claim that we can reduce to the case where Kσv (y, r) is replaced by Kσ

v (y, r) ∩ KGσv =

yK1,v(prv)y−1 ∩KGσ

v . Indeed, this double integral is

(15.7)

∫KGσv

volU (U ∩Kσv (ky, r)) dk =

∫KGσv

i(k) volU (U ∩Kσv (ky, r) ∩KGσ

v ) dk,

where

i(k) =[U ∩Kσ

v (ky, r) : U ∩Kσv (ky, r) ∩KGσ

v

]6 [Kσ

v (ky, r) : Kσv (ky, r) ∩KGσ

v ]

= [Kσv (y, r) : Kσ

v (y, r) ∩KGσv ] 6 [Kσ

v (y) : Kσv (y) ∩KGσ

v ].

Therefore the expression in (15.7) bounded by

(15.8) [Kσv (y) : Kσ

v (y) ∩KGσv ]

∫U(Fv)

∫KGσv

1yK1,v(prv)y−1∩KGσ

v(k−1uk) dk du.

Continuing, it now follows from (15.3) that

[Kσv (y) : Kσ

v (y) ∩KGσv ] |DG(σ)|−Cv qbBv .

It remains to bound the double integral in (15.8).We first note that the level exponent of yK1,v(p

rv)y−1 ∩ KGσ

v is at least r. That is to say,yK1,v(p

rv)y−1 ∩KGσ

v cannot contain KGσv (pr−1

v ). This follows, for example, from the fact that thecentral element 1+$r−1

v lies in KGσv (pr−1

v ) but not in y−1K1,v(prv)y∩KGσ

v . In light of this, we mayapply Proposition 15.3, with Hv = Gσ(Fv), to find θ, ε > 0 such that∫

KGσv

1yK1,v(prv)y−1∩KGσ

v(k−1uk) dk q−θrv

whenever λGv (u) < εr. The double integral in (15.8) is therefore bounded by

volu ∈ U(Fv) ∩KGσv : λGv (u) > εr+ q−θrv volu ∈ U(Fv) ∩KGσ

v : λGv (u) 6 εr,

We bound the second volume factor trivially by volu ∈ U(Fv)∩KGσv = 1. Finally, an application

of Proposition 15.4 (with H = Gσ) shows that the first volume factor is majorized by q−εrv , finishingthe proof.

49

15.2.4. Invariant orbital integrals. In view of Lemma 15.6, it now remains to establish good boundson the invariant orbital integrals of 1K1,v(prv) and 1B(tσ). It suffices to estimate the unnormalizedorbital integral

Oσ(φv) =

∫Gσ(Fv)\G(Fv)

φv(x−1γx)dµσ,v(x),

since JGG (σ, φv) = |DG(σ)|1/2v Oσ(φv).We first handle the invariant orbital integral for 1B(tσ). If σ /∈ So and the residue characteristic

of Fv is > n! then tσ 1, and we may apply [45, Theorem A.1] to deduce that Oσ(1B(tσ)) 1 inthis case. If either the residue characteristic of Fv is 6 n! or γ ∈ So then we proceed as follows.For every λ ∈ X+

∗ (T0) let τ(λ) denote the associated Hecke operator, namely the characteristicfunction of Kvλ($v)Kv. Then

Oσ(1B(tσ)) =∑

λ∈X+∗ (T0)

‖λ‖6logqv tσ

Oσ(τλ).

Note that there are only O(lognqv tσ) cocharacters λ satisfying the bound in the sum. For each of theabove orbital integrals, it follows from [45, Theorem 7.3] (see also [45, Theorem B.1] for a strongerresult) that there are constants B,C > 0 such that Oσ(τλ) tBσ |DG(σ)|−Cv . Inserting this intothe above expression (and recalling the definition of tσ from Lemma 15.6) we deduce the boundOσ(1B(tσ)) qBv |DG(σ)|−Cv in this case. We conclude that in all cases we have

Oσ(1B(tσ)) qbBv |DG(σ)|−Cv ,

where b = 0 or 1 according to whether the residue characteristic of Fv is > n! or not.It remains then to estimate the invariant orbital integral Oσ(1K1,v(prv)) uniformly in the level prv

and the semisimple element σ. We accomplish this in the next lemma; our presentation followsclosely that of [4, Proposition 5.3].

Lemma 15.7. There are constants B,C, θ > 0 such that the following holds. Let v be a finiteplace. For any r > 0 and semisimple σ ∈ G(Fv), σ /∈ Zv, we have

Oσ(1K1,v(prv)) qaB−θrv |DG(σ)|−Cv ,

where a = 1 or 0 according to whether v ∈ So or not.

Proof. Letting Cσ,Gv denote the conjugacy class of σ, we have

Oσ(1K1,v(prv)) = µσ,v(Cσ,Gv ∩K1,v(prv)).

Now Cσ,Gv is closed since σ is semisimple. The compact set Cσ,Gv ∩ K1,v(prv) is then a disjoint

union of finitely many (open) Kv-conjugacy classes Cxi,Kv meeting K1,v(prv). This gives

Oσ(1K1,v(prv)) =t∑i=1

µσ,v(Cxi,Kv ∩K1,v(prv))

µσ,v(Cxi,Kv)µσ,v(Cxi,Kv).

From the definition of the quotient measure, for any x ∈ Kv, we have

µσ,v(Cx,Kv ∩K1,v(prv)) =

µG,v(k ∈ Kv : k−1xk ∈ K1,v(prv))

µGx,v(Gx,v ∩Kv).

Using (15.2), we may deduce from Proposition 15.3 that if x ∈ Kv is semisimple and non-central,and |DG(x)|v q−εrv for some ε > 0, then

µGv(k ∈ Kv : k−1xk ∈ K1,v(prv)) q−(1−ε)r

v .

50

Thus for every ε > 0 there is θ > 0 such that if |DG(σ)|v = |DG(xi)|v q−εrv then

µσ,v(Cx,Kv ∩K1,v(prv))

µσ,v(Cx,Kv)= µGv(k ∈ Kv : k−1xk ∈ K1,v(p

rv)) q−θrv .

In this case we obtain

Oσ(1Kv) q−θrv

t∑i=1

µσ,v(Cxi,Kv) = q−θrv Oσ(1Kv),

sincet∑i=1

µσ,v(Cxi,Kv) =

t∑i=1

µσ,v(Cxi,Gv ∩Kv) = µσ,v(Cσ,v ∩Kv) = Oσ(1Kv).

If, on the other hand, |DG(σ)|v q−εrv (so that 1 q−θrv |DG(σ)|−θ/εv ) then we may apply thetrivial bound Oσ(1K1,v(prv)) 6 Oσ(1Kv) to obtain

Oσ(1K1,v(prv)) q−θrv |DG(σ)|−θ/εv Oσ(1Kv).

If v /∈ So then Oσ(1Kv) = 1. If v ∈ So we apply the bound Oσ(1Kv) qBv |DG(σ)|−Cv of [45,Theorems 7.3 and B.1]. This proves the desired estimate in either case.

15.3. Proof of Proposition 15.2. The statement of Proposition 15.2, without the explicateddependency in R, follows from the proof of [25, Theorem 1.8] (in the case of v real) and [24] (inthe case of v complex). To prove Proposition 15.2 it therefore suffices to explicate the dependencein R in these works. For simplicity, we shall concentrate on the real case here.

It suffices to take L = G, since on one hand the constant term map f 7→ f (Q) takes C∞c (G(Fv)1)R

to C∞c (L(Fv)1)cR (see, for example, [24, Lemma 7.1 (iii)]), and on the other the factor δ

1/2Q is

bounded by O(ec′R) on KL

vALRKL

v . Here c, c′ > 0 are constants depending only on n.We would like to explicate the dependency in C(f1) in [25, Theorem 1.8] on both ‖f1‖∞ and

the support of f1. (Note that, since we do not seek any savings in the spectral parameter, our

interest is in η = 0.) It is clearly enough to bound the modified weighted orbital integral JGM (γ, f),where the weight functions are replaced by their absolute values. The dependency on ‖f1‖∞ is easyenough to explicate, for in the proof of [25, Theorem 1.8] (see Propositions 6.6, 7.5, and 7.6 of loc.cit.), one replaces the function f1 with a majorizer of the characteristic function of its support. Wecan thus assume that f is the characteristic function of KvA

1RKv.

We now supplement a few of the lemmas and propositions leading up to the proof of [25, Theorem1.8], pointing out how the dependency in R can be explicated.

• The constants c and C in [25, Lemma 6.3] can be taken to be of the form eκR for someκ = κ(n) > 0. To see this, first note that the constants in Lemmas 3.6 and 3.7 dependonly on n. It can then readily be seen that each of the constants ai in the proof of Lemma6.3 can be taken to be of the form eκiR, for κi = κi(n). (For example, a1 = cec1R, wherec = c(n) > 0 and c1 = c1(n) > 0 are given in Lemma 3.6.)

• This then implies that the constant c1 in [25, Lemma 6.8] can be taken to be of the formeκR, for some κ = κ(n) > 0. (As the authors point out just before §6.6, one can takec2 = 2n and c3 = 1 in Lemma 6.8.) Their proof divides into two subcases, according towhether dimU2 > 1 or not.

– If dimU2 > 1, then their integral∫b+2

1b+2

r(γs)(Y )BM2

b2(Y ) dY is bounded by a polynomial

expression in r(γs), the latter quantity being logarithmic in c1. Furthermore, their

integral∫U2

1U2

R(γs)(u) du (recall that η = 0) is bounded by a polynomial expression in

R(γs), the latter quantity having a linear dependence in c1.

51

– If dimU2 = 0, then the integral∫b+2

1b+2

r(γs)(Y )BM2

b2(Y ) dY is bounded polynomially in

r(γs), thus logarithmically in c1.

• To explicate the dependence in R in the proof of [25, Proposition 6.6], one then appliesLemma 6.8 in the way we have just explicated, and Lemma 6.9 with s = eκR.

• We now consider the proof of [25, Proposition 6.10]. One is led to consider integrals of theform

∫V 1KvA1

RKv(v)| log |p(v)||k dv, where V is the unipotent radical of a proper parabolic

of G(Fv), k > 1 is an integer, and p : V → R is a polynomial function on the coordinates.(Once again, recall that η = 0.) This can be bounded by aRb

∫V 1KvA1

RKv(v) dv, where

a > 0 and b > 0 depend on p, k, and n. The latter integral is O(ecR).

16. Construction of test functions

In this section, we construct (in certain cases) explicit realizations of the test functions f δ,µR ∈C∞c (G(Fv))R, and provide bounds on these in terms of their spectral transforms. These boundswill be important in the estimation of the associated orbital integrals.

Proposition 16.1. Let n > 1. Let M∞ be a cuspidal standard Levi subgroup of G(F∞), δ ∈E2(M1

∞), µ ∈ h∗M∞,C, and R > 0. Suppose that either

(1) n 6 2, or(2) n > 1 is arbitrary, but M∞ = T∞ is the diagonal torus and δ is the trivial character of T 1

∞.

Let hδ,µR be the spectral localizer about the pair (δ, µ), as defined in Section 9. Let τ(δ) ∈ Π(K∞)be the minimal K∞-type of πδ,µ, as defined in [47, 48]; it is independent of µ. Write Πτ(δ) for theorthogonal projection onto the τ(δ)-isotypic component of πδ,µ.

Then there exists f δ,µR ∈ H(G(F∞)1)R such that

(16.1) πσ,λ(f δ,µR ) =

hδ,µR (λ) · 1

dim τ(δ)Πτ(δ), σ = δ;

0, else,

and

(16.2) ‖f δ,µR ‖∞ 6 2‖hδ,µR ‖L1(µpl∞).

Remark 16.1. It follows from (16.1) that trπσ,λ(f δ,µR ) = hδ,µR (λ) when σ = δ, and is 0 otherwise.

Remark 16.2. The existence of such a test function f δ,µR satisfying (16.1) is a consequence ofthe Paley–Wiener theorem of Clozel–Delorme, as stated in Section 9.4, which is of course validwithout assuming either (1) or (2). We were not able to extract the bound (16.2) from the proofof Clozel–Delorme. We have therefore restricted ourselves to the cases in (1) or (2), where we havean explicit inversion map.

16.1. Reduction to a fixed archimedean place. Let Z1∞ = Z∞ ∩G1

∞. Note the group decom-position

(16.3) G1∞ = Z1,nc

∞∏v|∞

G1v,

where Z1,nc∞ = Z1

∞/(Z1∞ ∩K∞) and G1

v = Gv/AGv . For π ∈ Π(G1∞) let ωnc

π denote the restriction

of the central character of π to Z1,nc∞ . Then π′ = ππ ⊗ (ωnc

π )−1 =∏v|∞ π

′v, where each π′v is a

representation of G′v. If a function f ∈ H(G1∞) factorizes as f = fZ

∏v|∞ f

′v according to (16.3)

thentrπ(f) = fZ(ωnc

π )∏v|∞

trπ′v(f′v).

52

Since the spectral localizers hδ,µR of Section 9 were taken to respect this decomposition, it suffices

to construct a test function fµZR ∈ C∞c (Z1,nc∞ ) such that

(16.4) fµZR (eλZ ) = hµZR (λZ),

and, for each v | ∞, a test function fδv ,µ0

vR ∈ C∞c (G1

v) satisfying the v-adic version of properties(16.1) and (16.2).

Finding fµZR satisfying (16.4) is a triviality: for µZ ∈ a0,∗G,C and R > 0 let gµZR ∈ C∞c (a0

G) be as in

Section 9.2. Then using Lie(Z1,nc∞ ) = a0

G, we set fµZR (z) = gµZR (log z).

16.2. Spherical transform of type τ : abstract theory. We begin by recalling the sphericalfunctions (and trace spherical functions) of a given Kv-type τ . These will then be used to definethe associated spherical transform on the τ -isotypic Hecke algebra. For these definitions, see, forexample, [49, §6.1] and [5].

Fix τ ∈ Π(Kv). For π ∈ Π(G(Fv)1), acting on the space Vπ, let Πτ be the canonical projection

onto the τ -isotypic subspace V τπ . Then the spherical function of type τ for π is defined by

(16.5) Φτπ(g) = Πτ π(g) Πτ .

Note that Φτπ(g) is an endomorphism of the finite dimensional space V τ

π , which is zero if τ is not aKv-type of π. Similarly, we may define the spherical trace function of type τ for π by

(16.6) ϕ τπ (g) = tr Φ τπ (g).

Note that φ τπ (e) = Πτ and ϕ τπ (e) = dimV τπ . Furthermore, since π is unitary, we have

(16.7) |ϕτπ(g)| 6 dimV τπ ‖Φτ

π(g)‖ 6 dimV τπ .

For τ ∈ Π(Kv) let ξτ denote the character of τ and write χτ = (dim τ) ξτ . We then letH(G(Fv)

1, τ) denote the space of functions f ∈ C∞c (G(Fv)1) such that

(1) f(kgk−1) = f(g) for all g ∈ G(Fv)1 and k ∈ Kv,

(2) χτ ∗ f = f = f ∗ χτ .

Then for any f ∈ H(G(Fv)1, τ) and any π ∈ Π(G(Fv)

1) we have Πτ π(f) Πτ = π(f); see, forexample, [5, Prop. 3.2]. In particular π(f) = 0 on H(G(Fv)

1, τ) unless τ is a Kv-type of π.We define the spherical transform of type τ of a function f ∈ H(G(Fv)

1, τ) by

H τ (f)(π) =

∫G(Fv)1

f(g)ϕτπ(g)dg.

It follows from the definitions (see [[5, (14)]) that, for f ∈ H(G(Fv)1, τ) we have

(16.8) π(f) = H τ (f)(π) · 1

dimV τπ

Πτ

and hence

(16.9) trπ(f) = H τ (f)(π).

The convolution algebra H(G(Fv)1, τ) is commutative if and only if τ appears with multiplicity

at most 1 in every π ∈ Π(G(Fv)1). This is the case, for example, for arbitrary Kv-types of

GL2(Fv)1, and for the trivial Kv-type for arbitrary G(Fv)

1; these are the two cases described inthe hypotheses (1) and (2) of Proposition 16.1. Whenever H(G(Fv)

1, τ) is commutative, we mayinvert the spherical transform of type τ . Indeed, it is shown in [5, p.43] that in this case one hasthe inversion formula

f(g) =1

dim τ

∫Π(G(Fv)1)

H τ (f)(π)ϕτπ(g−1)dµplv (π)

53

for all f ∈ H(G(Fv)1, τ). In such situations we see, using (16.7) and the equality dimV τ

π = dim τvalid in this case, that

(16.10) ‖f‖∞ 6 ‖H τ (f)‖L1(µpl

v ).

16.3. Reformulation using the subquotient theorem. Using the Casselman subquotient the-orem, we may complement the abstract theory of the previous section to give an explicit integralrepresentation of τ -spherical functions, and explicit formulae for the associated τ -spherical trans-forms and (in the commutative case) their inversions.

We begin by extending the definitions (16.5) and (16.6) to the principal series representationsπ(η, ν) associated with pairs (η, ν) ∈ E 2(T 1

v ) × h∗Tv ,C. These representations are not necessarily

unitary nor irreducible. For τ ∈ Π(Kv), we write

Φ τη,ν(g) = Πτ π(η, ν)(g) Πτ ,

where Πτ is the projection of I(η, ν) onto its τ -isotypic component I(η, ν)τ . Here, I(η, ν) is thespace on which π(η, ν) acts. Similarly, we put

ϕ τη,ν(g) = tr Φ τη,ν(g).

We have the integral representation of Harish-Chandra (see [49, Corollary 6.2.2.3])

ϕτη,ν(g) =

∫Kv

(χτ ∗ η)(κ(k−1gk))e〈ν−ρ,H(kg)〉 dk,

where dk is the probability Haar measure on Kv and ρ is the half-sum of positive roots. Moreover,if τ ∈ Π(Kv) appears as a Kv-type of π(η, ν), we associate with a function f ∈ H(G(Fv)

1, τ) thetransform

H τ (f)(η, ν) =

∫G(Fv)1

f(g)ϕ τη,ν(g)dg.

For R > 0 we let H(G(Fv)1, τ)R = H(G(Fv)

1, τ) ∩ H(G(Fv)1)R. Note that for a fixed σ and any

τ ∈ Π(Kv) appearing as a Kv-type in π(δ, 0), whenever f ∈ H(G(Fv)1, τ)R the assignment

ν 7→H τ (f)(η, ν)

lies in the Wσ-invariants of the Paley–Wiener space P(h∗Tv ,C)Wη

R , where Wη is the stabilizer in Wof η.

The relation between the τ -spherical functions ϕτπ and transforms H (f)(π) defined on unitaryrepresentations π ∈ Π(G(Fv)

1) and the above related objects associated with pairs (σ, λ) is given bythe Casselman subquotient theorem. This theorem states (in particular) that for any π ∈ Π(G(Fv)

1)there is η = η(π) ∈ E 2(T 1

v ) and ν = ν(π) ∈ h∗Tv ,C such that π is infinitesimally equivalent to a

subquotient of the principal series representation π(η, ν). Thus for πδ,µ ∈ Π(G(Fv)1) appearing as

a subquotient of π(η, ν), and for any τ ∈ Π(Kv) appearing as a Kv-type of πδ,µ, we have

(16.11) ϕ τπ = ϕ τη,ν and H τ (f)(πδ,µ) = H τ (f)(η, ν).

Note the importance of the assumption that τ ∈ Π(Kv) appearing as a Kv-type of πδ,µ for thisformula to hold: if τ is not a Kv-type of πδ,µ, then both ϕ τπ and H τ (f)(πδ,µ) are zero, whereasthis is not necessarily the case for ϕ τη,ν and H τ (f)(η, ν).

Now assume H(G(Fv)1, τ) commutative. Following [5, (46)], we write the inverse spherical trans-

form of type τ more explicitly. For any f ∈ H(G(Fv)1, τ) we have

(16.12) f(g) =1

dim τ

∑P

cP∑δ

∫ih∗Mv

H τ (f)(η, ν)ϕ τη,ν(g−1)βv(δ, µ)dµ,

where P runs over all cuspidal standard parabolic subgroups, cP > 0 are constants, the sum overδ ∈ E 2(MP ) is restricted to those form which τ appears as a Kv-type in π(δ, 0), and the parameters

54

(η, ν) ∈ E 2(T 1v )× h∗Tv ,C are chosen so that πδ,µ is the unique irreducible subquotient of π(η, ν) (this

choice is not unique).

16.4. Proof of Proposition 16.1 for G(Fv)1 spherical. Let Mv = T0,v and write Tv = T0,v

and hv = hTv . For the trivial Kv-type 1 ∈ Π(Kv), and the trivial character σtriv ∈ E 2(T 1v ) we

write ϕλ = ϕ1σtriv,λ

for the associated spherical function. Moreover, for f ∈ H(G(Fv)1,1), we write

h(λ) = H 1(f)(σtriv, λ) for the associated spherical transform. Then the inversion formula (16.12)becomes

f(g) =

∫ih∗v

h(λ)ϕ−λ(g)βv(λ)dλ.

For µ ∈ h∗v,C and R > 0 let hµR ∈ P(h∗v,C)WR be as in Section 9. Let fµR ∈ H(G(Fv)1,1)R be the

inverse spherical transform (of trivial Kv-type) of hµR. Explicitly,

fµR(g) =

∫ih∗v

hµR(λ)ϕ−λ(g)βv(λ)dλ.

In any case, recall the general fact that for any f ∈ H(G(Fv)1,1) the operator πσ,λ(f) acts on

Vπδ,λ as zero on whenever πσ,λ does not contain the trivial Kv-type. Thus πσ,λ(f) = 0 unless

σ = σtriv. Moreover, writing πλ = πσtriv,λ, we deduce from (16.8) that πλ(fµR) = hµR(λ)Π1. Thesetwo observations together imply (16.1). The bound (16.2) follows simply from (16.10).

16.5. Proof of Proposition 16.1 for GL2(C)1. We now consider the case of GL2(C).For an integer k ∈ Z let pk denote the character of C× given by z 7→ (z/|z|)k. For integers

(k, `) ∈ Z2 we let δk,` denote the character of TC sending diag(z1, z2) to pk(z1/z2)p`(z1z2). Then

E2(T 1C) = δk,` : (k, `) ∈ Z2.

The maximal compact KC of GL2(C) is U(2), and we have

Π(U(2)) = τn,m : (n,m) ∈ Z2, n > 0,where τn,m = Symn⊗detm. The U(2)-type decomposition of the induced representation π(k, `; s) =π(δk,`, s) can be computed by Frobenius reciprocity. One obtains

(16.13) ResU(2)π(k, `; s) =⊕n>|k|

n≡kmod 2

τn, `−n2.

Letting τ(k, `) denote the lowest U(2)-type of π(k, `; s), we have τ(k, `) = τ|k|,(`−|k|)/2.

For τn,m ∈ Π(U(2)) and f ∈ H(GL2(C)1, τn,m) we write hk(s) = H τm,n(f)(π(k, 2m + n; s))whenever |k| 6 n. From the decomposition (16.13) and the identity (16.9) we deduce that

trπ(k, `; s)(f) =

hk(s), |k| 6 n and k ≡ n mod 2 and ` = 2m+ n;

0, else.

We now explicate the inversion spherical transform in (16.12). Recall that for GL2(C) theonly cuspidal parabolic is M = T ; we identify h∗Tv ,C = C. For notational simplicity we write

ϕm,nk,s (g) = ϕτn,mηk,2m+n,s(g). There is a constant a > 0 such that for any integers (n,m) ∈ Z2 with

n > 0 and any function f ∈ H(GL2(C)1, τn,m) we have

f(g) =a

n+ 1

∑|k|6n

k≡nmod 2

∫iRϕm,nk,s (g−1)hk(s)βC(k, 2m+ n; s)ds.

With these preliminaries behind us, we now come to the construction of the test functions of

Proposition 16.1 for GL2(C)1. For δ ∈ E2(T 1C), µ ∈ h∗v, and R > 0 let hδ,µR denote the spectral

55

localizer of Section 9. Let τ(δ) be the lowest U(2)-type of π(δ, µ). Then, as explicated above, ifδ = ηk,` then τ(δ) = τ|k|,(`−|k|)/2.

First assume |k| 6 2. Then take f δ,µR ∈ H(GL2(C)1, τ(δ))R to be the inverse τ(δ)-spherical

transform of hδ,µR . Then similarly to the G(Fv)1 spherical case, this test function satisfies the two

stated properties of Proposition 16.1.Now assume |k| > 2. Write τ(δ)old = τ|k|−2,(`−|k|)/2+1) for the next lowest U(2)-type of the

same parity. Let f δ,µR,+ ∈ H(GL2(C)1, τ(δ))R be the inverse τ(δ)-spherical transform of hδ,µR and

f δ,µR,old ∈ H(GL2(C)1, τ(δ)old)R the inverse τ(δ)old-spherical transform of hδ,µR . We put

f δ,µR = f δ,µR,+ − fδ,µR,old.

Then clearly f δ,µR ∈ H(GL2(C)1)R. From (16.8) we see that

πσ,λ(f δ,µR ) = hδ,µR (σ, λ)

(1

|k|+ 1Πτ(δ) −

1

|k| − 1Πτ(δ)old

).

This is the zero operator whenever πσ,λ contains either both τ(δ) and τ(δold) or neither. What

remains is when πσ,λ contains τ(δ) as a lowest U(2)-type, i.e., σ = δ. On πδ,λ the operator πσ,λ(f δ,µR )

acts as hδ,µR (σ, λ) · 1|k|+1Πτ(δ). Together these observations show that f δ,µR verifies property (16.1).

Finally, we deduce from (16.10), applied to both terms in f δ,µR , that (16.2) holds.

16.6. The case of GL2(R)1. We now specialize the above discussion to GL2(R)1, with the goal ofconstructing the test functions of Proposition 16.1 in this case, and proving their stated properties.

16.6.1. Description of E2(M1v ). When M = T we have

E2(T 1R) = (sgnε1 , sgnε2) : εi ∈ 0, 1 .

Note that the principal series representation I(1, sgn; 0) is irreducible; it is the limit of discreteseries representation, often denoted D1.

We next take M = G. In this case

E2(GL2(R)1) = Dk : k > 2.

Here Dk = Dk, where (for k > 2) Dk is the discrete series representation for GL2(R)1 appearing asthe unique irreducible subquotient of I(1, sgnε; (k− 1)/2), where ε ≡ k mod 2. Namely, there is anexact sequence

(16.14) 1 −→ Dk −→ I(1, sgnε, (k − 1)/2) −→ Symk−2 −→ 1.

16.6.2. Kv-type decompositions. To begin the description of Kv-type decomposition, we considerthe principal series representations π(σ, λ).

We parametrize the irreducible dual of Kv = O(2) as follows. For k > 1 we put τk =

IndO(2)SO(2)(e

ikθ), a two dimensional representation. Then Π(O(2)) = 1, det ∪ τkk>1.

We let σ denote (sgnε1 , sgnε2), where εi ∈ 0, 1. Then for any λ ∈ h∗Tv ,C = C we have

(16.15) ResO(2)π(σ, λ) =

sgnε ⊕

⊕n>2n even

τn, ε1 = ε2 = ε,

⊕n>1n odd

τn, ε1 6= ε2;

56

Let τ(σ) denote the lowest O(2)-type of π(σ, λ). Then

(16.16) τ(σ) =

1, ε1 = ε2 = 0,

det, ε1 = ε2 = 1,

τ1, ε1 6= ε2.

From the exact sequence (16.14) we deduce that for any k > 2 we have

(16.17) ResO(2)(Dk) =⊕n>k

n≡kmod 2

τn.

Letting τ(Dk) denote the minimal O(2)-type of Dk, we have τ(Dk) = τk.

16.6.3. The distributional character of the discrete series. In this paragraph we explicate the τ -spherical transform for the discrete series representations, from which we deduce its distributionalcharacter.

We first agree to the following notational convention regarding the τ -spherical functions associ-ated with principal series parameters (σ, λ) ∈ E 2(T 1

R)× C:

• If σ = (1, 1), τ = 1 ∈ Π(O(2)), and f ∈ H(GL2(R)1,1) we write

ϕ0+,λ = ϕ1(1,1),λ, h0+(λ) = H 1(f)(π((1, 1), λ));

• If σ = (sgn, sgn), τ = det ∈ Π(O(2)), and f ∈ H(GL2(R)1, det) we write

ϕ0−,λ = ϕdet(sgn,sgn),λ, h0−(λ) = H det(f)(π((sgn, sgn), λ));

• If σ = (sgn, 1), τ = τ1 ∈ Π(O(2)), and f ∈ H(GL2(R)1, τ1) we write

ϕ1,λ = ϕτ1(sgn,1),λ, h1(λ) = H τ1(f)(π((sgn, 1), λ)).

We next explicate (16.11) for the discrete series Dk of GL2(R)1. Using (16.14) and (16.17) wefind that for k > 2 and j > k with j ≡ mod 2 we have

ϕτjDk

=

ϕτj(1,1),(k−1)/2, k even;

ϕτj(sgn,1),(k−1)/2, k odd.

Moreover, for k > 2 even and f ∈ H(GL2(R)1, τj) we have

trDk(f) =

h0+((k − 1)/2), j 6 k, j even;

0, else,

and for k > 3 odd and f ∈ H(GL2(R)1, τj) we have

trDk(f) =

h1((k − 1)/2), j 6 k, j odd;

0, else.

16.6.4. Explicit τ -spherical inversion. We now explicate the inversion formula (16.12) for each Kv-type τ . There are constants a, b > 0 such that the following holds:

• if τ = 1 then for any f ∈ H(GL2(R)1,1) we have

f(g) = a

∫iRϕ0+,−λ(g)h0+(λ)βR(λ)dλ;

• if τ = det then for any f ∈ H(GL2(R)1, det) we have

f(g) = a

∫iRϕ0−,−λ(g)h0−(λ)βR(λ)dλ;

57

• if τ = τ1 then for any f ∈ H(GL2(R)1, τ1) we have

f(g) = a

∫iRϕ1,−λ(g)h1(λ)βR(1, λ)dλ;

• if k > 2 is even then for any f ∈ H(GL2(R)1, τk) we have

f(g) = a∑±

∫iRϕ0±,−λ(g)h0±(λ)βR(λ)dλ+ b

∑26j6kj even

ϕτkDj (g−1)h0+((k − 1)/2);

• if k > 3 is odd then for any f ∈ H(GL2(R)1, τk) we have

f(g) = a

∫iRϕ1,−λ(g)h1(λ)βR(1, λ)dλ+ b

∑36j6kj odd

ϕτkDj (g−1)h1((k − 1)/2).

16.6.5. Proof of Proposition 16.1 for GL2(R)1. We now construct the test functions fµ,δR for GL2(R)1

having the properties described in Proposition 16.1.We begin by taking M = T . Fix δ ∈ E2(T 1

R) and µ ∈ h∗R. With τ(δ) as in (16.16), we define

f δ,µR ∈ H(GL2(R)1, τ(δ))R as the τ(δ)-spherical inverse transform of hδ,µR . Using the above explicitformulae, this gives

f δ,µR (g) =

a∫iR ϕ0+,−λ(g)hδ,µR (λ)βR(λ)dλ, if τ = 1;

a∫iR ϕ0−,−λ(g)hδ,µR (λ)βR(λ)dλ, if τ = det;

a∫iR ϕ1,−λ(g)hδ,µR (λ)βR(1, λ)dλ, if τ = τ1.

To see that this yields (16.1), it suffices to show that H (f δ,µR )(σ, λ) = 0 for any σ 6= δ. Notethat H τ (f)(σ, λ) = 0 for f ∈ H(G(Fv)

1), τ) whenever I(σ, λ) does not contain τ as a Kv-type.It then follows from the O(2)-type description (16.15) that for any f ∈ H(GL2(R)1, τ(δ)) we haveH (f)(σ, λ) = 0 for any σ 6= δ.

We next take M = G. For each δ = Dk, where k > 2, we let hδ ∈ P(C)R be such thathδ((k − 1)/2) = 1. Here we have identified h∗TR,C with C in the Paley–Wiener space. Recalling

that τ(δ) = τk is the lowest O(2)-type of δ = Dk, we let f δ+ ∈ H(GL2(R)1, τ(δ))R be the inverse

τ(δ)-spherical transform of hδ. Define τ(δ)old to be 1 if k = 2, sgn if k = 3, and τk−2 if k > 4; welet f δold ∈ H(GL2(R)1, (δ)old)R be the inverse τ(δ)old-spherical transform of hδ. We then put

f δ = f δ+ − f δold ∈ H(GL2(R)1)R.

A similar argument to the GL2(C)1 case shows that f δ satisfies all properties in Proposition 16.1.

17. Controlling the Eisenstein contribution

The Arthur trace formula is a distributional identity

(17.1) Jspec = Jgeom

on H(G(A)1), where the distribution Jspec is a sort of regularized traceWe first recall Arthur’s description of the spectral expansion. We have

Jspec =∑M∈L

Jspec,M ,

for distributions Jspec,M (φ) to be described in more detail later. In any case, note that

(17.2) Jspec,G = Jdisc.

58

Our aim in this section is to bound, uniformly in Nq, δ, µ, and R, the continuous contribution

JEis(εK1(q) ⊗ fδ,µR ) =

∑M 6=G

Jspec,M (εK1(q) ⊗ fδ,µR ),

for archimedean test functions f δ,µR ∈ H(G(F∞)1)R giving rise to the spectral localizers of Section9. We shall accomplish this for GL2, and for GLn in the archimedean spherical setting.

Theorem 17.1. Let G = GLn where n > 1. Let M∞ be a cuspidal standard Levi subgroup ofG(F∞) and δ ∈ E2(M1

∞). Suppose that either

(1) n 6 3, or(2) M∞ = T∞ and δ is the trivial character of T 1

∞.

Let µ ∈ h∗M,C and R > 0. Let f δ,µR be the function associated with this data by Proposition 16.1.Then there are constants κ, θ > 0, and for every ε > 0 a constant c > 0, such that for any integralideal q with 0 < R < c log Nq we have

JEis(εK1(q) ⊗ fδ,µR )ε (1 + log(1 + ‖τ(δ)‖))κNqn−θ+εβ(δ, µ).

We obtain the following important consequence, which is simply a reformulation of Theorem 1.2.

Corollary 17.2. Let n > 1. Let M ∈ L∞ be cuspidal and δ ∈ E2(M1∞). Assume that either

(1) n 6 2, or(2) M = T∞ and δ is the trivial character of T 1

∞.

Then Property (ELM) holds with respect to this data.

Proof. Using the equalities (17.1) and (17.2), the statement follows from Lemma ?, Theorem 14.1,Proposition 16.1, and Theorem 17.1.

We deduce the following result.

Corollary 17.3. Let G = GLn for n > 1. Let M ∈ L∞ be cuspidal and δ ∈ E2(M1∞). Assume that

either

(1) n 6 3; or(2) M = T and δ is the trivial character of T 1

∞.

Then there are positive constants κ, θ > 0, and for any ε > 0 a constant c > 0, such that for anyintegral ideal with q, spectral parameter λ ∈ h∗M , KM

∞-type τ ∈ Π(KM∞) such that [τ : τ(δ)] > 0, and

real parameter 0 < R < c log Nq we have∑π∈Πdisc(M(A)1)δλπ∞∈B1(λ)

eπR·Re(λπ∞ ) dimVKM

1 (q),τπ ε (1 + log(1 + ‖τ‖))κNqn−θ+εβM (δ, λ).

Proof. Let M = M1× · · ·×Mr be the block decomposition of M , where Mi ' GLni and n1 + · · ·+nr = n and r > 2. Then KM

1 (q) ' KM1f × · · · ×K

Mr−1

f × KMr1 (q). The statement follows from

Lemma 10.1 applied to each Mi, together with ϕnr(q) 6 ϕn−1(q) Nqn−1.

17.1. The distributions Jspec,M . To proceed we need to describe in more detail the distributionsJspec,M . This will necessitate a great deal of notation; we borrow essentially from [11, §4]. For thisparagraph (and the next) we shall not have need to specialize to the two cases of Proposition 17.1.

Let M be a proper standard Levi of G and P a parabolic subgroup of G having M as Levisubgroup. Let A2(P ) be the space of all complex-valued functions ϕ on UP (A)M(F )\G(A)1

such that for every x ∈ G(A) the function ϕx(g) = δP (g)−1/2ϕ(gx), where g ∈ M(F ), lies inL2(AMM(F )\M(A)). We require that ϕ be z-finite and Kf -finite, where z is the center of the

59

universal enveloping algebra of gC. Let A2(P ) be the Hilbert space completion of A2(P ). For everyλ ∈ a∗M,C the space A2(P ) receives an action by G(A)1 given by

(ρ(P, λ, y)(ϕ))(x) = ϕ(xy)e〈λ,HP (xy)−HP (x)〉,

which makes it isomorphic to the induced representation

Ind(L2disc(AMM(F )\M(A))⊗ e〈λ,HM (·)〉).

For P,Q ∈ P(M) and λ ∈ a∗M,C let MQ|P (λ) : A2(P ) → A2(Q) be the standard intertwining

operator; then MP |Q(λ) is unitary for all λ ∈ ia∗M . Let WM = NG(M)/M be the Weyl groupof M ; we can view it as a subgroup of the Weyl group W = W (G,T ) of T . Then WM acts onP(M) by sending P to wsPw

−1s , where ws ∈ NG(T ) is a representative. This gives rise to a map

on P -induced automorphic forms s : A2(P )→ A2(sP ) given by left-translation by w−1s . We write

M(P, s) for the composition MP |sP (0) s : A2(P ) → A2(P ). Then M(P, s) is a unitary operatorwhich for λ ∈ ia∗Ls intertwines ρ(P, λ) with itself, where Ls denotes the smallest Levi subgroupof G containing ws. For a description of the logarithmic derivatives of the intertwining operators,denoted ∆XLs (β) and associated to certain finite combinatorial data β ∈ BP,Ls , we prefer to sendthe reader to [9, §2] or [11, §4]. For any s ∈W (aM ) and β ∈ BP,Ls , we put

Jspec,M (φ; s, β) =

∫ia∗Ls

tr(

∆XLs (β)(P, λ)M(P, s)ρ(P, λ, φ))dλ,

where the operators are of trace class and the integrals are absolutely convergent [30]. Finally, letιs = | det(s− 1)

aLsM|−1.

With the above notation, then

Jspec,M (φ) =1

|WM |∑s∈WM

ιs∑

β∈BP,Ls

Jspec,M (φ; s, β);

see [9, Corollary 1] or [11, Theorem 4.1].

17.2. Local and global input for general GLn. In this paragraph, we continue to work inthe general case of GLn. Our goal here is to conveniently package two of the major inputs thatwill be necessary for the proof of Theorem 17.1. The first concerns the norm of the operators∆XLs (β) : A2(P ) → A2(P ) and is encapsulated in Lemma 17.4 below. The second, recorded inLemma 17.5, bounds the dimension of the space of oldforms (with fixed K∞-type) of an inducedautomorphic representation in terms of the corresponding dimension of the inducing data.

We first need to introduce some more notation. For π ∈ Πdisc(M(A)1), let A2π(P ) denote the

subspace of A2(P ) consisting of ϕ such that, for each x ∈ G(A)1, the function ϕx transforms underM(A)1 according to π. For a compact open subgroup Kf of G(Af ) and a K∞-type τ ∈ Π(K∞)

we let Aπ(P )Kf ,τ be the finite dimensional subspace of Kf -invariant functions, transforming underK∞ according to τ . Finally, for an irreducible admissible representation π∞ of M∞ of infinitesimalcharacter λπ∞ and minimal KM,∞-type τπ∞ , we write, following [11, §5],

ΛM (π∞) = 1 + λ2π∞ + ‖τπ∞‖2.

Lemma 17.4 (Finis-Lapid-Muller, Lapid, Matz). Let q be an integral ideal and τ ∈ Π(K∞). LetM ∈ L be a proper Levi subgroup of G and L ∈ L(M). Then for all π ∈ Πdisc(M(A)1) and λ ∈ ia∗Lthe integral ∫

B(λ)∩ia∗L‖∆X (P, ν)|A2

π(P )K1(q),τ ‖dν

is bounded by

O((1 + log Nq + log(1 + ‖λ‖) + log(1 + ΛM (π∞)) + log(1 + ‖τ‖))2rL),

60

where rL = dim aL.

Proof. For τ the trivial K∞-type, this is [24, Lemma 14.3]. The proof is based on two importantcontributions from Finis-Lapid-Mueller. The first a strong form of the Tempered Winding Numberproperty for GLn, established in [11, Proposition 5.5]. The second is the Bounded Degree property;in [10, Theorem 1] it is shown that GLn over p-adic fields satisfies this property and in the appendixto [30] (see also [10, Theorem 2]) the same is shown for arbitrary real groups.

It therefore remains to extend the proof of [24, Lemma 14.3] to arbitrary K∞-types. The onlyinstance in which the trivial K∞-type hypothesis is invoked in that proof is in [24, Lemma 14.4],when the Bounded Degree property in the archimedean case is applied. Using the notation of thatpaper, we claim that, when v is archimedean, for any irreducible unitary representation πv of Mv,any KM,v-type τ , and all T ∈ R we have

(17.3)

∫ T+1

T‖RQ|P (πv, it)

−1R′Q|P (πv, it)|τ‖dt 1 + log(1 + ‖τ‖).

Here, the restriction is to the τ -isotypic subspace of πv. We adapt the argument of [11, §5.4] toour situation, which differs from theirs in that our integral is over a bounded interval and does notcontain a factor of (1 + |s|2)−1.

As a first step, we modify the statement of Lemma 5.19 of [11] to fit our setting. With thenotation and same hypotheses of that result, we claim that∫ T+1

T‖A′(it)‖dt

m∑j=1

max1, |uj |−1.

To see this, we set φw(z) = (z + w)/(z − w) for w = u+ iv ∈ C− iR; then∫ T+1

T|φ′w(it)|dt max1, |u|−1.

Indeed we have φ′w(it) = 2|u|/(u2 + (t − v)2), and extending the integral over all R by positivityyields the result after computation.

Arguing as in the proof of Proposition 5.16 in [11], we shall apply the above bound with theunitary operator A = RQ|P (πv, is)|τ . Then the set wj = uj + ivj consists of the poles of thematrix coefficient (RQ|P (πv, is)ϕ1, ϕ2), where ϕ1, ϕ2 are unit vectors in the τ -isotypic componentof πv. We shall need some basic information on these poles wj , to be found in Lemma A.1 andProposition A.2 of [30]. We can deduce from these results that there are integers K,L ∈ N satisfyingK 1, L 1 + ‖τ‖, a real number δ ∈ [0, 1/2], and complex numbers ρk, for k = 1, . . . ,K, suchthat the poles of (RQ|P (πv, is)ϕ1, ϕ2) are given by

ρk − ` : 1 6 k 6 K, dRe(ρk) + δe 6 ` 6 bRe(ρk) + Lc.

Inserting this into the above estimate yields∫ T+1

T‖R′Q|P (πv, it)|τ‖dt 6 2

K∑k=1

bRe(ρk)+Lc∑`=dRe(ρk)+δe

max1, |Re(ρk)− `|−1.

Using the bounds on K and L, we deduce (17.3).

We next relate the dimension of space of invariants of the P -induced automorphic forms onG to the dimension of the corresponding space of invariants of the inducing representation. The

following notation will be useful: for M ∈ L, π ∈ Π(M(F∞)1), and τ ∈ Π(KM∞), we write V

τ+O(1)π

for a sum over KM∞-isotypic components of Vπ whose highest weights lie in an O(1)-ball about τ .

61

Lemma 17.5. Let L∞ ∈ L∞ be cuspidal and let δ ∈ E2(L1∞). Let M ∈ L be a proper Levi subgroup

of G = GLn, and π ∈ Πdisc(M(A)1)δ. For any integral ideal q we have

dimA2π(P )K1(q),τ(δ)

dim τ(δ)ε Nqε dimV

KM1 (q),τ(π∞)+O(1)

π ,

where τ(π∞) is the minimal KM∞-type of π∞.

Proof. Let HP (πf ) and HP (π∞) be the Hilbert spaces of the induced representations IG(Af )

P (Af ) (πf )

and IG∞P∞ (π∞). By [31, (3.5)], for any τ ∈ Π(K∞) we have

dimA2π(P )K1(q),τ = dimHP (πf )K1(q) dimHP (π∞)τ .

Here, we have implicitly used the multiplicity one theorem [18, 44, 36] for the cuspidal spectrumof GLn, and the description of the residual spectrum of GLn by [28]. Together, these results statethat the multiplicity with which π ∈ Πdisc(M(A)1) appears in the right-regular representationL2(M(F )\M(A)1) is 1.

By [31, (3.6)], for any τ ∈ Π(K∞) we have

dimHP (π∞)τ 6 dim τ∑

τ ′∈Π(KM∞)

[τ|KM∞

: τ ′] dimV τ ′π∞ .

Classical multiplicity one results for successive unitary and orthogonal groups imply the bound[τ|KM

∞: τ ′] = O(1), the implied constant depending only on n. Note that since δπ∞ ' δ the minimal

K∞-type of IG∞P∞ (π∞) is τ(δ). Let Λ and λ denote the the highest weights of τ(δ) and τ(π∞),respectively. The minimal K∞-type formula of Knapp [21] states that both Λ and λ differ fromthe Blattner parameter of δ by a cocharacter depending only on K∞ and KM

∞ . We conclude thatthe τ ′ contributing the above sum are in an O(1)-ball about τ(π∞).

As for the finite places, it suffices to consider v | q. Let f be the conductor exponent of the

irreducible tempered generic representation IG(Fv)P (Fv) (πv) of = G(Fv). From the dimension formulae

of [6, 37] we have

dimHP (πv)K1(pr) =

(n+r−f

n

), r > f ;

0, else.

Let M = M1 × · · · ×Mr be the block decomposition of M , with Mi ' GLni and n1 + · · ·+ nr = n.Say πv = πv,1 ⊗ · · · ⊗ πv,r and write fi for the conductor exponent of each πv,i. Then, writing

KMi1 (q) = K1(q) ∩Mi(Af ), we once again have the dimension formulae

dimVKM

1 (pr)πv =

∏i

dimVKMi1 (pr)

πv,i =

∏i

(ni+r−fi

ni

), r > maxi fi;

0, else.

We may use the Local Langlands Correspondence [14, 15, 43] to compare the conductors of πv and

IGvPv (πv). Indeed, if φ is the Langlands parameter of IG(Fv)P (Fv) (πv) and φi that of πv,i then φ = ⊕iφi.

From this it follows that f = maxi fi. Continuing, note that for r > f = maxi fi we have(n+ r − f

n

)6

(n+ r − f

n

)r=∏i

ai

(ni + r − fi

ni

),

where ai =(n+r−f

n

)/(ni+r−fi

ni

). Since ai 6

(n+rn

)n (1 + r)n we deduce from the above discussion

that

dimHP (πv)K1(prv) n (1 + r)n dimV

KM1 (prv)

πv .

Taking the product over all v | q finishes the proof.

62

17.3. Proof of Theorem 17.1. We are now ready to prove Theorem 17.1. We argue by inductionon n. For n = 1 there is no continuous spectrum so Theorem 17.1 is trivially true in that case.Now assume the result for GLm for all m < n. The induction hypothesis will be used after havingmade various simplifications coming from Proposition 16.1 and the general results from the previousparagraph.

It will be enough to bound Jspec,M (εK1(q) ⊗ fδ,µR ; s, β) for a given s ∈ W (aM ) and β ∈ BP,Ls .

For convenience, we agree to drop the dependence of s and β from the notation. In particular, wewrite L in place of Ls, M(P ) in place of M(P, s), and X in place of XLs(β).

Expanding over Πdisc(M(A)1), we find

(17.4)∑

π∈Πdisc(M(A)1)

∫ia∗L

tr (∆X (P, π, ν)M(P, π)ρ(P, ν, π, φ)) dν,

where ∆X (P, π, ν), M(P, π), and ρ(P, ν, π, φ) denote the restrictions of the corresponding opera-tors to A2

π(P ). For τ ∈ Π(K∞) we let ΠK1(q),τ denote the orthogonal projection of A2π(P ) onto

A2π(P )K1(q),τ . From Proposition 16.1 it follows that

ρ(P, ν, π, εK1(q) ⊗ fδ,µR ) =

hδ,µR (λπ∞ + ν) · 1

dim τ(δ)ΠK1(q),τ(δ), δπ∞ ' δ;0, else.

yielding

(17.5)1

dim τ(δ)

∫ia∗L

hδ,µR (λπ∞ + ν)tr(∆X (P, π, ν)M(P, π)ΠK1(q),τ(δ)

)dν

for the integral in (17.4) whenever π ∈ Πdisc(M(A)1)δ, and 0 otherwise. Using the unitarity ofM(P, π), as well as the upper bound ‖A‖1 6 dimV ‖A‖ for any linear operator A on a finitedimensional Hilbert space V , the expression in (17.5) is bounded in absolute value by

(17.6)dimA2

π(P )K1(q),τ(δ)

dim τ(δ)

∫ia∗L

|hδ,µR (λπ∞ + ν)|‖∆X (P, π, ν)|K1(q),τ(δ)‖dν.

We now proceed to break up the sum-integral in (17.4), so that the infinitesimal character of π andthe continuous twisting parameter ν lie in balls centered at lattice points. We write aLM = aM ∩ aLand let (aLM )⊥ be the annihilator of aLM in h∗. For ν ∈ i(aLM )⊥ we write ν = νM + νL according to

the decomposition i(aLM )⊥ = ih∗M ⊕ ia∗L. We choose a lattices ΛM ⊂ ih∗M and ΛL ⊂ ia∗L such that

Λ = ΛM ⊕ ΛL ⊂ i(aLM )⊥ satisfies ⋃λ∈Λ

(B1(λ) ∩ i(aLM )⊥) = i(aLM )⊥.

We deduce that (17.4) is bounded above by

(17.7)∑λ∈Λ

∑π∈Πdisc(M(A)1)δλπ∞∈B1(λM )

dimA2π(P )K1(q),τ(δ)

dim τ(δ)

∫B1(λL)∩ia∗L

|hδ,µR (λπ∞ + ν)|‖∆X (P, π, ν)|K1(q),τ(δ)‖dν.

We now use the Paley-Wiener estimate

hδ,µR (λπ∞ + ν)N eπR·Re(λπ∞ )(1 + ‖Im(λπ∞ + ν − µ)‖)−N

for any ν ∈ ia∗L, along with Lemma 17.4 and∑λL∈ΛL

(1 + ‖λM + λL − µ‖)−N (1 + log(1 + ‖λL‖))κ (1 + ‖λM − µM‖)−N (1 + log(1 + ‖µM‖))κ,

63

valid for large enough N , to bound (17.7) by

(17.8)

(1 + logNq + log(1 + ‖τ(δ)‖))κ(1 + log(1 + ‖µM‖))κ

×∑

λM∈ΛM

(1 + ‖λM − µM‖)−N∑

π∈Πdisc(M(A)1)δλπ∞∈B1(λM )

eπR·Re(λπ∞ ) dimA2π(P )K1(q),τ(δ)

dim τ(δ).

It follows from Lemma 17.5, the induction hypothesis, and Corollary 17.3 that, for 0 < R < c log Nq,∑π∈Πdisc(M(A)1)δλπ∞∈B1(λM )

eπR·Re(λπ∞ ) dimA2π(P )K1(q),τ(δ)

dim τ(δ)ε (1 + log(1 + ‖τ(δ)‖))κNqn−θ+εβM (δ, λM ).

We deduce that the quantity in (17.8) is bounded by

(1 + log(1 + ‖τ(δ)‖))κ(1 + log(1 + ‖µM‖))κNqn−θ+ε∑

λM∈ΛM

(1 + ‖λM − µM‖)−NβM (δ, λM ).

The latter sum is O(βM (δ, µM )), concluding the proof of Theorem 17.1.

References

1. James Arthur, The trace formula in invariant form, Ann. of Math. (2) 114 (1981), no. 1, 1–74. MR 625344 412. , The local behaviour of weighted orbital integrals, Duke Math. J. 56 (1988), no. 2, 223–293. MR 932848

413. V. V. Batyrev and Yu. I. Manin, Sur le nombre des points rationnels de hauteur borne des varietes algebriques,

Math. Ann. 286 (1990), no. 1-3, 27–43. MR 1032922 34. F. Brumley and S. Marshall, Lower bounds for Maass forms on semisimple groups, ArXiv e-prints (2016). 45, 495. Roberto Camporesi, The spherical transform for homogeneous vector bundles over Riemannian symmetric spaces,

J. Lie Theory 7 (1997), no. 1, 29–60. MR 1450744 12, 52, 536. William Casselman, On some results of Atkin and Lehner, Math. Ann. 201 (1973), 301–314. MR 0337789 4, 17,

617. L. Clozel and P. Delorme, Le theoreme de Paley-Wiener invariant pour les groupes de Lie reductifs, Invent. Math.

77 (1984), no. 3, 427–453. MR 759263 5, 258. T. Finis and E. Lapid, An approximation principle for congruence subgroups, ArXiv e-prints (2013). 8, 12, 38,

43, 44, 45, 489. Tobias Finis, Erez Lapid, and Werner Muller, On the spectral side of Arthur’s trace formula—absolute convergence,

Ann. of Math. (2) 174 (2011), no. 1, 173–195. MR 2811597 5910. , On the degrees of matrix coefficients of intertwining operators, Pacific J. Math. 260 (2012), no. 2,

433–456. MR 3001800 12, 6011. , Limit multiplicities for principal congruence subgroups of GL(n) and SL(n), J. Inst. Math. Jussieu 14

(2015), no. 3, 589–638. MR 3352530 8, 12, 58, 59, 6012. Roger Godement and Herve Jacquet, Zeta functions of simple algebras, Lecture Notes in Mathematics, Vol. 260,

Springer-Verlag, Berlin-New York, 1972. MR 0342495 1713. Benedict H. Gross, On the motive of a reductive group, Invent. Math. 130 (1997), no. 2, 287–313. MR 1474159

3, 4, 15, 1614. Michael Harris and Richard Taylor, The geometry and cohomology of some simple Shimura varieties, Annals of

Mathematics Studies, vol. 151, Princeton University Press, Princeton, NJ, 2001, With an appendix by VladimirG. Berkovich. MR 1876802 61

15. Guy Henniart, Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique, Invent. Math.139 (2000), no. 2, 439–455. MR 1738446 61

16. James E. Humphreys, Conjugacy classes in semisimple algebraic groups, Mathematical Surveys and Monographs,vol. 43, American Mathematical Society, Providence, RI, 1995. MR 1343976 41

17. H. Iwaniec and P. Sarnak, Perspectives on the analytic theory of L-functions, Geom. Funct. Anal. (2000), no. Spe-cial Volume, Part II, 705–741, GAFA 2000 (Tel Aviv, 1999). MR 1826269 2, 18

18. H. Jacquet and R. P. Langlands, Automorphic forms on GL(2), Lecture Notes in Mathematics, Vol. 114, Springer-Verlag, Berlin-New York, 1970. MR 0401654 61

19. H. Jacquet, I. I. Piatetski-Shapiro, and J. Shalika, Conducteur des representations du groupe lineaire, Math. Ann.256 (1981), no. 2, 199–214. MR 620708 4, 17

64

20. H. Jacquet, I. I. Piatetskii-Shapiro, and J. A. Shalika, Rankin-Selberg convolutions, Amer. J. Math. 105 (1983),no. 2, 367–464. MR 701565 18

21. A. W. Knapp, Minimal K-type formula, Noncommutative harmonic analysis and Lie groups (Marseille, 1982),Lecture Notes in Math., vol. 1020, Springer, Berlin, 1983, pp. 107–118. MR 733463 61

22. E. Landau, Einfuhrung in die elementare und analytische Theorie der algebraischen Zahlen und der Ideale.,Leipzig: B. G. Teubner. VII u. 143 S. 8 (1918)., 1918. 35

23. Erez Lapid and Werner Muller, Spectral asymptotics for arithmetic quotients of SL(n,R)/SO(n), Duke Math. J.149 (2009), no. 1, 117–155. MR 2541128 9, 41

24. J. Matz, Weyl’s law for Hecke operators on GL(n) over imaginary quadratic number fields, ArXiv e-prints (2013).8, 12, 38, 39, 40, 42, 43, 45, 46, 47, 48, 50, 60

25. J. Matz and N. Templier, Sato-Tate equidistribution for families of Hecke-Maass forms on SL(n,R)/SO(n), ArXive-prints (2015). 8, 12, 38, 39, 40, 42, 43, 50, 51

26. Jasmin Matz, Bounds for global coefficients in the fine geometric expansion of Arthur’s trace formula for GL(n),Israel J. Math. 205 (2015), no. 1, 337–396. MR 3314592 40

27. , Distribution of Hecke eigenvalues for GL(n), Families of automorphic forms and the trace formula.Proceedings of the Simons symposium, Puerto Rico, January 26 – February 1, 2014., Cham: Springer, 2016,pp. 327–350. 23

28. C. Mœglin and J.-L. Waldspurger, Le spectre residuel de GL(n), Ann. Sci. Ecole Norm. Sup. (4) 22 (1989), no. 4,605–674. MR 1026752 29, 61

29. Carlos J. Moreno, Analytic proof of the strong multiplicity one theorem, Amer. J. Math. 107 (1985), no. 1,163–206. MR 778093 3

30. W. Muller and B. Speh, Absolute convergence of the spectral side of the Arthur trace formula for GLn, Geom.Funct. Anal. 14 (2004), no. 1, 58–93, With an appendix by E. M. Lapid. MR 2053600 59, 60

31. Werner Muller, Weyl’s law for the cuspidal spectrum of SLn, Ann. of Math. (2) 165 (2007), no. 1, 275–333.MR 2276771 61

32. D. G. Northcott, A further inequality in the theory of arithmetic on algebraic varieties, Proc. Cambridge Philos.Soc. 45 (1949), 510–518. MR 0033095 3

33. , An inequality in the theory of arithmetic on algebraic varieties, Proc. Cambridge Philos. Soc. 45 (1949),502–509. MR 0033094 3

34. Marc R. Palm, Explicit GL(2) trace formulas and uniform, mixed Weyl laws., Gottingen: Univ. Gottingen (Diss.),2012 (English). 8

35. Emmanuel Peyre, Hauteurs et mesures de Tamagawa sur les varietes de Fano, Duke Math. J. 79 (1995), no. 1,101–218. MR 1340296 3

36. I. I. Piatetski-Shapiro, Multiplicity one theorems, Automorphic forms, representations and L-functions (Proc.Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII,Amer. Math. Soc., Providence, R.I., 1979, pp. 209–212. MR 546599 61

37. Mark Reeder, Old forms on GLn, Amer. J. Math. 113 (1991), no. 5, 911–930. MR 1129297 17, 6138. Zeev Rudnick and Peter Sarnak, Zeros of principal L-functions and random matrix theory, Duke Math. J. 81

(1996), no. 2, 269–322, A celebration of John F. Nash, Jr. MR 1395406 (97f:11074) 1939. P. Sarnak, S. W. Shin, and N. Templier, Families of L-functions and their Symmetry, ArXiv e-prints (2014). 2,

6, 7, 1040. Peter Sarnak, Statistical properties of eigenvalues of the Hecke operators, Analytic number theory and Diophantine

problems (Stillwater, OK, 1984), Progr. Math., vol. 70, Birkhauser Boston, Boston, MA, 1987, pp. 321–331.MR 1018385 2

41. , Definition of families of L-functions, (2008), preprint available at http://publications.ias.edu/

sarnak/paper/507. 242. Stephen Hoel Schanuel, Heights in number fields, Bull. Soc. Math. France 107 (1979), no. 4, 433–449. MR 557080

343. Peter Scholze, The local Langlands correspondence for GLn over p-adic fields, Invent. Math. 192 (2013), no. 3,

663–715. MR 3049932 6144. J. A. Shalika, The multiplicity one theorem for GLn, Ann. of Math. (2) 100 (1974), 171–193. MR 0348047 6145. Sug Woo Shin and Nicolas Templier, Sato-Tate theorem for families and low-lying zeros of automorphic L-

functions, Invent. Math. 203 (2016), no. 1, 1–177, Appendix A by Robert Kottwitz, and Appendix B by RafCluckers, Julia Gordon and Immanuel Halupczok. MR 3437869 10, 43, 45, 46, 49, 50

46. J. T. Tate, Fourier analysis in number fields, and Hecke’s zeta-functions, Algebraic Number Theory (Proc.Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 305–347. MR 0217026 17

47. David A. Vogan, Jr., The algebraic structure of the representation of semisimple Lie groups. I, Ann. of Math. (2)109 (1979), no. 1, 1–60. MR 519352 51

65

48. , The unitary dual of GL(n) over an Archimedean field, Invent. Math. 83 (1986), no. 3, 449–505.MR 827363 51

49. Garth Warner, Harmonic analysis on semi-simple Lie groups. II, Springer-Verlag, New York-Heidelberg, 1972,Die Grundlehren der mathematischen Wissenschaften, Band 189. MR 0499000 52, 53

Institut Galilee, Universite Paris 13, 93430 Villetaneuse, FranceE-mail address: brumley@math.univ-paris13.fr

Bryn Mawr College, Department of Mathematics, 101 North Merion Avenue, Bryn Mawr, PA 19010,U.S.A.

E-mail address: dmilicevic@brynmawr.edu