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AN INTRODUCTION TO p-ADIC L-FUNCTIONS II:

MODULAR FORMS

by

Chris Williams

Contents

General overview, continued. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

I. Adelic automorphic forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

I.1. Dirichlet characters as automorphic forms for GL(1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

I.2. Automorphic forms for GL(2): modular forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

II. Iwasawa theory for modular forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

II.1. Recapping GL(1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

II.2. Analogues for GL(2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

II.3. Iwasawa theory for elliptic curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

II.4. Further generalisations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

II.5. What do we cover in these lectures?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

II.6. Prerequisites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

II.7. Organisation of lectures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

II.8. Acknowledgements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Part III: p-adic L-functions for modular forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

9. The L-function of a modular form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

9.1. Modular forms 101. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

9.2. Hecke operators and algebraic structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

9.3. p-stabilisation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

9.4. The L-function of a modular form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

10. Modular symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

10.1. Modular symbols in weight 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

10.2. Abstract modular symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

11. Algebraicity of L-values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

11.1. Motivation: a proof strategy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

11.2. The EichlerShimura isomorphism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

11.3. Plus-minus spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

11.4. Connection to L-values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

11.5. EichlerShimura with algebraic coecients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

11.6. Periods and algebraic modular symbols for newforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

11.7. Algebraic modular symbols for general eigenforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

11.8. Algebraicity of L-values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Interlude: a change of language. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

12. Overconvergent modular symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

12.1. Analytic distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

12.2. Overconvergent modular symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

12.3. Stevens' control theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

12.4. Additional remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

13. Proof of Stevens' control theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

13.1. An example: improving with Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

13.2. Step 1: Passing to integral coecients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

13.3. Step 2: The integral α−1Up-operator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

13.4. Step 3: Improving with Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

13.5. Explicit lifting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

14. The p-adic L-function of a modular form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2 CHRIS WILLIAMS

14.1. Locally analytic distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

14.2. The p-adic L-function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

15. Growth properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

15.1. Norms on distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

15.2. Uniqueness properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

This set of notes is a continuation of `An introduction to p-adic L-functions', written bythe author jointly with Joaquín Rodrigues Jacinto. The rst notes gave an introductionto p-adic L-functions, with a focus on p-adic L-functions for GL(1), and in particular thep-adic analogue of the Riemann zeta function. The main areas of study were p-adic measuresand Iwasawa algebras, cyclotomic units, and the Iwasawa main conjecture for GL(1). Theprevious notes contained a general overview (section 1), Part I (sections 25) and Part II(sections 68), as well as an appendix of classical background results. They are not requiredas a prerequisite to the present notes, though sections 13 at least would provide usefulmotivation for the topics we consider here. Throughout, they are cited as [RJW17].

The present notes should be considered as the second half of a rst course on p-adicL-functions. In them, we look at the analytic side of this picture for GL(2), studying p-adicL-functions of modular forms. We consider the theory of p-adic distributions (a more generalnotion of measures), and give a construction of the p-adic L-function of a modular form dueto Pollack and Stevens.

Recommended reading: The material on modular forms is explained in far more detailin [DS05], particularly in 5, where the properties of the L-function are explored. All of thematerial on p-adic analysis and distributions can be found in a paper of Colmez [Col10],a complete and very readable account of the theory (in French). The construction of thep-adic L-function of a modular form via overconvergent modular symbols is from the originalpapers of Pollack and Stevens [PS11, PS12], and is explained beautifully in their ArizonaWinter School lectures [Pol11], which can be viewed online through the AWS page. Almostall of the material here, and lots more there was not time to cover, also appears in Bellaïche'sexcellent (but sadly unnished) book [Bel].

General overview, continued

In the rst half of these notes, we studied only the very simplest kinds of L-functions,those attached to Dirichlet characters, whilst alluding to the more general picture. It isnatural to ask in what generality L-functions, and the results of Iwasawa theory, should live.We now sketch some of the more general aspects of the theory.

AN INTRODUCTION TO p-ADIC L-FUNCTIONS II: MODULAR FORMS 3

I. Adelic automorphic forms

The philosophy of the Langlands program says that `every L-function should come froman automorphic form(1).' Such objects are analytic functions on adelic groups that are highlysymmetric for a group action.

I.1. Dirichlet characters as automorphic forms for GL(1). The most basic exam-ples of automorphic forms, again, are Dirichlet characters. We would like a uniform way toconsider all such characters, which in their standard denition are all dened on dierentnite multiplicative groups. With that in mind, let D =

∏p prime p

rp be a positive integer,noting that for all but nitely many primes, rp = 0. Consider a primitive Dirichlet character

χ : (Z/DZ)× −→ C×

modulo D. Decomposing, one can consider this as a product of characters

χ =

Ñ ∏p prime

χp

é:∏

(Z/prpZ)× −→ C×;

and then, lifting under the natural maps Z×p → (Z/prpZ)×, as a character

χ :∏p

Z×p −→ C×.

Ultimately, this means we can consider every Dirichlet character as an element

χ ∈ Homcts

(∏p

Z×p ,C×

).

We now have a contribution from every nite place of Q. In the spirit of L-functions, weusually go further, and include the innite place as well. Recall the identication

C ∼−→ Homcts(R>0,C×)

s 7−→ [x 7→ xs].

We may then consider elements

(s, χ) ∈ Homcts

(R>0 ×

∏p

Z×p , C×

).

Observe that this is exactly the input one gives an L-function! In particular, we can considerall Dirichlet L-functions at once as a single measure on this space R>0 ×

∏p Z×p . Such a

viewpoint was outlined in Tate's thesis.

This is usually packaged into a nicer form by introducing the ideles

A× = R× ×∏p

′Q×p

..= (x∞, x2, x3, x5, ...) : xp ∈ Z×p for all but nitely many p,

the∏′

denoting `restricted product'. Note that there is a unique rational number q suchthat x∞/q ∈ R>0 and xp/q ∈ Z×p for all p. Indeed, any rational number is determined byits valuation at every prime p and its sign; the valuation of q at p is exactly vp(xp), and thesign is equal to the sign of x∞. Note that this is well-dened by the condition that xp ∈ Z×pfor almost all p. We thus get a decomposition

A× ∼= Q× ×R>0 ×∏p

Z×p , (I.1)

(1)Or more properly, an automorphic representation, where one considers the space of all such automorphic

forms under an adelic group action.

4 CHRIS WILLIAMS

and we can consider Dirichlet characters as functions

χ : A× −→ C×

which are invariant under pre-multiplication by Q×, where the multiplication is by

q · (x∞, x2, x3, ...) = (qx∞, qx2, qx3, ...).

To complete this transition to automorphic forms, now observe that Q× = GL1(Q).Further, the notation A× is suggestive, and indeed this is the group of units in a ring

A ..= R×′∏p

Qp

= (x∞, x2, x3, x5, ...) : xp ∈ Zp for all but nitely many p.

In this regime, we then see the Dirichlet character χ as a function

χ : GL1(A) −→ C×

which is:

invariant under GL1(Q); invariant under R>0; and invariant under the action of a nite index subgroup K(D) ⊂

∏p Z×p .

Here K(D) is dened place by place to be

K(D)p =

ß1 + prpZp : rp > 0

Z×p : rp = 0.

We normally again parcel this up by dening

Z = lim←−N≥1

(Z/NZ) =∏p

Zp,

then seeing K(D) as an open compact subgroup of GL1(Z) =∏p Z×p .

This leads us to consider χ as an automorphic form for G = GL1 of level D, that is, acomplex function

χ : G(A) −→ C

which is (left-)invariant under G(Q) and (right-)invariant under an open compact subgroup

K ⊂ G(Z).

Remark (Weights and the idelic norm). The Dirichlet characters we consider arenite order, and thus are trivial on the component R>0. These correspond to automorphicforms of `weight 0'. More generally, the weight of an automorphic form is always encodedin its archimedean part. Automorphic forms for GL1 of weight k are those functions whoserestriction to R>0 is of the form

χ∞ : R>0 −→ C

x 7−→ xk.

Note, however, that the function

χk : A× −→ C×

(x∞;xf ) 7−→ xk∞ · χ(xf ),

where xf = (x2, x3, x5, ...) denotes the nite part of an idele x, is not automorphic in thesense above: it is not invariant under pre-multiplication by Q. If we multiply by the prime5, for example, then

χk(5 · x) = 5k · χk(x)

= |5|−k5 · χk(x).


We can x this, then, by also multiplying by | · |kp at every nite prime. This translates intomultiplying by a power of the idelic norm function

| · |A× : A× −→ R>0,

(x∞;x2, x3, ...) 7−→ |x∞|R ×∏p

|xp|p.

One checks that |Q×|A× = 1, and hence that the function

χ| · |kA× : A× −→ C×

preserves the property of invariance under Q×, that is, it is a weight k automorphic formfor GL(1).

It turns out that all algebraic weight k automorphic forms for GL(1) are a power of thenorm times a nite-order character.

Remark (Further conditions at innity). For a more general group G, the innitepart R>0 should be replaced by the connected component G+(R) of the identity in G(R).We should then also impose a symmetry condition under the maximal compact subgroupK∞ ⊂ G+(R). Note that the maximal compact subgroup of R>0 is just 1, so that forG = GL1, this is an empty condition.

Finally, we should more generally also consider an analyticity condition at the inniteplace, which we ignore for now.

I.2. Automorphic forms for GL(2): modular forms. In the second half of thesenotes, we turn our attention to what is perhaps the next most natural (and most studied!)case, that of GL2. The natural generalisation of the above is to consider complex functions

f : GL2(A) −→ C

that are:

left-invariant under GL2(Q),

right-invariant under an open compact subgroup K ⊂ GL2(Z), transforms like x 7→ x−k on the centre A× ⊂ GL2(A) of diagonal matrices, and satises a symmetry condition under the maximal compact subgroup

K∞ = SO2(R) ⊂ GL+2 (R).

Here the connected component GL+2 (R) is just the group of real matrices of positive deter-

minant.

It turns out these (algebraic) adelic automorphic forms admit a much simpler description.Indeed, the strong approximation theorem for GL2 tells us that

GL2(A) = GL2(Q) ·GL+2 (R) ·K,

a higher-dimensional analogue of (I.1). Now, we cut down our space.

The left-invariance under GL2(Q), and the right-invariance under K, means these func-tions are uniquely determined by their values on GL+

2 (R). These functions have specied transformation properties under the centre R>0 and

the maximal compact SO2(R), so are uniquely determined by their values on the quotientGL+

2 (R)/[R>0 · SO2(R).

Lemma. Let H ..= R×R>0 = z = x+ iy ∈ C : y > 0. Then we have

H ∼= GL+2 (R)

/[R>0 · SO2(R)

]via the map x+ iy 7→ ( y x0 1 ) .

6 CHRIS WILLIAMS

In particular, any function f as above descends to a function

f : H −→ C

with induced transformation properties under a matrix group. One can check that the com-bination of being right-invariant underK, left-invariant under G(Q) and weight k symmetricunder R>0 · SO2(R) translates into the condition that this function f satises

f(γz) = (cz + d)k+2f(z)

for all

γ =(a bc d

)∈ Γ ..= K ∩ SL2(Z).

In particular, it becomes exactly the transformation condition dening weight k+2 modularforms of level Γ. Of course, the analyticity condition we have been ignoring translates inthis formulation to exactly the statement that f is holomorphic on H and bounded at thecusps. This is all explained in [Wei71].

The upshot, then, of all of the above calculations is that modular forms are (algebraic)automorphic forms for GL(2), and are the natural 2-dimensional generalisation of Dirichletcharacters. For the rest of these lectures, we turn to this case.

II. Iwasawa theory for modular forms

Suppose now that we have a modular form f with attached L-function L(f, s) (whichwe recap in 9). As we hopefully made clear in [RJW17], Iwasawa theory is founded onthe study of special values of L-functions. It is natural to ask how much of the theory wedescribed for GL(1) has has an analogue for modular forms.

II.1. Recapping GL(1). Ultimately, in [RJW17] we described three dierent con-structions of the KubotaLeopoldt p-adic L-function ζp.

(1) In 2-4, we gave an analytic construction, a p-adic measure ζanp interpolating special

values L(χ,−k) of Dirichlet L-functions attached to characters of p-power conductor.

(2) In 6, we gave an arithmetic construction. We dened the Coleman map

Col : lim←−ÄO×Qp(ζpn )

ä−→ Λ(Z×p )

from p-adic towers of local units into measures on Z×p , and dened ζarithp as the image under

Col of the family of cyclotomic units.

(3) Finally, in 7 we gave an algebraic construction. We dened a Galois group X +∞ , and

explained that it was a torsion Λ(Z×p ) module, which thus has a well-dened characteristic

ideal ζalgp ⊂ Λ(Z×p ) by the structure theory of Λ-modules.

We showed in 6 that the analytic and arithmetic constructions agree, that is, thatζanp = ζarith

p . The Iwasawa main conjecture is exactly the statement that the algebraicconstruction agrees with the others, that is, that

ζalgp = ζan

p I(Z×p ) ⊂ Λ(Z×p ).

II.2. Analogues for GL(2). Ultimately, versions of all of the above theory are knownfor suciently nice modular forms. Let f be a cuspidal Hecke eigenform of weight k+ 2 andlevel Γ0(N), with p|N , and let L(f, s) be its attached L-function. There are three ways ofassociating a p-adic L-function to f .


II.2.1. Analytic. There exists a range of `critical' values of the complex L-functionL(f, s), namely the values L(f, χ, j + 1) for χ any Dirichlet character and 0 ≤ j ≤ k.These values are those that the BlochKato conjecture suggests should relate to arithmeticinformation arising from f .

The analytic p-adic L-function is an element Lanp (f) in a p-adic analytic space D(Z×p )

which interpolates these critical values. In particular, we have the following:

Theorem A. Let αp denote the Up eigenvalue of f . If vp(αp) < k+ 1, then there existsa unique locally analytic distribution Lan

p (f) on Z×p such that

Lanp (f) has growth of order vp(αp),

and for all Dirichlet characters χ of conductor pn, and for all 0 ≤ j ≤ k, we have

Lanp (f, χ, j + 1) =

∫Z×p

χ(x)xj · Lanp (f)

= −α−np ·Å

1− χ(p)pj

αp

ã· G(χ) · j! · pnj

(2πi)j+1· L(f, χ, j + 1)

Ω±f.

The proof of this theorem is the main aim of these notes. In particular, over the courseof the lectures we will explain all of the concepts and terms appearing in the theorem. Theinterpolating constant, whilst signicantly more involved than the analogous constant forGL(1), is very natural, and largely corresponds to `completing the L-function at innity.'For now, the important thing to take from this theorem is that there exists a single p-adicanalytic object which should `see' the data predicted by the BlochKato conjecture.

II.2.2. Arithmetic. The arithmetic side hinges on Galois representations and Euler sys-tems. In [RJW17, 6.6], we explained that:

The KubotaLeopoldt p-adic L-function is attached to the Galois representationQp(1),the cyclotomic character.

The tower of cyclotomic units forms an Euler system for Qp(1), a norm-compatiblefamily of Galois cohomology classes.

The Coleman map is the Perrin-Riou big logarithm map LogQp(1) for Qp(1), whichtakes values in p-adic measures.

Attached to a modular form f , we have a Galois representation Vf , in which we can picka Galois-stable integral lattice Tf . The arithmetic p-adic L-function is then given by thefollowing deep theorem of Kato, proved in his magisterial paper [Kat04].

Theorem B. There exists an Euler system

zm(f) ∈ H1(Q(µm), Tf )

attached to Tf .

In the yoga described in [RJW17, 6.6], we then consider the system of classes overQ(µpn), localise to get classes over Qp(µpn), and then take the inverse limit to obtain anattached element of the Iwasawa cohomology

zIw(f) ∈ H1Iw(Qp, Vf ).

The arithmetic p-adic L-function is then the image of zIw under the Perrin-Riou big loga-rithm map

LogVf : H1Iw(Qp, Vf ) −→ D(Z×p )

zIw 7−→ Larithp (f).

Here D(Z×p ) is the space of locally analytic distributions from Theorem A in which Lanp (f)

lives.

The second deep theorem of [Kat04] in this setting is the following explicit reciprocitylaw.

8 CHRIS WILLIAMS

Theorem C. There is an equality

Lanp (f) = Larith

p (f)

of distributions on Z×p .

For more on the theory of Euler systems, [Rub00] is a comprehensive account. The readeris also encouraged to use the resources from Loeer and Zerbes' course at the 2018 ArizonaWinter School; there are video recordings of their lectures in addition to their lecture notes[LZ18].

II.2.3. Algebraic. In the case of GL(1), the group X +∞ is a Selmer group for ρ, a Galois

cohomology group cut out by a family of local conditions. This is described in [Gre89, 1].We may make an analogous denition in general, dening a Selmer group Xp∞(Vf ) attachedto the representation Vf at p.

When f is a p-ordinary modular form, that is, when the eigenvalue αp has vp(αp) = 0,then Xp∞(Vf ) is naturally a module over the Iwasawa algebra Λ(Z×p ) of Z×p . Again, in[Kat04], Kato proved that this is a torsion Λ-module, and thus has a characteristic ideal

Lalgp (f) ..= chΛ(Z×p )(Xp∞(Vf )),

the algebraic p-adic L-function of f .

When f is p-ordinary, the analytic/arithmetic p-adic L-function is actually a measure onZ×p , and hence lives in the subspace Λ(Z×p ) ⊂ D(Z×p ). The Iwasawa main conjecture forsuch f is then the following theorem.

Theorem D (Iwasawa main conjecture for f). Under some mild additional technicalhypotheses, we have

Lalgp (f) = (Lan

p (f)) ⊂ Λ(Z×p ).

This is a theorem of Kato [Kat04] and SkinnerUrban [SU14]. Kato proves one divisi-bility, that Lalg

p |(Lanp ), without requiring the additional hypotheses. These hypotheses were

used in SkinnerUrban's proof of the other divisibility; they involve conditions like residualirreducibility of the Galois representation Vf and technical conditions on ramied primes.There has since been much further work weakening the required hypotheses, including ana-logues for non-ordinary modular forms.

II.3. Iwasawa theory for elliptic curves. Perhaps the most important aspects ofthe Iwasawa theory of modular forms come through the applications to elliptic curves. TheTaniyamaShimura conjecture, now a theorem due to the ground-breaking work of Wiles[Wil95], TaylorWiles [TW95] and BreuilConradDiamondTaylor [BCDT01], is thestatement that every rational elliptic curve is modular in the sense that its L-function isequal to the L-function of a weight 2 modular form. In this sense, the Iwasawa theory ofelliptic curves is a proper subset of the Iwasawa theory of modular forms, and indeed almosteverything we know today about L-functions of elliptic curves goes through the modularinterpretation.

As we outlined in the introduction of [RJW17], Iwasawa theory has really providedthe best available results towards the Birch and Swinnerton-Dyer (BSD) conjecture. Inparticular, the Iwasawa main conjecture for elliptic curves can be viewed as a p-adic versionof BSD.

Loosely, an application to classical BSD takes the following, extremely vague, shape.Suppose L(E, 1) 6= 0. Then through the connection between the classical L-function andthe analytic p-adic L-function, this gives a lower bound on the size of the ideal (Lan

p ) in

Λ(Z×p ). For this ideal to be big causes the corresponding Selmer group Xp∞(E) to be small.But this Selmer group can be thought of as a proxy for the Q(µp∞)-rational points on E,


via the Kummmer exact sequence for elliptic curves. In particular, at nite level m over anumber eld F we have a short exact sequence

0→ E(F )/mE(F ) −→ Selm(E/F ) −→X(E/F )[m]→ 0,

where X(E/F ) is the (conjecturally nite) TateShafarevich group. From this, we deducethat the rank of E in the tower Q(µpn) is bounded, a theorem of Mazur. We also geteven ner control at all stages, which allows us to deduce that E(Q) itself is small, and inparticular that the rank is 0, giving `weak BSD in analytic rank 0'.

By being more precise, one may also deduce the p-part of the leading term formula instrong BSD. There are also results in analytic rank 1, and partial results in analytic rank2, that arise directly from knowledge of the Iwasawa main conjecture (see, for example,[JSW17]).

More details on all of this, and the more general Iwasawa theory of modular forms, arecontained in Skinner's 2018 Arizona Winter School lectures [Ski18].

II.4. Further generalisations. The three constructions above, and the equalities be-tween them, are expected to go through in very wide generality, but there are very fewcases in which the whole picture has been completed. We sketch this here. Suppose ρ is aGalois representation, arising from a motive M , and corresponding under Langlands to anautomorphic representation.

(1) (Analytic). There should be an element Lanp (ρ) in a p-adic analytic space D which

interpolates special values of L(ρ, s). The criterion to be a `special value' was predicted byDeligne [Del79], and the exact form of this analytic p-adic L-function is subject to a preciseconjecture of CoatesPerrin-Riou [Coa89, CPR89].

In practice, this is already dicult, and there are many fundamental cases where sucha construction is not known. For example, we've seen the cases GL(1) and GL(2); but atpresent, there is no construction that works for GL(3). Much more is known for generalisa-tions in other directions (for example, working over number elds, or working with dierentalgebraic groups such as unitary or symplectic groups).

(2) (Arithmetic). We also expect Euler systems to exist in great generality, but knownexamples are scarcer still. Until very recently, Kato's Euler system and the cyclotomic unitswere two of only three examples of Euler systems, the other being the system of elliptic units(though the system of Heegner points is closely related). There has been a recent increaseof activity in the area, stemming from Lei, Loeer and Zerbes' construction of the Eulersystem of BeilinsonFlach elements [LLZ14], for ρ the RankinSelberg convolution of twomodular forms.

Where an Euler system exists, one can apply a Perrin-Riou logarithm map and extractan arithmetic p-adic L-function; but proving an explicit reciprocity law is harder still, andat present the only known examples of the equality Lan

p = Larithp are those presented above

and the RankinSelberg case [KLZ17].Examples of Euler systems where the explicit reciprocity laws are not yet known are given

in [LLZ15, LLZ18, LSZ17]. For the p-adic L-function, one may also try and constructthe Iwasawa cohomology class zIw directly, without constructing a full Euler system; this isthe approach taken in [CRJ18].

(3) (Algebraic). One also expects Iwasawa main conjectures to hold in wide generality,at least in ordinary settings, and there are many partial results towards this too. Wheneverone has an Euler system with the equality Lan

p = Larithp , for example, one has that the

corresponding Selmer group is torsion and the divisibility Lalgp |(Lan

p ).

II.5. What do we cover in these lectures? Since the case of GL(2) is signicantlymore involved than that of GL(1), in these lectures a treatment of the Iwasawa main con-jecture for modular forms will be out of reach. Instead, we focus on proving the existence ofthe analytic p-adic L-function Lp(f)an for modular eigenforms f . In particular, we will give

10 CHRIS WILLIAMS

a proof of Theorem A above, as well as developing the `right' p-adic analytic framework that of p-adic distributions in which this result lives. We hope this will provide motivationto explore further aspects of the Iwasawa theory of modular forms, or the analytic theoryin more general settings.

II.6. Prerequisites. In [RJW17], we attempted to make the treatment as self-contained as possible, and included relevant background material in a series of appendices.The resulting notes should have been accessible to a student with a thorough grounding inundergraduate algebraic number theory. In the second half of these notes, we necessarilyassume that the reader is comfortable with the theory of modular forms, including thetheory of Hecke operators; a standard rst course in the subject should be sucient.

Even then, the treatment will not be entirely self-contained, and at times we borrowdeep theorems from automorphic representation theory. Most notably, among results thatmight not typically appear in a rst course on modular forms, we do not prove algebraicityof Hecke eigenvalues, and we do not include a proof of the EichlerShimura isomorphismbetween modular forms and cohomology/modular symbols.

II.7. Organisation of lectures. These notes are organised as follows.

In 9 we recall basic facts about modular forms, dene their L-functions, and deriveintegral formulae for them in terms of their period integrals.

In 10 we recast all of this more algebraically through the theory of (classical) modularsymbols, discarding the analytic data inherent in the denition of modular forms in favourof a purely algebraic denition.

In 11, we describe the EichlerShimura isomorphism between spaces of modular formsand modular symbols, and prove that certain `special/critical values' of the L-function of amodular form are algebraic multiples of complex periods.

In 12, we describe the theory of overconvergent modular symbols, innite-dimensionalspaces of p-adic modular symbols built using spaces of distributions from p-adic analysis.We state Stevens' control theorem, which controls the space of classical modular symbols asa quotient of the space of overconvergent modular symbols.

In 13, we give Greenberg's proof of Stevens' control theorem. The proof is computa-tionally eective and uses ltrations on distributions.

In 14, we use all of the above to construct the p-adic L-function attached to a smallslope modular form f as a locally analytic distribution on Z×p , and prove that it interpolatesalgebraic parts of the critical L-values of f .

In 15, we dene growth properties of locally analytic distributions, and prove that thep-adic L-function has growth of order equal to its slope. As a corollary, we deduce that thep-adic L-function is uniquely determined by its interpolation property.

II.8. Acknowledgements. I am grateful to Timo Keller, Marc Masdeu and Rob Rock-wood for pointing out a number of corrections on earlier drafts of these notes. Furthercomments and corrections are welcome: please send them to christopher.d.williams@

warwick.ac.uk.


Part III: p-adic L-functions for Modular Forms

9. The L-function of a modular form

9.1. Modular forms 101. We start Part III by recalling some relevant facts aboutmodular forms. We treat this only briey, as it is a vast topic in its own right; an excellentreference for much of this material is [DS05], particularly chapters 1 and 5.

Denition 9.1. Dene the upper half-plane to be

H ..= x+ iy ∈ C : y > 0= R×R>0.

The boundary of H is the set of cusps P1(Q), where a cusp r ∈ Q is viewed inside C asr + 0i on the real line, and where ∞ is viewed to be at the `end' of the positive imaginaryaxis in C.

Thoughout, we will work with the congruence subgroup Γ0(N) ⊂ SL2(Z), and for conve-nience will henceforth always denote this by Γ. The reader should consider N to be xedthrough the rest of these notes. When considering the p-adic behaviour, we will impose theadditional condition that N is divisible by p. We have the standard action of SL2(Z) onH ∪P1(Q) by Å

a bc d

ã· z =

az + b

cz + d,

noting that this preserves both H and P1(Q).

Denition 9.2. A modular form of weight k and level Γ is a holomorphic function

f : H −→ C

which:

(1) satises the invariance property

f(γz) = (cz + d)kf(z)

for all γ ∈ Γ,(2) and is holomorphic (has bounded growth) as z tends to any cusp.

We write Mk(Γ,C) for the (nite-dimensional) complex vector space of modular forms ofweight k and level Γ.

We say f is a cusp form if it vanishes at every cusp, and write Sk(Γ,C) ⊂Mk(Γ,C) forthe sub-vector-space of cusp forms.

Proposition 9.3. There is a direct sum decomposition

Mk(Γ,C) ∼= Sk(Γ,C)⊕ E(Γ,C),

where Ek(Γ,C) is the space of Eisenstein series (see [DS05, 4]).

Since ( 1 10 1 ) ∈ Γ, any modular form satises

f(z + 1) = f(z),

and thus admits a Fourier expansion

f(z) =∑n≥0

an(f)qn

in the variable q = e2πiz. We call this the q-expansion of f . If f is a cusp form, thena0(f) = 0. More precisely, this is the Fourier expansion at the cusp ∞ ∈ P1(Q); there areanalogous Fourier expansions around any other cusp r ∈ P1(Q), and f is a cusp form if andonly if the constant term of the Fourier expansion at r vanishes for each cusp r ∈ P1(Q).

12 CHRIS WILLIAMS

The notion of q-expansions allow us to dene natural algebraic structures on the space ofmodular forms.

Denition 9.4. For F ⊂ C, let

Mk(Γ, F ) =f ∈Mk(Γ,C) : f(z) ∈ F [[q]]

,

a nite-dimensional F -vector space. As before, we have a decomposition

Mk(Γ, F ) = Sk(Γ, F )⊕ Ek(Γ, F )

into cuspidal and Eisenstein series.

Remark 9.5. As F [[q]]⊗F C ⊂ C[[q]], we clearly have

Mk(Γ, F )⊗F C →Mk(Γ,C).

From the denition, it is far from obvious that this space is non-zero for any F . Oneremarkable consequence of the theory of Hecke operators, as recalled in the next section,is that there is always a number eld F (which depends on Γ and k) such that this is anisomorphism, that is, all modular forms of weight k and level Γ are dened over F .

9.2. Hecke operators and algebraic structure. The denition of modular formsis inherently analytic, and appears to be implicitly transcendental in nature. One of themajor reasons modular forms are so important is that they admit an incredibly rich algebraicstructure, arising from the action of Hecke operators, a collection of commuting operatorsbuilt from arithmetic. The space of modular forms admits a basis of eigenforms for theseoperators, each with algebraic eigenvalues. We say thatMk(Γ,C) has a good spectral theoryunder the Hecke operators.

For a full denition of Hecke operators, see [DS05, 5]. For our purposes, we recall onlythe very basics.

Denition 9.6. For γ ∈ GL+2 (Q), dene the weight k operator

f |kγ ..= det(γ)k−1(cz + d)−kf(γz).

Dene Hecke operators as follows:

Let q be a prime not dividing N . There is an operator

Tq : Mk(Γ,C)→Mk(Γ,C)

dened by

Tqf(z) =

q−1∑a=0

f

∣∣∣∣k Å1 a0 p

ã(z) + f

∣∣∣∣k Åp 00 1

ã(z).

For q|N , we have an analogous operator

Uqf(z) =

q−1∑a=0

f

∣∣∣∣k Å1 a0 p

ã(z).

These denitions extend in a natural way to composite n; there is a recurrence formulafor Tqn [DS05, 5.3], and for m,n coprime to N and each other, we have Tmn = TmTn. Forq|N , we have Uqn = Unq . All of these operators commute [DS05, Prop. 5.2.4].

Denition 9.7. Let HN denote the Hecke algebra, the free Z-algebra generated by theTq and Uq operators.

Denition 9.8. An eigenform for HN is f ∈Mk(Γ,C) such that

Tf = λ(T )f for all T ∈ HN ,

where λ : HN → C is a homomorphism. We call λ an eigenpacket.

The following theorems encode the algebraicity of modular forms. Recall that an eigen-form is a newform at level N if it `does not come from any lower level M |N '; see [DS05,5.6] for further details.


Theorem 9.9. (1) Let f be an eigenform for the Hecke operators, corresponding toan eigenpacket λ. There exists a number eld Kf such that

λ : HN −→ Kf ⊂ C

factors, that is, λ is valued in Kf .(2) Sk(Γ,C) has a basis of eigenforms.(3) Let f be a newform, and let

Sk(Γ,C)[f ] ..=g ∈ Sk(Γ,C) : Tg = λ(f)g for all T ∈ HN

,

the f -eigenspace for HN in Sk(Γ,C). This eigenspace is 1-dimensional over C.

Proof. Part (1) is proved (in the case of weight 2) in [DS05, 6.5]. The idea is touse the Jacobian JΓ of the modular curve XΓ = Γ\[H ∪ P1(Q)]; in particular, there is anidentication between JΓ and the quotient of the dual space S2(Γ)∨ by the integral homologyH1(XΓ,Z). The Jacobian and hence this quotient admits a natural Hecke action, which iscompatible with the action on cusp forms under the above identication; but then the Heckeoperators stabilise H1(XΓ,Z), which forces their characteristic polynomials to be integral,and the eigenvalues to be algebraic integers. The higher weight case is more involved.

For (2), see [DS05, Thm. 5.8.3]. For (3), see [DS05, Thm. 5.8.2].

Theorem 9.10. Let (f, λ) be a cuspidal eigenform that is normalised in the sense thata1(f) = 1. Then

λ(Tq) = aq(f),

λ(Up) = ap(f).

In particular, f ∈Mk(Γ,Kf ).

Proof. The action of Hecke operators on q-expansions can be computed (see [DS05,(5.3)]), and reduces to

a1(Tqf) = aq(f).

But f is an eigenform, so

a1(Tqf) = a1(λ(Tq) · f)

= λ(Tq)a1(f) = λ(Tq).

Corollary 9.11. There exists a number eld F such that Mk(Γ, F )⊗F C ∼= Mk(Γ,C).

Proof. Take a (nite) basis of eigenforms fii∈I , and let F be the compositum of thenumber elds Kfi for i ∈ I.

9.3. p-stabilisation. In the theory of p-adic variation later on, we will always take pto divide the level N , and the Up operator will be of central importance. Indeed, if p doesnot divide the level, then there can be no theory of p-adic variation: we will return to thistheme in Remark 12.11. Fortunately it is always possible to force p into the level by passingto an appropriate oldform. All of this is explained in [DS05, 5.6].

Suppose now we have a newform g ∈ Sk(Γ0(N)), with N prime to p, with associatedeigenpacket λg. Note that we have two modular forms

g(z), g(pz) ∈ Sk(Γ0(Np));

these functions are both holomorphic on C and at the cusps, and it is simple to see that theytransform as expected under Γ0(Np). The characteristic polynomial of Up on the span ofg(z) and g(pz) is X2 − ap(g)X + pk−1. When the roots α, β of this polynomial are distinct,then we may diagonalise Up on this space, and obtain two Up-eigenforms

fα(z) ..= g(z)− βg(pz), fβ ..= g(z)− αg(pz);

14 CHRIS WILLIAMS

then Upfα = αfα and Upfβ = αfβ . Since Up commutes with the other Hecke operators, andg(z), g(pz) are easily seen to be Hecke eigenforms for all the Tq, Uq with q 6= p, we obtaintwo eigenforms fα, fβ , and corresponding eigenpackets λα, λβ respectively, where

λα(Tq) = λg(Tq), q - Np;λα(Uq) = λg(Uq), q|N ;

λα(Up) = α

(and similarly for λβ).

Recall Theorem 9.9(iii), which said that Sk(Γ0(N))[f ] is one-dimensional (for the newformg). When α 6= β, then using strong multiplicity one we can deduce the analogous statementfor the p-stabilisations.

Proposition 9.12. Let g be a newform of level N , and let α, β be the roots of X2 −ap(g)X + pk−1. If α 6= β, then

dimCSk(Γ0(Np))[fα] = dimCSk(Γ0(Np))[fβ ] = 1. (9.1)

Proof. The subspace of Sk(Γ0(Np)) on which the Hecke operators Tq for q - Np act asλg(Tq) is 2-dimensional, spanned by g(z) and g(pz), by the results of [DS05, 5]. But thenUp is diagonal with distinct eigenvalues on this 2-dimensional space, so decomposes into thetwo one-dimensional eigenspaces claimed.

If k ≥ 2, then conjecturally α 6= β always; this is a form of Maeda's conjecture.

Remark 9.13. If α = β the so-called `p-irregular' case then it is still always possibleto take a p-stabilisation fα of f to level Np which is an eigenform. Morevoer, Proposition9.12 can be seen to be true as well. This is far more subtle: the generalised eigenspace isnow 2-dimensional, but a simple calculation (see [RS17, 9.2]) shows that the Up-action onthis is not semisimple, and hence the actual eigenspace is only 1-dimensional.

9.4. The L-function of a modular form. The central object in Part III of thesenotes is the L-function of a modular form, which we now dene.

Denition 9.14. Let f =∑n≥1 an(f)qn ∈ Sk(Γ,C). For Re(s) > k/2 + 1, dene the

L-function of f to be

L(f, s) =∑n≥1

an(f)n−s.

If χ is a Dirichlet character, then dene the twist of L(f, s) by χ to be

L(f, χ, s) =∑n≥1

an(f)χ(n)n−s.

To see that this makes sense in the region of denition, we use:

Lemma 9.15. There exists a constant C such that |an| ≤ Cnk/2 for all n. Thus L(f, s)is absolutely convergent for Re(s) > k/2 + 1.

Proof. See [DS05, Prop. 5.9.1]. This is essentially a consequence of f being holomorphicat the cusp innity.

9.4.1. An integral formula. Recall we described the Γ-function in [RJW17, Intro.] as

Γ(s) =

∫ ∞0

e−tts−1dt.

Previously, we used the transformation t↔ nt to extract an integral description of n−s. Inthis setting, given the variable q = e2πiz, it is more convenient to normalise dierently. Weobtain:


Lemma 9.16. We have

n−s =(2π)s

Γ(s)

∫ ∞0

e−2πntts−1dt.

Proof. Make the substitution t↔ 2πnt, and rearrange.

Proposition 9.17. Let f ∈ Sk(Γ) be a cusp form, and dene

g(s) =

∫ i∞

0

f(z)zs−1dz.

Then for Re(s) 0, we have

g(s) =Γ(s)

(−2πi)sL(f, s).

Proof. We compute directly that

L(f, s) =∑n≥1

an(f)n−s

=(2π)s

Γ(s)

∑n≥1

an(f)

∫ ∞0

e−2πntts−1dt

using the expression for n−s in Lemma 9.16. For Re(s) 0, this converges absolutely, andwe may swap the sum and product. We also write −t = i×(it), so as to recover the exponent2πi, and obtain

=(2π)s

Γ(s)

∫ ∞0

∑n≥1

an(f)e2πni(it)ts−1dt

=(2π)s

Γ(s)

∫ ∞0

f(it)ts−1dt

=(−2πi)s

Γ(s)

∫ ∞0

f(it)(it)s−1d(it),

where in the last step we multiply by (−i)sis = 1 and make a basic change of variables. Weconclude by setting z = it.

Corollary 9.18. The L-function L(f, s) has an analytic continuation to the whole com-plex plane C. It admits a functional equation relating the values at s and k − s.

Proof. To prove the analytic continuation, one shows that the integral is everywhere con-vergent, and thus gives the extension of the Dirichlet series to all ofC. See the accompanyingexercise sheets for further details.

By specialising at the value s = j + 1, we immediately obtain the following special valueformula, recalling that the integer values Γ(j + 1) = j! of Γ are the factorials.

Corollary 9.19. Let j ∈ Z≥0. We have∫ i∞

0

f(z)zjdz =j!

(−2πi)j+1L(f, j + 1).

We also have twisted versions of this. Let χ : (Z/DZ)× → Q×be a Dirichlet character

of conductor D, and recall the Gauss sums from [RJW17, 4.1].

Lemma 9.20. We have

χ(n)n−s =(−2πi)s

Γ(s)G(χ)

∑a∈(Z/DZ)×

χ(a)

∫ ∞a/D

e2πinz(z − a

D

)s−1

dz.

16 CHRIS WILLIAMS

Proof. We have the identity

1

G(χ)

∑a∈(Z/DZ)×

χ(a)e2πina/D =

ßχ−1(n) = χ(n) : (n,D) = 1

0 : otherwise

of Gauss sums, which we used in [RJW17, Rem. 4.3] when studying Dirichlet p-adic L-functions. We may consider this uniformly by extending χ to Z in the natural way, thatis, letting χ(n) = 0 if n is not coprime to the conductor D. Multiplying the expression ofLemma 9.16 by this identity, we obtain

χ(n)n−s =(2π)s

Γ(s)G(χ)

∑a∈(Z/DZ)×

χ(a)e2πina/D

∫ ∞0

e−2πntts−1dt

=(2π)s

Γ(s)G(χ)

∑a∈(Z/DZ)×

χ(a)

∫ ∞0

e2πin(it+ aD )ts−1dt

=(−2πi)s

Γ(s)G(χ)

∑a∈(Z/DZ)×

χ(a)

∫ ∞0

e2πin(it+ aD )(it)s−1d(it),

multiplying by 1 = (−i)s · is−1 · i. This is

=(−2πi)s

Γ(s)G(χ)

∑a∈(Z/DZ)×

χ(a)

∫ ∞a/D

e2πinz(z − a

D

)s−1

dz

after making the substitution z = it+ a/D.

Corollary 9.21. We have

L(f, χ, s) =(−2πi)s

Γ(s)G(χ)

∑a∈(Z/DZ)×

χ(a)

∫ ∞a/D

f(z)(z − a

D

)s−1

dz.

In particular, the twisted L-function also admits analytic continuation to all of C, andsatises a functional equation.

Proof. We argue almost exactly as in the proof of Proposition 9.17, only now with twistedcoecients.

The following will not be used during these lectures, but the statement is included forcompleteness.

Theorem 9.22. f is a normalised eigenform if and only if

L(f, s) =∏

p prime

(1− app−s + pk−1−2s)−1.

Proof. [DS05, Thm. 5.9.2].

10. Modular symbols

As we've seen, modular forms are dened in an inherently analytic way, but via Heckeoperators, they turn out to have deep underlying algebraic structure. Further than that, viathe q-expansion, we can recover a cuspidal eigenform f purely from its Hecke eigenvalues,as ap(f) = λ(Tp). Moreover, the L-function is then also dened in terms of the Heckeeigenvalues. It is natural to ask: was it possible to dene this algebraic structure directly,without resorting the analytic denitions at all?

The theory of modular symbols, which we now describe, does exactly this, by algebraicallyencoding values of integrals of f (sometimes called periods of f). In particular, the space ofmodular symbols can be viewed as `throwing away' all the analytic properties of the modularform, leaving only the algebraic skeleton underneath.


10.1. Modular symbols in weight 2. Let f ∈ S2(Γ) be a modular form. We saw inthe previous section that the integrals∫ s

r

f(z)dz, r, s ∈ P1(Q)

are interesting, and linear combinations of them give twisted values L(f, χ, 1) of the L-function attached to f . Let us now consider them all at once.

Denition 10.1. Let ∆ = Div(P1(Q)) = Z[P1(Q)] be the free abelian group generatedby symbols [r] for r ∈ P1(Q) a cusp. There is a natural degree map

deg : ∆ −→ Z,∑r

cr[r] 7−→∑r

cr,

and we dene

∆0 = ker(deg)

to be its kernel. Then ∆0 is generated by terms of the form [r]− [s] for r, s ∈ P1(Q).

We view [r] − [s] as (any) simple path r → s in H, and then ∆0 as the (additive) spaceof such paths. We will encode this principle in notation by writing [r]− [s] as r → s.

Denition 10.2. Dene

Symb(C) = Hom(∆0,C).

By analogy with our motivation, a homomorphism φ ∈ Symb(C) should be viewed as a`system of integrals over paths r → s'.

Example 10.3. Let f ∈ S2(Γ,C). Then

φf : r → s 7→∫ s

r

f(z)dz

gives an element of Symb(C). Note that this integral is well-dened by the properties of f :in particular, it is independent of the choice of simple path r → s since f holomorphic, andconverges as f is a cusp form, hence has exponential decay at every cusp.

This construction gives a map S2(Γ,C) → Symb(C). But Symb(C) is far too big tomaintain a nice spectral theory of Hecke operators: in particular, it is innite dimensional.We need to build in the remaining property satised by modular forms, that is, a notion ofΓ-invariance.

Lemma 10.4. Let f ∈ S2(Γ,C). Then

φfγr → γs = φfr → s.

for all γ ∈ Γ.

Proof. We compute directly that

d(γz)

dz=

d

dz

Åaz + b

cz + d

ã=a(cz + d)− c(az + b)

(cz + d)2

=1

(cz + d)2,

as ad− bc = 1. Thus under the change of variables z ↔ γz, we have

d(γz) = (cz + d)−2dz.

18 CHRIS WILLIAMS

Then since f has weight 2, we get f(γz)d(γz) = f(z)dz, and

φfγr → γs =

∫ γs

γr

f(z)dz =

∫ s

r

f(γz)d(γz)

=

∫ s

r

f(z)dz = φfr → s.

Denition 10.5. The space of C-valued modular symbols for Γ is the space

SymbΓ(C) ..= HomΓ(∆0,C)

of Γ-invariant homomorphisms (that is, homomorphisms satisfying φγr → γs = φr →s).

In particular, Lemma 10.4 shows that

φf ∈ SymbΓ(C).

This is now a purely algebraic space, containing elements corresponding to f . Moreover,the symbol φf (more or less by denition) computes special values of the L-function of f ;by Corollary 9.19 we have

φf0→∞ = − 1

2πiL(f, 1).

Remark 10.6. The −1 here appears because we are considering∫∞

0rather than

∫ 0

∞,as in [Pol11].

Moreover, φf also sees all the twisted L-values at 1.

Denition 10.7. Let χ : (Z/DZ)× → Q×be a (primitive) Dirichlet character. Dene

Evχ : SymbΓ(C) −→ C

φ 7−→∑

a∈(Z/DZ)×

χ(a)φf

∣∣∣∣Å1 a0 D

ã0→∞

=∑

a∈(Z/DZ)×

χ(a)φf

aD→∞

.

We abuse notation here, and henceforth, and write a ∈ (Z/DZ)× to mean a set of integerrepresentatives of the unit classes modulo D.

(We will dene a more general version of this map, with general coecients, in Denition10.16 below).

Proposition 10.8. Let f ∈ S2(Γ,C), and χ : (Z/DZ)× → Q×be a Dirichlet character.

Then

Evχ(φf ) = −G(χ)

2πiL(f, χ, 1).

Proof. This follows directly by specialising the integral formula for L(f, χ, s) given inCorollary 9.21; since s = 1, the polynomial term z − a/D does not appear.

10.2. Abstract modular symbols. All of the above was valid only for weight 2. Wewant an analogous theory for higher weight modular forms.

Remark 10.9. Henceforth, it will be convenient to always consider forms to have weightk + 2, instead of weight k. (One always has to contend with this shift by 2; some authorschoose to continue to work with weight k forms, at the cost of introducing coecients ofindex k− 2). We remark that this shift by 2 arises because weight 2 is the lowest weight forwhich modular forms are cohomological, that is, weight 2 modular forms have trivial or`weight 0' cohomological coecients.


Let f ∈ Sk+2(Γ,C). Then the computations above show that

f(γz)d(γz) = (cz + d)kf(z)dz. (10.1)

Thus the same denition of modular symbol does not work, as we no longer have invarianceunder the action of Γ. To x this, we need to consider modular symbols with coecients, sothat we can kill the additional action of (cz + d)k through the coecients.

Denition 10.10. Let V be a right Γ-module. Then dene an action of Γ onHom(∆0, V ) by

φ|γr → s = φγr → γs|γ.

The space of V -valued modular symbols for Γ is

SymbΓ(V ) ..= HomΓ(∆0, V ).

The following turns out to be a good choice of V .

Denition 10.11. For a commutative ring R, let Vk(R) be the space of polynomials inX over R of degree ≤ k.

Equivalently, we could see this as homogeneous polynomials in two variables X,Y ofdegree k, which can be identied with the symmetric kth power Symk(R2). In this sense,it is apparent that this carries a plethora of natural actions of GL2(R) through its actionson R2. It is important for us to choose a very specic action, which will not be the mostnatural, but will translate into precisely cancelling the factor (cz + d)k from (10.1).

We will consider the function

φf : ∆0 −→ Vk(C),

r → s 7−→∫ s

r

f(z)(zX + 1)kdz.

To equip Vk(C) with an action that renders this Γ-invariant, we consider the polynomial(zX + 1)k, and compute that

(γ(z)X + 1)k =

Åaz + b

cx+ dX + 1

ãk=

ï(az + b)X + (cz + d)

òk(cz + d)−k.

In particular, up to some action on X that we are yet to dene, this cancels o the errantfactor (cz+d)k. To determine the action on X, note that φf is an element of SymbΓ(Vk(C))if and only if φfγr → γs = φfr → s|γ−1 for all γ ∈ Γ. Our computations show that

φfγr → γs =

∫ γs

γr

f(z)(zX + 1)kdz

=

∫ s

r

f(γz)(γ(z)X + 1)kd(γz)

=

∫ s

r

f(z)

ï(az + b)X + (cz + d)

òkdz

=

∫ s

r

f(z)

ï(aX + c)z + (bX + d)

òkdz

So γ−1 sends X to aX + c, and 1 to bX + d. Then γ should send X to dX − c, and 1 to−bX + a.

Denition 10.12. Equip Vk with an action of GL2(R) by

P

∣∣∣∣Åa bc d

ã(X) = (−bX + a)kP

ÅdX − c−bX + a

ã.

20 CHRIS WILLIAMS

From the way we've dened this action, the following is immediate.

Proposition 10.13. We have φf ∈ SymbΓ(Vk(C)).

Moreover, we retain the connection to L-values: and indeed, we see a wider range L-values.

Proposition 10.14. We have

φf0→∞ =

k∑j=0

Åkj

ãïj!

(−2πi)j+1· L(f, j + 1)

òXj ∈ Vk(C).

Proof. Expanding out (zX + 1)k, we obtain

φf0→∞ =

k∑j=0

Åkj

ãï∫ ∞0

f(z)zjdz

òXj .

The result follows from Corollary 9.19.

Remark 10.15. Unlike in Part I, when we considered values of the Riemann zetafunction at negative integers, we see only a nite range of `critical' values here namely, thevalues s = j + 1 for 0 ≤ j ≤ k. In particular, for weight 2 forms, the modular symbolsees only one value, namely s = 1. This is not surprising, and in fact tallies exactly withexpectations surrounding these L-values going back to Deligne [Del79]. He predicted thatany L-function that is motivic (which, conjecturally, is all L-functions) should admit a setJ of critical integers, dened in terms of `Euler factors at innity', and a controlled set ofcomplex periods such that the values of the L-function at j ∈ J are algebraic multiples ofone of the periods.

In the case of the Riemann zeta function, the critical values are all the negative oddintegers, and the period can be taken to be 1 ∈ C, since the values are already algebraic (infact, rational).

For a modular form of weight k, Deligne's conjecture translates into the statement thatJ = j + 1 : 0 ≤ j ≤ k, and thus the modular symbol is seeing all of these critical values.In the next section, we use the modular symbol to show that these L-values are algebraicmultiples of periods.

There are, naturally, analogous results computing the twisted L-values in the criticalstrip through values of the modular symbol. The following map Evχ is a direct generali-sation of that given for trivial coecients in Denition 10.7.

Denition 10.16. Let χ be a Dirichlet character of conductor D. Dene a twistingmap

SymbΓ(Vk(C)) −→ SymbΓ(Vk(C))

by

φ 7−→ φχ ..=∑

a∈(Z/DZ)×

χ(a)φ

∣∣∣∣Å1 a0 D

ã.

Dene the evaluation at χ map by

Evχ : SymbΓ(Vk(C)) −→ Vk(C)

φ 7−→ φχ0→∞

=∑

a∈(Z/DZ)×

χ(a)φ

∣∣∣∣Å1 a0 D

ã0→∞.

Remark 10.17. This twisting map is very closely related to the one we dened onmeasures in [RJW17, 4]. Indeed, we relate the two twisting maps in 14 below.

We have the following.


Proposition 10.18. Let f ∈ S2(Γ,C), and χ : (Z/DZ)× → Q×be a Dirichlet charac-

ter. Then

Evχ(φf ) =

k∑j=0

Åkj

ãïG(χ) · j! ·Dj

(−2πi)j+1· L(f, χ, j + 1)

òXj .

Proof. We compute:

φf

∣∣∣∣Å1 a0 D

ã0→∞ = φf

aD→∞

∣∣∣∣Å1 a0 D

ã=

ñ∫ ∞a/D

f(z)(zX + 1)kdz

ô ∣∣∣∣Å1 a0 D

ã=

∫ ∞a/D

f(z)

Åz

DX

−aX + 1+ 1

ãk(−aX + 1)kdz

=

∫ ∞a/D

f(z) (zDX + (−aX + 1))kdz

=

∫ ∞a/D

f(z)

ï(zD − a)X + 1

òkdz

=

k∑j=0

Åkj

ãñ∫ ∞a/D

f(z)(Dz − a)j

)dz

ôXj

=

k∑j=0

Åkj

ãDj

ñ∫ ∞a/D

f(z)(z − a

D

)jdz

ôXj .

Multiplying by χ(a) and summing over all a ∈ (Z/DZ)×, then, the coecient of Xj inEvχ(φf ) is thereforeÅ

kj

ãDj

∑a∈(Z/DZ)×

χ(a)

ñ∫ ∞a/D

f(z)(z − a

D

)jdz

ô=

Åkj

ãDj · G(χ)Γ(j + 1)

(−2πi)j+1· L(f, χ, j + 1),

where the last equality follows from specialising Corollary 9.21 at s = j + 1.

Denition 10.19. We dene

Evχ,j : SymbΓ(Vk(C)) −→ C

to be the composition of Evχ, as dened in Denition 10.16, with projection to the jthco-ordinate.

In particular, we've shown:

Corollary 10.20. We have

Evχ,j(φf ) = Cχ,j · L(f, χ, j + 1),

where

Cχ,j ..=

Åkj

ãDj · G(χ) · j!

(−2πi)j+1.

For later use, we summarise all of this in the following diagram:

22 CHRIS WILLIAMS

φf1 ''∈ SymbΓ(Vk(C))

Evχ,j // C 3 Cχ,j · L(f, χ, j + 1)

f

ee ;;

Figure 10.1. L-values from modular symbols: complex coecients

11. Algebraicity of L-values

By analogy with the KubotaLeopoldt p-adic L-function, to study the Iwasawa theoryof a modular form we wish to construct a p-adic version of L(f, s). The most naturalgeneralisation is to look for a `p-adic measure' Lp(f) on Z×p satisfying an interpolationproperty of the shape ∫

Z×p

χ(x)xj · Lp(f) = (∗)L(f, χ, j + 1),

where (∗) is some explicit factor including Euler factors at p.

The values of this integral, however, will be p-adic numbers; and the L-function L(f, s)is valued in C. Whilst it is possible to x an isomorphism C ∼= Qp, to do so renders suchan interpolation formula essentially meaningless from an arithmetic standing. Instead, wewant to view the right-hand side algebraically. That this is possible is the main result ofthis section. In particular, we will explain the following theorem.

Theorem 11.1 (Manin, Shimura). Let f ∈ Sk+2(Γ,C) be an eigenform. There existcomplex periods Ω+

f ,Ω−f ∈ C× such that for any Dirichlet character χ, and any 0 ≤ j ≤ k,

we haveL(f, χ, j + 1)

(2πi)j+1 · Ω±f∈ Q.

Here the sign is determined by χ(−1)(−1)j = ±1.

In other words, at certain special values, the L-function can only be transcendental in avery controlled manner. It is then meaningful to p-adically interpolate the `algebraic parts'of these L-values.

We will prove this rst for newforms, where we have tight control on the dimensions ofHecke eigenspaces. At the end of the section we will describe the generalisation to arbitraryeigenforms via p-stabilisation.

Remark 11.2. If we take the periods Ω±f to be dened by this property, then they

are only well-dened as elements of C×/Q×. The question of normalising these periods

correctly is an important one, particularly for Iwasawa theory, where one is particularlyinterested in the p-adic valuation of the algebraic part of the L-value. Without furtherinput, renormalising the period by an arbitrary power of p will change this p-adic valuationto any specied value, and render its study meaningless. Since the aim of these lectures isto construct Lp(f), rather than describe its Iwasawa-theoretic properties, we will not lookat this in detail.

When f is weight 2, and attached to an elliptic curve, then one may construct canonicalperiods Ω±f from the usual period of the elliptic curve.

11.1. Motivation: a proof strategy. We outline a strategy for the proof.

Assume rst that f is a newform. Then by Theorem 9.9(iii), the Hecke eigenspaceSk+2(Γ,C)[f ] is one-dimensional; and normalising to have algebraic coecients, the same is


true of Sk+2(Γ, F )[f ], where F ⊂ C is any number eld containing the Hecke eld Kf overwhich the eigenpacket λf of f is dened.

Sk+2(Γ, F )[f ]

⊂

? // : SymbΓ(Vk(F ))

⊂

Evχ,j // Q

1-dimensional

11

.. Sk+2(Γ,C)[f ] // SymbΓ(Vk(C))Evχ,j // C

Now, suppose we could prove the same result for modular symbols; that is, that we couldprove that for a natural Hecke action, we have

dimCSymbΓ(Vk(C))[f ]?= 1

?= dimFSymbΓ(Vk(F ))[f ]. (11.1)

One would hope that the C-space is spanned by the canonical modular symbol φf we sawabove. If we then choose a generator φf,F of the analogous F -space, then necessarily this

is also in SymbΓ(Vk(C)), and hence a complex multiple Ω−1f · φf of φf . In particular, we

would then have

Evχ,j

ÅφfΩf

ã∼ L(f, χ, j + 1)

(2πi)j+1Ωf∈ Q.

In practice, however, (11.1) does not hold, so this strategy does not work on the nose.Fortunately, it can be salvaged with a little modication: the spaces in (11.1) are eachtwo-dimensional, decomposing into two one-dimensional ±-eigenspaces under a natural in-volution ι. This is why we end up with two periods Ω+

f ,Ω−f , one for each ι-eigenspace, and

a sign condition on the algebraicity result.

To enact this, we need the following ingredients:

a Hecke action on modular symbols, for which f 7→ φf is Hecke-equivariant; control on the size of SymbΓ(Vk(C)) in terms of spaces of modular forms; both of the above steps to be dened with coecients in a suciently large number

eld F .

These steps are the content of the EichlerShimura isomorphism. In the remainder of thissection, we make all of this precise.

11.2. The EichlerShimura isomorphism. We rst describe the Hecke action onmodular symbols. Each Hecke operator on modular forms has a double coset decomposition,which translates into the sum of the actions of a nite set of matrices. As each of thesematrices is an element of the monoid Σ0(p), we can dene their action on modular symbolsas well; for example, we may dene

φ|Up =

p−1∑a=0

φ

∣∣∣∣Å1 a0 p

ã,

and similarly for all the T`'s.

Proposition 11.3. The map f 7→ φf is Hecke equivariant.

Proof. This is another computation in change of variables. Recall the weight k+2 actionof GL+

2 (Q) on modular forms is by f |kγ = detk+1(γ)(cz + d)−k−2f(γz). Let γ = ( 1 a0 ` ) for

some prime `, then compute

φf |γr → s =

∫ s

r

(zX + 1)kf |kγdz

=

∫ s

r

(zX + 1)k`k+1(0z + `)−k−2f(z + a

`

)dz

= `−1

∫ γs

γr

((`y − a)X + 1)kf(y)(`dy),

24 CHRIS WILLIAMS

under the substitution y = z+a` , with z = `y − a and dz = `dy. We further compute that

(yX + 1)k|γ = (1− aX)kÅy

`X

1− aX+ 1

ãk= ((`y − a)X + 1)k,

and substituting this in, our expression becomes

=

ñ∫ γs

γr

(yX + 1)kf(y)dy

ô∣∣∣∣∣ γ= φf |γr → s.

The Up operator is comprised of a sum of matrices of this form, so we deduce that φf |Up =

φf |Up. For the T` operator, it remains to check only the matrix ( ` 00 1 ) . Let γ = ( ` 0

0 1 ); wecheck directly that

φf |γr → s =

∫ s

r

`k+1f(`z)(zX + 1)kdz

=

∫ γs

γr

f(y)

ñ`kÅyX

`+ 1

ãkôdy

= φf |γr → s,

where we use that P |γ(X) = `kP (X/`). Thus f 7→ φf is equivariant for each matrix usedin the T` operator as well, and we conclude the desired Hecke equivariance.

Remark 11.4. In passing from f (an analytic object) to φf (a purely algebraic one) weare a priori throwing away all the analytic data. However, via its q-expansion and Theorem9.10, one can always reconstruct f solely from the data of its Hecke eigenvalues. This Hecke-equivariance statement, therefore, can be summarised as saying `no information is lost indiscarding the analytic data': we can reconstruct f from its modular symbol φf .

This remark means that the map f 7→ φf is injective. It is natural to ask if it is alsosurjective. This is not true, but it fails in a controlled manner. As well as the usual spaceof holomorphic cusp forms, we have modular symbols attached to:

Eisenstein series. If f is an Eisenstein series, then the association f 7→ φf aboveneeds adapting, as we no longer have absolute convergence of the integrals

∫ sr. However,

there is still an easily identied Eisenstein subspace in the space of modular symbols. Inparticular, consider the space HomΓ(∆, Vk(C)). As a Z[Γ]-module, ∆ = Div(P1(Q)) isfreely generated by the (nite set of) cusps of Γ, so any such homomorphism is just the dataof an element of Vk(C) at each cusp. There is a natural restriction map

res : HomΓ(∆, Vk(C)) −→ HomΓ(∆0, Vk(C)) = SymbΓ(Vk(C)),

where res(φ)r → s ..= φ(s)− φ(r). We dene

BSymbΓ(Vk(C)) ..= Im(res) ⊂ SymbΓ(Vk(C)).

The space BSymbΓ(Vk(C)) can be canonically and Hecke-equivariantly identied with thespace of cusp forms of weight k + 2 and level Γ (see [Bel, III.2.8]); in line with the above,we write φg for the symbol corresponding to an Eisenstein series g.

Antiholomorphic cusp forms. Every cusp form f ∈ Sk+2(Γ,C) has an anti-

holomorphic cousin f(z) ..= f(z), its complex conjugate. We write Sk+2(Γ,C) for thespace of antiholomorphic cusp forms. Each of these antiholomorphic cusp forms also givesa modular symbol φf by

φfr → s ..=∫ s

r

f(z)(zX + 1)kdz,

and this association is again Hecke-equivariant for the natural Hecke action on Sk+2(Γ,C).Importantly, if f is a (holomorphic) Hecke eigenform, then f is also an eigenform, with thesame Hecke eigenvalues.


EichlerShimura says that the space of modular symbols is exhausted by the contributionsof (holomorphic) cusp forms, antiholomorphic cusp forms, and Eisenstein series.

Theorem 11.5 (EichlerShimura). There is a Hecke-equivariant isomorphism

Sk+2(Γ,C)⊕ Sk+2(Γ,C)⊕ Ek+2(Γ,C) ∼= SymbΓ(Vk(C))

given by the map

(f, f , g) 7→ φf + φf + φg.

Proof. This theorem is very important in the sequel, but its proof is quite involved, andtakes us quite far from the p-adic L-functions we are aiming to study. Thus, in these notes,we give only a sketch of the proof. The main steps are as follows:

Rephrase the theorem in cohomology. Let YΓ..= Γ\H denote the open modular curve.

By [AS86], there is an isomorphism

SymbΓ(Vk(C)) ∼= H1c(YΓ, Vk(C))

between modular symbols and the compactly supported cohomology; this is proved us-ing homological algebra, and identifying SymbΓ(V ) with the degree 0 group cohomologyH0(Γ,Hom(∆0, V )).

Recall the the Petersson product on modular forms. There is also a natural cup productpairing

H1c(YΓ, Vk(C))×H1(YΓ, Vk(C))→ C.

One nds an explicit relationship between these two pairings. Use the relationship between the two pairings, and the properties of the Petersson

product, to show that the map is injective. Show that both sides have the same dimension, using standard dimension results in the

theory of modular forms (for the left-hand side) and algebraic topology (for the right-handside). Such formulae involve topological invariants of YΓ, for example the genus.

11.3. Plus-minus spaces. Let rst f ∈ Sk+2(Γ,C) be a newform.

Denition 11.6. Let λf : HN → F ⊂ C be the eigenpacket attached to f . If M is aspace with an action of the Hecke algebra HN , then recall we write

M [f ] = f -eigenspace of HN inside M

= m ∈M : Tm = λf (T )m for all T ∈ HN.

In particular, Theorem 9.9(3) is the statement that

dimC Sk+2(Γ,C)[f ] = 1 = dimC Sk+2(Γ,C)[f ], (11.2)

where the last equality follows from the fact that f and its antiholomorphic cousin f havethe same Hecke eigenvalues.


dimC SymbΓ(Vk(C))[f ] = 2.

Proof. Under Corollary 11.5, we may identify this space with Sk+2(Γ,C)⊕Sk+2(Γ,C)⊕Ek+2(Γ,C). The eigenpacket λf cannot occur in Ek+2(Γ,C), since the T`-eigenvalue of anEisenstein series at a prime ` is of size approximately `k, whilst for cusp forms, we have theestimate a`(f) ≤ C`k/2 of Lemma 9.15. It appears exactly once in both Sk+2(Γ,C) andSk+2(Γ,C) by (11.2), and hence in the space of symbols with multiplicity two.

We cut this down to a direct sum of two 1-dimensional eigenspaces with the action of anatural involution.

26 CHRIS WILLIAMS

Denition 11.8. Let ι be the involution given by the action of(

1 00 −1

)on SymbΓ(Vk(C)).

Note that

ι = [Γ(

1 00 −1

)Γ]

has the structure of a double coset operator for Γ, and hence ι commutes with the Heckealgebra HN .

Proposition 11.9. We have a Hecke-stable decomposition

SymbΓ(Vk(C)) ∼= Symb+Γ (Vk(C))⊕ Symb−Γ (Vk(C))

into the ±1 eigenspaces of the involution ι.

Proof. Attached to ι, we have projectors 1±ι2 which naturally land in the ±1-eigenspaces.

This induces the required direct sum decomposition.

Note that it is not immediately obvious that both components are non-zero at f .

11.4. Connection to L-values. Recall Proposition 10.14, where we derived the ex-pression

φf0→∞ =

k∑j=0

Åkj

ãïj!

(−2πi)j+1L(f, j + 1)

òXj . (11.3)

This also decomposes under the involution ι. Let

φ±f..=

ι± 1

2(φf ),

so that φf = φ+f + φ−f ∈ Symb+

Γ (Vk(C))⊕ Symb−Γ (Vk(C)).


φ+f 0→∞ =

k∑j=0j even

Åkj

ãïj!

(−2πi)j+1L(f, j + 1)

òXj .

Similarly, φ−f 0→∞ is the sum of the terms in (11.3) for j odd.

Proof. The involution ι xes both 0 and ∞, and acts on polynomials by sending

Xj |ι ..= Xj

∣∣∣∣Å1 00 −1

ã= (−1)jXj .

It follows that ι±12 acts as the identity on the terms Xj where (−1)j = ±1, and 0 on others,

from which the proposition follows immediately.

As usual, we have an analogous statement for twisted L-values, which (as ever) makes agood exercise. The below should be compared to Proposition 10.18 and Corollary 10.20.

Proposition 11.11. Let χ : (Z/DZ)× → Q×be a Dirichlet character of conductor D,

and let 0 ≤ j ≤ k. Then

Evχ(φ±f ) =

k∑j=0

χ(−1)(−1)j=±1

Åkj

ãïG(χ) · j! ·Dj

(−2πi)j+1· L(f, χ, j + 1)

òXj .

In particular, for Cχ,j as in Corollary 10.20, we see that

Evχ,j(φ±f ) =

ßCχ,j · L(f, χ, j + 1) : χ(−1)(−1)j = ±1

0 : else.


Proof. This is a straightforward calculation. Note that ι · a/D = −a/D. Thus in thecalculations that go into the proof of Proposition 10.18, one brings out a factor of χ(−1)to obtain χ(−a/D) in the sum, rescaling the choice of representatives for (Z/DZ)×. Whencombined with the action of ι on Xj described above, we nd that

coecient of Xj in Evχ(φ|ι) = χ(−1)(−1)j × coecient of Xj in Evχ(φ).

In particular, the projector pr± acts as the identity on the coecients where χ(−1)(−1)j =±1, and kills the others.

Corollary 11.12. For each sign, we have

dimC Symb±Γ (Vk(C))[f ] = 1.

Proof. A theorem of Rohrlich [Roh91] says that for all but nitely many Dirichlet char-acters χ, the L-value

L(f, χ, 1) 6= 0.

In particular, there exists at least one even χ and one odd χ with this non-vanishing property.But if χ is even (resp. odd), then a non-zero multiple of L(f, χ, 1) appears as the coecientof Evχ(φ+

f ), a sum of values of φ+f (resp. φ−f ). It follows that φ±f 6= 0 for both choices of

sign. We then conclude since they live in a 2-dimensional ambient space which is the directsum of the ±-eigenspaces of ι.

Remark 11.13. Rohrlich's theorem is a deep result in analytic number theory. If theweight k+ 2 of f is at least 4, then there is a proof that doesn't invoke this. Indeed, in thiscase the value L(f, χ, k + 1) appears as a linear combination of values of φf . But this isin the region of absolute convergence for the L-function, where we have the Euler product,which converges absolutely to a non-zero value. We then deduce φ±f 6= 0 as before.

11.5. EichlerShimura with algebraic coecients. In all of the above, we workedwith complex coecients. We saw in the previous sections that there is a natural alge-braic structure on the space of modular forms by looking at q-expansions with algebraiccoecients, and moreover that there exists a number eld F/Q such that

Mk+2(Γ,C) ∼= Mk+2(Γ, F )⊗F C, (11.4)

with the Hecke action dened over F . There is also a natural algebraic structure on modularsymbols: we can simply take modular symbols with coecients in Vk(F ). We have theanalogy of (11.4) for modular symbols.

Proposition 11.14 (Shimura). There is a (Hecke-equivariant) isomorphism

SymbΓ(Vk(F ))⊗F C ∼= SymbΓ(Vk(C)).

Remark 11.15. We omit the proof; this is proved, for example, in [Hid88, 7] and[Hid94, 8].

Note that this is not automatic without appealing to deeper results in algebraic ge-ometry. We have seen that there are canonical one-dimensional Hecke-stable subspacesSymb±Γ (Vk(C))[f ] attached to f , and moreover we know that the eigenpacket λ : HN →F ⊂ C takes values in F . It follows easily that we can exhibit a (non-canonical) F -rationalstructure on Symb±Γ (Vk(C)); this is purely a linear algebra statement on the abstract vectorspaces. Sometimes, it is claimed that this rational structure then coincides with the spaceSymb±Γ (Vk(F ))[f ]. To this author, however, this appears far from obvious without further(cohomological) input.

As an example of this, suppose we are in the weight 3 situation, so that we consider thespace SymbΓ(CX⊕C). As the complex EichlerShimura isomorphism is Hecke-equivariant,the modular symbol φf attached to any eigenform f is an eigensymbol, and hence the F -line F · φf is preserved by the Hecke operators. We may always scale φf by a complexscalar such that, for example, φf0 → ∞ = aX + b, with a ∈ F . This process does not apriori also control b, however, which without further input may by horribly transcendental.In addition, it says nothing about the other values φfr → s. It may be possible to say

28 CHRIS WILLIAMS

something directly using symmetries under Γ and the Hecke action, but this quickly becomesa ddly exercise without much conceptual basis.

One may x this by instead arguing on the identication of modular symbols with coho-mology mentioned above. One can prove that the Hecke algebra is naturally dual to modularforms, and also via complex EichlerShimura to modular symbols. From the study ofmodular forms, we obtain an F -rational structure on the Hecke algebra, and we may thentransfer this to the cohomology, showing that the rational structure outlined above doescoincide with the F -rational modular symbols. This is the approach taken in the referencesgiven above. Alternatively, one may observe that the modular curve has a model over Q(as in [DS05, 7], and hence its cohomology admits Q-rational structure; so SymbΓ(Vk(C))admits a C-basis of modular symbols valued in Vk(Q) (though this will not in general be abasis of eigensymbols).

Allowing the proposition, we can descend EichlerShimura to algebraic coecients. Theargument from here now proceeds via elementary linear algebra.

Lemma 11.16. Let M and N be F -vector spaces with actions of the Hecke algebra H.Suppose that M ⊗F C ∼= N ⊗F C as H-modules. Then

M ∼= N

(non-canonically) as H-modules.

Proof. Choose bases for M and N over F . A linear map M → N is determined by itsmatrix (xij) in these bases, and since the Hecke action is dened over F (even Q), the xijare elements of F . Then:

the statement `the map is an isomorphism' is determined by a linear equation in thexij (for example, det 6= 0 for M,N nite dimensional of the same dimension),

and for each T ∈ H, the statement `the map is T -equivariant' cuts out another linearequation in the xij .

The hypothesis that M ⊗F C ∼= N ⊗F C implies this this system of equations has a simul-taneous solution over C. But since every equation in this system is dened over F , thisimplies there exists a solution (xij) with each xij ∈ F ; and then by construction the map(xij) denes the required H-equivariant isomorphism.

Applying this with M and N the F -rational structures on modular forms and modularsymbols respectively, with the HN -module isomorphism on the complexications given byTheorem 11.5, we obtain the following.

Corollary 11.17. There is a Hecke-equivariant isomorphism

Sk+2(Γ, F )⊕ Sk+2(Γ, F )⊕ Ek+2(Γ, F ) ∼= SymbΓ(Vk(F )).

Note, however, that this map is now non-canonical. This non-canonicity will manifestitself in an ambiguity in complex periods in the next section.

11.6. Periods and algebraic modular symbols for newforms. The algebraicEichlerShimura isomorphism shows that all of the dimension results of 11.3 descend toalgebraic coecients. In particular, we have the following analogue of Corollary 11.12, fromwhich we can enact the strategy of 11.1.

Proposition 11.18. For each sign, for the newform f we have

dimF Symb±Γ (Vk(F ))[f ] = 1.

Let φ±f,F be generators of these two 1-dimensional vector spaces. These are non-canonical,

well-dened only up to multiplication by an element of F×. We clearly have an inclusion

Symb±Γ (Vk(F )) ⊂ Symb±Γ (Vk(C)),


of 1-dimensional vector spaces, with the F -valued modular symbols spanning an F -line insideC (see Figure 11.1). In this larger C-space, we have canonical generators, namely φ±f , the

±-classes projected from the canonical class φf . As any element of Symb±Γ (Vk(F )) is thenin the C-span of φ±f , we deduce that there exist Ω±f ∈ C× such that

φ±f,F =φ±f

Ω±f∈ Symb±Γ (Vk(F )).

We call the Ω±f the ±-periods of f .

Figure 11.1. Algebraic structures on modular symbols

We summarise this discussion in the following proposition.

Proposition 11.19. For any eld F ⊂ C containing the Hecke eld Kf of the newform

f , there exist complex periods Ω±f ∈ C×, well-dened up to multiplication by F×, such that

φ±f


11.7. Algebraic modular symbols for general eigenforms. Let now f ∈Sk+2(Γ0(N)) be an arbitrary eigenform; we wish to deduce an analogue of this for f .From the general theory of eigenforms, there exists a unique newform g ∈ Sk+2(Γ0(M))(furnished by [DS05, Prop. 5.8.4] for some M |N such that f is a linear combination ofmodular forms of the form g(mz), where m is a divisor of N/M (via [DS05, Thm. 5.8.3]).

Proposition 11.20. The modular symbol attached to the modular form hm(z) ..= g(mz)is

φhm..= m−k−1φg |(m 0

0 1 )

Proof. We have hm(z) = g(mz) = m−k−1g| (m 00 1 ) (z). But we already showed in the

proof of Proposition 11.3 that the association g 7→ φg is equivariant for the action of (m 00 1 )

on both sides, from which the result follows immediately.

Let ϕ±g..= φ±f /Ω

±f ∈ Symb±Γ0(M)(Vk(F )) be the algebraic symbols given by Proposition

11.19. We may trivially consider these symbols as elements of the space Symb±Γ (Vk(F )).Then for any m dividing N/M , attached to hm(z) = g(mz) we have

φ±hmΩ±g

=m−k−1 · φ±g | (m 0

0 1 )

Ω±g∈ Symb±Γ (Vk(F )).

30 CHRIS WILLIAMS

Write f =∑m|(N/M)Amhm as a linear combination of the hm's. After possibly renormalis-

ing, it is always possible to take the Am's to be algebraic; and we may enlarge F to containthem. Then:

φ±f

Ω±g=

∑m|(N/M)

Amφ±hmΩ±g∈ SymbΓ(Vk(F )).

So we may dene Ω±f..= Ω±g to be the periods of the associated newform, after which we

obtain:

Proposition 11.21. Let f be an eigenform. There exists a suciently large numbereld F and complex periods Ω±f ∈ C×, well-dened up to multiplication by F×, such that

φ±f


11.8. Algebraicity of L-values. We are nally in a position to prove Shimura's the-orem.

Proof. (Theorem 11.1). Let χ be a Dirichlet character of conductor D, and let 0 ≤ j ≤ k,as in the statement of Theorem 11.1. Let F (χ) be the nite eld extension of F obtainedby adjoining the values of χ (which will all be roots of unity, since χ has order dividing thetotient φ(D)).

Recall the map Evχ,j from Denitions 10.16 and 10.19. These were composed of a `twistby χ' map, then evaluation at 0→∞, then projection to the jth co-ordinate. In particular,if φ is valued in Vk(F ), then

Evχ(φ) ∈ Vk(F (χ)), and Evχ,j(φ) ∈ F (χ).

We end up with the following commutative diagram:

Evχ,j : Symb±Γ (Vk(C))Evχ //

⊃

Vk(C)jth coe //

⊃

C

⊃

Evχ,j : Symb±Γ (Vk(F ))Evχ // Vk(F (χ))

jth coe // F (χ).

By Proposition 11.11, we know that

Evχ,j(φ±f ) =

Åkj

ãDj · G(χ)Γ(j + 1)

(−2πi)j+1· L(f, χ, j + 1)

if χ(−1)(−1)j = ±1 (and is equal to 0 otherwise). Since φ±f /Ω±f ∈ Symb±Γ (Vk(F )), we

deduce that when this sign condition is satised, we have

Evχ,j

Çφ±f

Ω±f

å=

Evχ,j(φ±f )

Ω±f

=

Åkj

ãDj · G(χ)Γ(j + 1)

(−2πi)j+1· L(f, χ, j + 1)

Ω±f∈ F (χ) ⊂ Q.

All of the terms in the interpolating factor are algebraic except (−2πi)j+1 and possibly Ω±f ,from which we deduce that

L(f, χ, j + 1)

(2πi)j+1 · Ω±f∈ Q,

as required. (We may always swap χ and χ = χ−1 to get the stated formulation).

Unlike the complex modular symbol φf , the algebraic modular symbols φ+f /Ω

+f and

φ−f /Ω−f each only see half of the critical values. We rectify this by dening:


Denition 11.22. Dene

ϕf ..=φ+f

Ω+f

+φ−f

Ω−f∈ SymbΓ(Vk(F )).

From Proposition 11.11, we immediately have

Evχ,j(ϕf ) = Cχ,j ·L(f, χ, j + 1)

Ω±f,

where the sign of the period is determined by χ(−1)(−1)j = ±1. The left-hand side now hasno dependence on the sign: that is, ϕf is algebraic and sees all the critical values, regardlessof sign. The following algebraic version of Figure 10.1 is then the starting point for thep-adic theory.

ϕf+ **∈ SymbΓ(Vk(F ))

Evχ,j // Q 3 Cχ,j · L(f,χ,j+1)

Ω±f

φf

± symbols, & divide by Ω±f

OO

+ **∈ SymbΓ(Vk(C))

Evχ,j // C 3 Cχ,j · L(f, χ, j + 1)

f

ee ;;

Figure 11.2. L-values from modular symbols: algebraic coecients

Interlude: a change of language

As we saw in [RJW17] where we frequently switched between measures and powerseries it sometimes pays to change the language in which you consider an Iwasawa-theoreticproblem. As a stand-alone theory, it makes far more sense to introduce modular symbolsto be valued in polynomials, as we did above. However, in the context of overconvergentmodular symbols and p-adic L-functions, it is more convenient to take the dual approach.

Ultimately, our aim is to construct a p-adic measure(2) Lp(f) a function on charactersof the form x 7→ χ(x)xj that interpolates the algebraic L-values above, that is, such that

Lp(f, j) =

∫Z×p

χ(x)xk · Lp(f) = (∗)L(f, χ, j + 1)

(2πi)j+1Ω±f

for all χ of p-power conductor, and for all 0 ≤ j ≤ k, where (∗) is an explicit algebraicconstant (of the form we are already seeing arising in integral formulae).

In this sense, we want:

an element dual to functions, such that the measure of zj the jth moment computes the L-value at j + 1.

Currently, our modular symbols are valued in Vk(F ), namely:

a module of (polynomial) functions, such that the jth coecient computes the L-value at j.

(2)In fact, the space of measures is not big enough to contain Lp(f) in general, and we must pass to the

space of p-adic distributions, which we introduce in the next section.

32 CHRIS WILLIAMS

Our change of perspective is to replace Vk(F ) with its dual space.

Denition 11.23. Let Vk(R) be the space of polynomials over R of degree at most k,with a new (left) action of Γ dened byÅ

a bc d

ã· P (X) = (a+ cX)kP

Åb+ dX

a+ cX

ã.

Denote the dual space by

V∨k (R) ..= Hom(Vk(R), R),

with the dual right action given by

µ|γ(P ) ..= µ(γ · P ).

This is a space of polynomial measures.

Note that V∨k (F ) has the (dual) basis X j : 0 ≤ j ≤ k, where X j is dened on monomialsby

X j : Xi 7→ß

1 : i = j0 i 6= j.

Lemma 11.24. There is an isomorphism

η : Vk(F ) ∼−→ V∨k (F )

(−1)jÅkj

ãXj 7−→ X j

of GL2(F )-modules.

Proof. The map is obviously an isomorphism of F -vector spaces, so the content of thislemma is entirely in the equivariance of the GL2(F )-action. One can check this explicitly,for which we give only a simple example by way of illustration. Note that

X j∣∣∣∣Å1 1

0 1

ã(Xi) = X j

((X + 1)i

)=

i∑n=0

Åin

ãX j(Xn) =

Åij

ã, (11.5)

so that

X j∣∣∣∣Å1 1

0 1

ã=

k∑i=j

Åij

ãX i.

We also have

(−1)jÅkj

ãXj

∣∣∣∣Å1 10 1

ã= (−1)j(X − 1)k−jXj

= (−1)jÅkj

ã k−j∑n=0

Åk − jn

ã(−1)k−j−nXn+j

=

k∑i=j

(−1)iÅki

ãÅij

ãXi, (11.6)

where we have used the identity Åkj

ãÅk − ji− j

ã=

Åki

ãÅij

ãin the nal step. Under the stated map, (11.6) is sent to (11.5). The general case is similar,just keeping track of additional notation.


In particular, under this isomorphism we have

[jth coecient of a polynomial P ] =

∫Xj · η(P ).

This is visibly closer to our goal. Putting this into the context of modular symbols, weobtain:

Corollary 11.25. There is an isomorphism

η : SymbΓ(Vk(F )) ∼= SymbΓ(V∨k (F ))

of Hecke-modules, and∫Xj · η(ϕf )0→∞ = − j!

(2πi)j+1· L(f, j + 1)

Ω±f.

Note in particular that as a consequence of this change of language, we have killed thebinomial coecient from the integral formula of Proposition 10.14, as well as (most of) thesign. (The remaining −1 is simply a matter of convention).

From now on:

we switch notation and replace ϕf with its image η(ϕf ) ∈ SymbΓ(V∨k (F )), and we redene

Evχ,j : SymbΓ(V∨k (F ))ϕ7→ϕχ−−−−→ SymbΓ(V∨k (F (χ)))

0→∞−−−−−→ V∨k (F (χ))∫zj

−−→ F (χ).

Corollary 11.26. If χ is a Dirichlet character of conductor D, then

Evχ,j(ϕf ) = Cχ,j · L(f, χ, j + 1),

where

Cχ,j =G(χ) · j! ·Dj

(2πi)j+1Ω±f.

There is still a glaring problem to overcome: we cannot compute beyond the kth mo-ment. In the case of GL(1), only when we passed to full measures on Zp did we see thecorrespondence to power series, to local units, and ultimately the statement of the Iwasawamain conjecture. To obtain analogues here, we must rigidify further with p-adic analysis.

12. Overconvergent modular symbols

Let L/Qp be a nite extension that contains all possible embeddings of F . We write Ofor the ring of integers in L. Henceforth, we consider ϕf as an element of SymbΓ(V∨k (L)).

We are aiming to construct the element Lp(f) at the top right of the diagram below.

[p-adic

distributions

]∫χ(x)xj

3 Lp(f)_

ϕf 33

∈ SymbΓ(V∨k (L))Evχ,j // Qp 3 Cχ,j · L(f, χ, j + 1)

Figure 12.1.

34 CHRIS WILLIAMS

The idea behind overconvergent modular symbols, as introduced by Glenn Stevens in[Ste94], is to instead produce the analogous objects on the left-hand side of the diagram,`interpolating' the spaces of modular symbols and the maps Evχ,j .

12.1. Analytic distributions. To access higher moments, we pass from polynomialfunctions to power series.

Denition 12.1. A function ψ : Zp → L is rigid analytic if we can write

ψ(x) =∑n≥1

anxn

with an ∈ L tending to zero as n → ∞; equivalently, it is a power series that converges onZp. We write A(L) = A(Zp, L) for the space of rigid analytic functions Zp → L.

The space A(L) is a Banach space over L, where the norm of f is the supremum of |f(x)|as x varies over the closed unit ball in Cp (upon which f converges by denition). Weexplore this in much more detail in 15.

Note that Vk(L) is contained in A(L) for all positive integers k; so this is the analyticversion of the module Vk. Dually, we get the following analytic version of V∨k .

Denition 12.2. A rigid analytic distribution is a continuous homomorphism

µ : A(L) −→ L.

We write D(L) = D(Zp, L) = Homcts(A(L), L).

Notation 12.3. The coecient eld does not play a major role in the construction, andwe sometimes drop it from the notation. Where we write A(L) or D(L) without specifyingthe domain, these are always functions or distributions on Zp.

Remarks 12.4. (1) Note that any rigid analytic function on Zp is continuous, andhence any measure µ ∈M (Zp, L) denes a rigid analytic distribution by restriction. More-over, this restriction is injective, as xj ∈ A(Zp, L) for all j, and we have seen that a measureis totally determined by its moments

∫xj · µ. Thus there is an inclusion

M (Zp, L) ⊂ D(Zp, L),

and the space of rigid analytic distributions is a huge space generalising the notion of mea-sures.

(2) If µ ∈ D(Zp, L), then since the binomial polynomials are rigid analytic, we may takethe Amice transform

Aµ(T ) =

∫Zp

(1 + T )x · µ

analogously to [RJW17, 2]. From the denition, this is an element of L[[T ]]. A distributionµ ∈ D lives in the smaller space of measures if and only if Aµ(T ) ∈ L⊗O O[[T ]].

(3) If µ ∈ D(Zp, L), then Aµ(T ) ∈ L[[T ]] converges on the open ball

B

Å0, p− 1p−1

ã..=¶x ∈ Cp : vp(x) > 1

p−1

©.

(see [Col10, Thm. II.2.2], with h = 0).

The convergence in (3) above imposes a growth condition on the coecients of Aµ(T ). Itis more convenient, however, to characterise distributions using a dierent, and equivalent,set of data.

Denition 12.5. Let µ ∈ D(L). Dene the moments of µ to be the sequence

µ(xn) : n ≥ 0.

Lemma 12.6. Taking moments induces a bijection

D(Zp, L) ∼−→ (bounded sequences in L).


Proof. Let µ ∈ D(Zp, L), and suppose that µ(xj) is an unbounded sequence. Let (bjm)be a subsequence tending to ∞, bjm = µ(xjm). Then

ψ(x) =∑m≥0

xjm

bjm∈ A(Zp, L)

is rigid analytic. But µ(ψ) doesn't converge; so µ was not rigid analytic.

Conversely, for any bounded sequence (bm), and any rigid analytic ψ(x) =∑amx

m, theexpression

µ(f) ..=∑

ambm

converges since am → 0. This then denes a distribution µ.

Corollary 12.7. Let

D(O) = D(Zp,O)

..= µ ∈ D(L) : µ(xj) ∈ O∀j.

Then

D(O)⊗O L ∼= D(L).

Proof. Let µ ∈ D(L). Since the moments are bounded, there exists c ∈ L such thatcµ(xn) ∈ O for all n. Then

µ = µ⊗ 1 = cµ⊗ c−1

∈ D(O)⊗ L.

In particular, up to scaling by L, we may pass freely between integral and rational dis-tributions.

12.2. Overconvergent modular symbols. The top left corner of Figure 12.1 is givenby modular symbols valued in D(L).

Denition 12.8. Let

Σ0(p) =

ßÅa bc d

ã∈M2(Zp) : p|c, a ∈ Z×p , ad− bc 6= 0

™.

When p|N , this monoid contains Γ = Γ0(N), as well as all the matrices required for a Heckeaction.

Remark 12.9. Note that Σ0(p) is generated by matrices of the form ( 1 0c 1 ) and

(a b0 d

),

since Åa bc d

ã=

Å1 0c/a 1

ã·Åa b0 ∆/a

ã,

where ∆ = ad− bc.

Lemma 12.10. Let k ≥ 0 be an integer. There is a well-dened left weight k action ofΣ0(p) on A, dened by Å

a bc d

ã· ψ(x) = (a+ cx)kψ

Åb+ dx

a+ cx

ã.

Proof. The only problem term is (a+ cx)−1. We compute that

1

a+ cx= a−1 1

1 + cxa

=∑n≥0

(−1)n(cxa

)n.

Since p|c and a is a p-adic unit, this converges, giving a well-dened power series. It followsthat

(a bc d

)· ψ ∈ A(L) using the algebra structure on power series.

36 CHRIS WILLIAMS

Remark 12.11. Note that it is crucial that p|c for this. In particular, it is not possibleto dene overconvergent modular symbols when p does not divide the level N . This doesnot cause a signicant obstacle: given, for example, a newform of level M prime to p, wemay always pass to an old eigenform of level Mp and consider overconvergent symbols atthis level instead.

Notation 12.12. We write Ak for the space A equipped with the weight k action. Wewrite Dk for D with the induced dual action

µ|γ(ψ) = µ(γ · ψ).

Denition 12.13. The space of overconvergent modular symbols of weight k and levelΓ is

SymbΓ(Dk) = HomΓ(∆0,Dk).

This space has a natural Hecke action dened in exactly the same manner as before; thatis, if T ∈ HN is dened on modular forms by

Tf =∑i

f |kγi,

then we dene TΦ =∑i Φ|γi for Φ an overconvergent modular symbol.

It is clear from the denitions that the injection Vk → Ak is equivariant for the Σ0(p)actions on both sides. Dualising this, we obtain a Σ0(p)-equivariant surjection

ρ : Dk −→ V∨k .

Corollary 12.14. There exists a Hecke equivariant specialisation map

ρ : SymbΓ(Dk) −→ SymbΓ(V∨k )

The Hecke equivariance follows immediately from Σ0(p)-equivariance.

12.3. Stevens' control theorem. Returning to our running diagram, we now havethe situation of Figure 12.2; we are aiming to construct the top right part, and we have justlled in the top left part.

SymbΓ(Dk(L))

ρ

[p-adic

distributions

]∫χ(x)xj

3 Lp(f)_

ϕf 33

∈ SymbΓ(V∨k (L))Evχ,j // Qp 3 Cχ,j · L(f, χ, j + 1).

Figure 12.2.

We want to lift ϕf to an overconvergent modular symbol. The space SymbΓ(Dk(L)) isenormous, however, whilst SymbΓ(Vk(L)) is nite-dimensional; thus necessarily the kernelof ρ is massive.

Stevens' control theorem says that the kernel can be completely controlled by studyingeigenspaces of the Up-operator. In particular, if we restrict to certain Up-eigenspaces, thenρ even becomes an isomorphism. The criterion we use is the slope of the Up-operator.


12.3.1. Slopes of modular forms.

Denition 12.15. Let f ∈ Sk+2(Γ) be an eigenform, with Up-eigenvalue αp. The slopeof f at p is vp(α) ∈ Q.

The form f has nite slope if αp 6= 0, and innite slope otherwise. There are tightconstrictions on the possible range of nite slopes.

Lemma 12.16. Suppose:

(1) f is new at p with αp 6= 0,(2) or f is old at p, the p-stabilisation of a form g at level M prime to p.

Then 0 ≤ vp(αp) ≤ k + 1.

Proof. Hecke eigenvalues are always algebraic integers. In particular, they are p-integral,so vp(αp) ≥ 0.

If f is new at p, then as αp 6= 0, we have p||N , corresponding to `multiplicative reduction';

then it is classically known that αp = ±pk/2 for some choice of sign, in which case andvp(αp) = k/2 ≤ k + 1.

Alternatively, suppose f is old at p, the p-stabilisation of a modular eigenform g of levelprime M prime to p. Suppose

Tpg = ap(g)g;

then the characteristic polynomial of Up at level N = Mp is X2 − ap(g)X + pk+1. Theeigenvalue α is a root of this polynomial. So α|pk+1 and hence vp(α) ≤ k + 1.

12.3.2. The control theorem.

Theorem 12.17 (Stevens). Let α ∈ L× with vp(α) < k + 1. Then the restriction

ρα : SymbΓ(Dk(L))Up=α ∼−→ SymbΓ(V∨k (L))Up=α

of ρ to the eigenspaces where Up acts as α is an isomorphism.

An immediate corollary is the following.

Corollary 12.18. Let f ∈ Sk+2(Γ) be an eigenform, Upf = αpf . If vp(αp) < k + 1,then

SymbΓ(Dk(L))[f ] ∼−→ SymbΓ(V∨k (L))[f ].

In particular, there exists unique eigenlift Φf ∈ SymbΓ(Dk)[f ] of ϕf .

In the next section, we give a proof of this theorem.

12.4. Additional remarks.

Remark 12.19. We see that the slope at p `detects' classical modular symbols in thespace of overconvergent modular symbols. In particular, let Φ be an overconvergent eigen-symbol, corresponding to an eigenpacket λ : HN → L.

If Φ has slope < k + 1, then λ is the eigenpacket of a classical modular symbol, andhence a classical modular form by EichlerShimura.

If Φ has slope > k + 1, then λ is never classical, by Lemma 12.16; If the slope is exactly equal to k+1, then the situation is ambiguous. There are examples

where ρα is an isomorphism, and examples where it is not. For more on this phenomenon,see [PS12] and [Bel12].

38 CHRIS WILLIAMS

Remark 12.20. This theorem has a close analogue in the theory of overconvergentmodular forms, due to Coleman [Col96]. In particular, an overconvergent modular formof weight k + 2 and slope < k + 1 is classical. The parallels go further: it is a theorem ofStevens [PS12] that there is a bijection between systems of eigenvalues in overconvergentmodular forms and overconvergent modular symbols (up to one exceptional system, denotedEcrit

2 op. cit., which occurs in modular symbols but not modular forms).

Remark 12.21. In terms of p-adic L-functions, we're on the right track. Let

µf = Φf0→∞ ∈ D.

For 0 ≤ j ≤ k, we have xj ∈ Vk ⊂ Ak, and hence the integral of xj against µf factorsthrough ρ : Dk → V∨k . Thus we have∫

Zp

xj · µf =

∫Zp

xj · ρ(µf )

=

∫Zp

xj · ϕf0→∞

= − j!

(2πi)j+1· L(f, j + 1).

In the context of Figure 12.2, we are completing the diagram with the top map beingevluation at 0→∞.

We cannot yet call µf the p-adic L-function, however. The space D is too big, as A istoo small; in particular, it does not contain any Dirichlet characters, which are not rigidanalytic, but instead are only locally analytic. We also need to restrict to Z×p . We treatboth of these issues in 14.

Remark 12.22. In addition to the construction of p-adic L-functions of modular forms,as we will see in this course, Stevens' control theorem has a huge range of other applications,a some of which we briey mention.

Stevens' original motivation was the study of p-adic families of modular forms, buildingon ideas of Serre, Katz and Coleman. Suppose f is an eigenform of weight k. Then onecan deform k to innitessimally p-adically close integer weights k+ pn. It turns out that forsuciently large n, there are eigenforms fn of weight p+ kn whose eigenvalues converge, asn → ∞, to the eigenvalues of f . One can see this from overconvergent modular symbols.In addition to being a beautiful in their own right, p-adic families of modular forms are acrucial tool in proving results such as the Iwasawa main conjecture.

In a dierent direction, Darmon has used the control theorem in his construction of StarkHeegner or Darmon points, p-adic analogues of Heegner points on elliptic curves [Dar01].In more recent work with Vonk [DV17], he has used it in a dierent context to give aconjectural approach to explicit class eld theory over real quadratic elds.

13. Proof of Stevens' control theorem

We now give a proof of Stevens' control theorem. This was not Stevens' original proof:this proof is due to Greenberg, and rst appeared in [Gre07].

Via the moments, the spaces of coecients can be viewed as:

Dk = data of µ(x0), µ(x1), ..., µ(xk), µ(xk+1), ... ∈ LN

V∨k = data of µ(x0), µ(x1), ..., µ(xk) ∈ Lk+1.

The map from Dk to V∨k is `forget the moments µ(xk+1), µ(xk+2), ...'. Thus to lift, we needto reconstruct these higher moments from:

the moments up to µ(xk), and the property of being a Up-eigensymbol.


Given this, there is only one sensible thing to do: we must iterate the Up operator.

13.1. An example: improving with Up. The idea of Greenberg's proof is that eachiteration of Up `improves' the information we have. Let us illustrate this with an example,indicating two key steps to make precise in the proof below.

Suppose for simplicity that f ∈ S2(Γ,Q) has weight 2 and rational coecients. Afterscaling, this gives rise to a modular symbol ϕf ∈ SymbΓ(Qp) with Upϕf = αϕf .

Step 1 (Integral coecients). Show that after possibly multiplying ϕf by a scalarin Qp, we may take

ϕf ∈ SymbΓ(Zp).

We may lift ϕ at a set-theoretic level to a function

Φ ∈ Maps(∆0,D(Zp)),

imposing no homomorphism or Γ-invariance conditions. This involves so much choice thatthe result, at the moment, is essentially complete nonsense.

Step 2 (An integral α−1Up operator). Make sense of the operator α−1Up withintegral coecients, so that

α−1UpΦ ∈ Maps(∆0,D)

is well-dened. (A priori this operator takes values only in α−1D(O), which is strictly largerthan D(O) if α is not a unit).

Where it is well-dened, α−1UpΦ is also a lift of ϕf , since α−1Up xes ϕf :

Φ //_

α−1UpΦ_

ϕ // ϕ.

For any r, s ∈ P1(Q), we then compute the rst moment to be

(α−1UpΦ)r → s(x) = α−1

p−1∑a=0

Φ

ßr + a

p→ s+ a

p

™(a+ px)

≡ α−1∑

Φ

ßr + a

p→ s+ a

p

™(a) (mod p)

≡ α−1∑

a · ϕßr + a

p→ s+ a

p

™(mod p),

which is well-dened and completely independent of our choice of lift Φ!

Thus: we seemingly started with no information about the rst moment of values of Φ,which was simply an arbitrary set-theoretic lift to aD-valued map. After a single applicationof Up, we have a well-dened value for the rst moment mod p: we have gained one digit ofp-adic precision. The proof makes steps 1 and 2 precise and then extends this observationto higher moments and arbitrary precision.

13.2. Step 1: Passing to integral coecients.

Proposition 13.1. ∆0 = Div0(P1(Q)) is a nitely generated Z[Γ]-module.

Proof. We have two niteness results:

Γ\P1(Q) = cusps(Γ) = [c1], ..., [cm], for a choice of representatives ci ∈ Q, and Γ = 〈γ1, ..., γn〉 is a nitely generated group.

40 CHRIS WILLIAMS

Let [r]− [s] ∈ ∆0. Then we may write r = g · ci and s = g′ · cj , for g, g′ ∈ Γ, as translatesof our chosen cusp representatives. Then we compute that

[r]− [s] =

Å[r]− [ci]

ã+

Å[ci]− [cj ]

ã+

Å[cj ]− [s]

ã= (g − 1)[ci] + ([ci]− [cj ]) + (1− g′)[cj ].

It follows that ∆0 is generated as a Z[Γ]-module by the (still innite) set(1− g)[cj ] : g ∈ Γ, 1 ≤ j ≤ m

∪

([ci]− [cj ]).

But for g1, g2 ∈ Γ, we have relations

(1− g1g2)[ci] = (1− g1)[ci]− g1(1− g2)[ci],

so we may rewrite in terms of our generators of Γ, and ∆0 is generated by the nite set(1− γi)[cj ] : 1 ≤ i ≤ n, 1 ≤ j ≤ m

∪

[ci]− [cj ] : 1 ≤ i < j ≤ m.

Corollary 13.2. Let D be an O[Γ]-module. Then

SymbΓ(D ⊗O L) ∼= SymbΓ(D)⊗O L.

In particular,

SymbΓ(Dk(L)) ∼= SymbΓ(Dk(O))⊗O L,

and similarly for coecients in V∨k .

Proof. Let d1, ..., dn be a nite set of generators for ∆0 as a Z[Γ]-module. Let ϕ ∈SymbΓ(D ⊗O L). Then there exists a scalar c ∈ L such that

cϕ(di) ∈ D

for all i. Then cϕ ∈ SymbΓ(D).

From Stevens' theorem with integral coecients, therefore, we can deduce the stated formby tensoring with L.

13.3. Step 2: The integral α−1Up-operator. From now in, in this proof we willwork with xed k and only with integral coecients. Thus for clarity of notation, for therest of 13 we x

D = Dk(O), V∨ = V∨k (O).

The condition that α−1Up is a well-dened operator on Φ is equivalent to saying that

UpΦ takes values in αD. To ensure this, we pass to a slightly smaller subspace.

Denition 13.3. Let

Vα ..=

ßµ ∈ V∨k (O) : µ(xj) ∈ αp−jO

™⊂ V∨.

Let

Dα ..=

ßµ ∈ Dk(O) : ρ(µ) ∈ Vαk

™⊂ D.

Lemma 13.4. Let µ ∈ Dα. Then

µ

∣∣∣∣Å1 a0 p

ã∈ αD.

Thus the operator α−1Up is well-dened on Maps(∆0,Dα).


Proof. We have

µ

∣∣∣∣Å1 a0 p

ã(xm) = µ((a+ px)m)

=

m∑j=0

Åmj

ãam−jpj · µ(xj).

Recall by Lemma 12.16 that vp(α) ≤ k + 1. If j > k, then vp(pj) ≥ k + 1 ≥ vp(α), and

accordingly we have Åmj

ãam−jpj · µ(xj) ∈ pjO ⊂ αO,

as all the other terms are in O. If j ≤ k, then µ(xj) = ρ(µ)(xj) ∈ αp−jO, andÅmj

ãam−jpj · µ(xj) ∈ pjαp−jO = αO.

Thus every term in the sum dening the mth moment, and hence the mth moment itself, isin αO. But m was arbitrary, which proves that µ|

(1 a0 p

)∈ αD, as required.

The last statement is an immediate corollary.

Lemma 13.5. The modules Vα ⊂ V∨k (O) and Dα ⊂ Dk(O) are stable under the actionof Σ0(p).

Proof. Since ρ is Σ0(p)-stable, it suces to check this for Vα. This is an explicit check.By Remark 12.9, we may at least reduce this check to matrices of the form ( 1 0

c 1 ) and(a b0 d

).

We show this only for the rst type, since the second is similar.

Compute that if µ ∈ Vα, then

µ

∣∣∣∣Å1 0c 1

ã(xj) = µ

[(1 + cx)k−jxj

]=

k∑n=j

Åk − jn− j

ãpn−jµ(xn).

The binomial coecients are p-integral, whilst pn−jµ(xn) ∈ pn−jαp−nO = αp−jO by theassumption that µ ∈ Vα. Thus µ| ( 1 0

c 1 ) (xj) ∈ αp−jO, and µ| ( 1 0c 1 ) ∈ Vα, as required.

In particular we may consider modular symbols with coecients in Vα. Note that Vαk ⊗L ∼= Vk(L) and Dα ⊗ L ∼= Dk(L). Hence after possibly scaling further, we may take ϕf ∈SymbΓ(Vα), and it suces to prove ρ induces an isomorphism

ρ : SymbΓ(Dα)Up=α ∼−→ SymbΓ(Vα)Up=α,

which is the form of the theorem we show.

Remark 13.6. Note that if vp(α) = 0, that is if α is a p-adic unit, then Vα = V∨ andDα = D. Thus in the case of ordinary modular forms we will prove a true integral controltheorem.

13.4. Step 3: Improving with Up. We now encode `improving with Up' via ltrations.In the example above, we started at what will be ltration step 0, and the Up-operator movedus to step 1; we give a denition that formalises `Up moves from step n to n+ 1'.

Denition 13.7. Let π be a uniformiser of O. For N ≥ 0, let

FN (D) =

ßµ ∈ D : µ(xj) ∈ πN−jO

™.

42 CHRIS WILLIAMS

Recalling that D = Dk(O), dene the Nth ltration step to be

FilN (Dα) = FN (D) ∩ ker(D→ V∨k ) ∩ Dα (13.1)

=

ßµ ∈ Dα : µ(xj) ∈ πN−jO∀j, µ(xj) = 0 for 0 ≤ j ≤ k

™.

Denition 13.8. Let AN (Dα) = Dα/FilN (Dα).

We obtain an inverse system (AN ). If α is a p-adic unit, this can be described as

Vαk = A0 ..= Ok+1

↑A1 ..= Ok+1 × Oπ↑A2 ..= Ok+1 × Oπ2 × Oπ↑A3 ..= Ok+1 × Oπ3 × Oπ2 × Oπ↑...↑

Dαk∼= lim←−A

i = Ok+1 ×O ×O ×O × · · · .

More generally, we get the same picture, only Vα is a nite-index subgroup of Ok+1,which remains constant up the system (with all higher moments unchanged).

Lemma 13.9. The system (AN ) is stable under the weight k action of Σ0(p).

Proof. This is equivalent to proving that the ltration steps FilN (Dα) are stable underΣ0(p). It suces to prove that each of the terms in the intersection 13.1 are stable.

Dα is Σ0(p)-stable by Lemma 13.5; ker(Dα → V∨) is Σ0(p)-stable, since the map is equivariant.

It remains to show that FN (D) is stable. This is similar to the proof of Lemma 13.5, andagain we don't give all of the details, checking only for matrices of the form ( 1 0

c 1 ). NoteÅ1 0c 1

ã· xj = xj · (1 + cx)k−j

=

∑m≥0

Åk − jm

ãcmxm+j : j ≤ k∑

m≥0 (−1)mÅj − k +m− 1

m

ãcmxm+j : j > k.

In this calculation, we have cm ∈ pmZp; and if µ ∈ FND, then µ(xm+j) ∈ πN−m−jO bydenition. Thus µ(cmxm+j) ∈ πN−jO and

µ| ( 1 0c 1 ) (xj) ∈ πN−jO,

as required.

The next lemma encodes the key step that Up moves us up the ltration, and marks the(rst and only) point at which we must assume that vp(α) < k + 1.

Lemma 13.10. Suppose vp(α) < k + 1. Let µ ∈ FilNDα. Then

µ

∣∣∣∣Å1 a0 p

ã∈ α · FilN+1Dα.


Proof. We have

µ

∣∣∣∣Å1 a0 p

ã(xm) =

m∑j=0

Åmj

ãam−jpjµ(xj).

As µ ∈ ker(Dα → V∨), terms with j ≤ k vanish. Then

j ≥ k + 1 > vp(α),

by assumption. As pj must be divisible by an integer power of π, we deduce that pj ∈ απO.As µ(xj) ∈ πN−jO, we have

pjµ(xj) ∈ απN+1−jO ⊂ απN+1−mO,

as m ≥ j. Thus µ∣∣∣∣Å1 a

0 p

ã(xm) ∈ λπN+1−mO, which immediately implies the result.

13.5. Explicit lifting. In the example of 13.1, we started with a symbol valued inA0, applied Up, and landed in A1. We now show this holds in general.

Let vp(α) < k + 1, and let ϕN ∈ SymbΓ(AN )Up=α be an α-eigensymbol of Up. DeneϕN+1 ∈ Hom(∆0, A

N+1) as follows:

(1) Choose a lift of ϕN to Φ ∈ Hom(∆0,Dα).

(2) Apply α−1Up.

(3) Reduce modulo FilN+1.

Remark 13.11. Note that lifting to a homorphism is always possible. As ∆0 is acountable torsion-free Z-module, we may choose a Z-basis, and then simply lift ϕN on eachgenerator and extend linearly to a homomorphism. However, even this is not necessary. Onemay lift to an arbitrary set-theoretic map Φ; then [Gre07, Prop. 12] shows that ϕN+1 isautomatically a homomorphism.

Φ ∈0 ''

Hom(∆0,Dα)

α−1Up // Hom(∆0,Dα)

α−1UpΦ

∈

\

vv

Hom(∆0, AN+1)

SymbΓ(AN )Up=α

OO

55

ϕN+1

∈

ϕN

∈

,

55

KK

Figure 13.1. Denition of ϕN+1

Proposition 13.12. The homomorphism ϕN+1, as dened by the above process, is:

independent of all choices, Γ-invariant, a Up-eigensymbol with eigenvalue α.

That is, we have

ϕN+1 ∈ SymbΓ(AN+1(Dα))Up=α.

44 CHRIS WILLIAMS

Proof. (1) To show independence of lift, let Φ,Φ′ be any two lifts to Hom(∆0,Dα).

As both have the same reduction module FilN , we have Φ − Φ′ ∈ FilN . Then by Lemma13.10, we have

α−1Up(Φ− Φ′) ∈ FilN+1.

Thus both α−1UpΦ and α−1UpΦ′ have the same image in AN+1.

(2) To see ϕN+1 is Γ-invariant: let γ ∈ Γ. Since the matrices(

1 a0 p

)are comprised of

representatives of the double coset [Γ(

1 00 p

)Γ], for each a we deduce the existence of γa ∈ Γ

such that

α−1 (Φ|Up) |γ = α−1∑a

ÅΦ|γa

ã ∣∣∣∣Å1 a0 p

ã.

As ϕ is Γ-invariant, Φ|γa is another lift of ϕ. By the same proof as (1) this is

= α−1∑a

Φ

∣∣∣∣Å1 a0 p

ã(mod FilN+1),

so ϕN+1 is Γ-invariant.

(3) To see that ϕN+1 is a Up-eigensymbol, we show it is xed by α−1Up. By denition,we have

α−1Up(ϕN+1) = (α−1Up)

2Φ (mod FilN+1)

= ϕN+1.

Here the last equality follows from (1), since (α−1Up)2Φ is also a lift of ϕN .

Finally, we put all of this together to give Greenberg's proof of Stevens' control theorem.

Proof. (of Stevens' control theorem). We are aiming to prove that ρ restricts to anisomorphism

ρ : SymbΓ(Dα)Up=α ∼−→ SymbΓ(Vα)Up=α.

To see that this map is surjective, let ϕ = ϕ0 be in the target. Iterating Proposition 13.12on ϕ gives elements ϕ1, ϕ2, and so on, until we get an element of the inverse limit

Φ ..= lim←−ϕN ∈ lim←− SymbΓ(AN )Up=α

∼= SymbΓ(Dα)Up=α,

where the last isomorphism is canonical and easily checked. By construction, Φ maps toϕ0 = ϕ under ρ.

It remains to show that the map is injective. Let Φ ∈ ker(ρ) be an α-eigensymbol. Bydenition of ρ, this implies that Φ ∈ SymbΓ(Fil0)Up=α. Using Lemma 13.10, we see that

Φ = α−1UpΦ ∈ SymbΓ(Fil1)Up=α.

Iterating, we deduce that Φ ∈ SymbΓ(FilN ) for all N . But it is easily checked that⋂N

FilN = 0;

indeed, any element µ of this intersection must have µ(xj) ∈ πN−jO for all j and N , whichcan only happen if µ = 0. It follows that Φ = 0, and ρ is injective, completing the proof.

14. The p-adic L-function of a modular form

In this section, we rene Stevens' control theorem to prove that the overconvergent liftsconstructed land in a smaller space of locally analytic distributions, which allow us to eval-uate their values at Dirichlet characters. We use this to construct the p-adic L-functionattached to a modular form, and prove the interpolation property it satises.


14.1. Locally analytic distributions. In Remark 12.21, we used Stevens' controltheorem to construct a distribution µf ∈ D(Zp, L) attached to f . To do this, we lifted themodular symbol ϕf to an overconvergent symbol Φf ∈ SymbΓ(Dk(L)), and then evaluatedat 0 → ∞. We showed that the jth moment of µf computed L(f, j + 1) for 0 ≤ j ≤ k.Unlike the Riemann zeta function, however, without twisting we only see nitely manyL-values. It is thus necessary to consider twisted L-values as well.

Let, then, χ be a Dirichlet character of conductor pn. As usual, we can lift this to acharacter χ on Z×p . But χ /∈ A is not a rigid analytic function on Zp, since it cannot bewritten as a single power series on all of Zp. It is, rather, a locally constant function.

As such, it is not clear that we may evaluate µf at χ at all, as our space A is too small tocontain the functions x 7→ χ(x)xj . As a result, the space D is too big. Fortunately, we mayexhibit more control: we now prove the overconvergent symbol takes values in the (muchsmaller) space of locally analytic distributions.

Denition 14.1. A function ψ : Zp → L is locally analytic of order r if for all b ∈ Zp,the restriction

ψ|b+prZp(x) =∑n≥0

an(b) · (x− b)n

is analytic on b + prZp (that is, can be written as a single convergent power series). Notethat for this to be well-dened, we must have |an(b)|p−n → 0 as n→∞. Write

A[r](L) = A[r](Zp, L) ⊂ C (Zp, L)

for the subspace of such functions.

Denition 14.2. A function ψ : Zp → L is locally analytic if for all b ∈ Zp, there is anopen ball U around b such that the restriction ψ|U is analytic (again, can be written as asingle convergent power series). Write A(L) for the space of locally analytic functions.

Remarks 14.3. (1) Since Zp is compact, if ψ is locally analytic, there must existsome r such that ψ is locally analytic of order r. Thus

A(L) =⋃r≥0

A[r](L)

can be written as a union. Henceforth, we use this description.

(2) Note that A(L) = A[0](L), and that for all r, we have A[r] ⊂ A[r + 1].

(3) Let χ be a Dirichlet character of conductor pr, and lift as usual to a character of Z×p .Then since χ is locally constant on the cosets a+ prZp, we have χ ∈ A[r] ⊂ A.

As before, we really want to consider the dual theory.

Denition 14.4. Dually, let

D[r](L) = Homcts(A[r](L), L)

be the space of locally analytic distributions on Zp of order r, and

D(L) = Homcts(A(L), L)

be the space of locally analytic distributions on Zp.

Remark 14.5. Note that any function ψ in A and A is continuous, and thus hasa Mahler expansion

ψ(x) =∑n≥0

an(ψ)

Åxn

ãas in [RJW17, Thm. 2.8]. In particular, any element of D or D is completely determinedby its values on the binomial polynomials, and thus by its Amice transform

Aµ(T ) =∑n≥0

ñ∫Zp

Åxn

ã· µ(x)

ôTn.

46 CHRIS WILLIAMS

Further, if ψ ∈ C (Zp, L), ner properties of ψ can be detected solely from how fast theMahler coecients of ψ decay. On the dual side, this translates into growth properties forthe coecients of the Amice transform. This is all explained in detail in [Col10, 2].

The inclusions A[r] → A[r + 1] give restriction maps

D[r + 1]→ D[r].

Since the binomial polynomials are elements of A[0], the rst part of this remark makes itclear that these restriction maps must be injective. We thus think of D[r+ 1] as a subset ofD[r].


D(L) = lim←−r

D[r] =⋂r≥0

D[r].

Proof. If µ is an element of the intersection, then it is dened on all A[r] and thus denedon A, so we get a map ∩D[r] → D. It is easy to check that this gives an inverse to thenatural restriction map D → ∩D[r].

These spaces all have natural Σ0(p)-actions as before.

Denition 14.7. The action of Σ0(p) on A[0] byÅa bc d

ã· ψ(x) = (a+ cx)kψ

Åb+ dx

a+ cx

ãextends immediately to A[r] for all r. We get an action on the union A, and dual actions onD[r] and D. As before, when considering these spaces with the weight k action we decorateeach space Ak[r], Ak, Dk[r], Dk with a subscript k.

We thus have the following chain of p-adic analytic spaces:

A = A[0] ⊂ A[1] ⊂ A[2] ⊂ · · · ⊂ A ⊂ C (Zp, L),

D = D[0] ⊃ D[1] ⊃ D[2] ⊃ · · · ⊃ D ⊃ M (Zp, L),

and each of these has a weight k action of Σ0(p).

The following proposition says that Up eigensymbols in SymbΓ(Dk) take values in themuch smaller space of locally analytic distributions. The key observation is that Up takesdistributions that are locally analytic of order r to order r + 1, and thus for a second,completely independent time iterating Up allows us to move up a system.

Proposition 14.8. Let α ∈ L×. There is an isomorphism

SymbΓ(Dk)Up=α ∼= SymbΓ(Dk)Up=α.

(Note the only condition we impose on α is that it is non-zero, that is we are in the caseof nite slope).

Proof. Since D ⊂ D, the inclusion from right to left is obvious.

Claim: The action of(

1 b0 p

)denes a map

A[r + 1] −→ A[r].

Proof of claim: Suppose rst ψ ∈ A[1], that is, it is analytic of form∑n an(b)(x − b)n

on each subset b+ pZp. Let x ∈ Zp. We haveÅ1 b0 p

ã· ψ(x) = ψ(b+ px) =

∑n≥0

an(b)(b+ px)n,

which now converges on all of Zp. Thus(

1 b0 p

)· ψ ∈ A[0].


The general case is proved identically.

Dualising this map, the action of(

1 b0 p

)denes a mapÅ

1 b0 p

ã: D[r] −→ D[r + 1] ⊂ D[r].

Since

Up =

p−1∑b=0

Å1 b0 p

ã,

the Up-operator thus denes a map

Up : SymbΓ(D[r])→ SymbΓ(D[r + 1]).

Now let Φ ∈ SymbΓ(Dk[0]) be a Up eigensymbol with non-zero eigenvalue α. Then

Φ = (α−rUrp )Φ ∈ SymbΓ(Dk[r])

is valued in the smaller space Dk[r] for all r. So

Φ ∈⋂

SymbΓ(Dk[r]) = SymbΓ(D)

is valued in the space D of locally analytic distributions.

Corollary 14.9. If vp(α) < k + 1, the natural restriction map

ρ : SymbΓ(Dk)Up=α ∼−→ SymbΓ(V∨k )Up=α

is an isomorphism.

Remark 14.10. In the proof of the claim, we saw that(

1 b0 p

)· ψ(x) depends only on

the restriction of ψ to b + pZp. More generally, we see in the same way that(

1 b0 pn

)· ψ(x)

depends only on the restriction of ψ to b+ pnZp. In particular, this action denes a mapÅ1 b0 pn

ã: A(b+ pnZp, L)→ A(Zp, L).

When we dualise, we see that the right action of this matrix then denes a mapÅ1 b0 pn

ã: D(Zp, L) −→ D(b+ pnZp, L),

that is, if µ is supported on Zp, then µ|(

1 b0 pn

)is supported on b+ pnZp. This is crucial in

the next section when proving interpolation formulae at Dirichlet characters.

14.2. The p-adic L-function. Let f ∈ Sk+2(Γ) be an eigenform, and supposevp(α) < k + 1, where Upf = αf . Corollary 14.9 gives a unique overconvergent liftΦf ∈ SymbΓ(Dk)Up=α of the classical modular symbol ϕf .

Denition 14.11. Let Lp(f) ..= Φf0→∞|Z×p , the restriction of Φf0→∞ to Z×p .

This is a locally analytic distribution on Z×p . We call Lp(f) the p-adic L-function of f . Ifχ is a Dirichlet character of conductor pn, and j ∈ Z, then let

Lp(f, χ, j + 1) ..=

∫Z×p

χ(x)xj · Lp(f).

We want this to satisfy an interpolation property. We now have the set-up of Figure14.1; lifting ρ, evaluating at 0 → ∞, restricting to Z×p and then integrating χ(x)xj givesLp(f, χ, j+ 1), whilst Evχ,j computes L(f, χ, j+ 1). We want to measure how far away thisdiagram is from being commutative at Φf , that is, to measure the failure of the equality of∫

Z×p

χ(x)xj · Φf0→∞ ∼ Evχ,j ρ(Φf ),

The following proposition shows that they dier only by a power of the eigenvalue αp whichdepends on the conductor of χ.

48 CHRIS WILLIAMS

Φf- ((∈ SymbΓ(Dk(L))

⊂

0→∞, res. to Z×p // D(Z×p , L)

∫χ(x)xj

3 Lp(f)_

SymbΓ(Dk(L))

ρ

QpOO

?

3 Lp(f, χ, j + 1)

ϕf_

OO

44∈ SymbΓ(V∨k (L))

Evχ,j // Qp 3 Cχ,j · L(f, χ, j + 1).

Figure 14.1. Construction of Lp(f)

Proposition 14.12. Let Φ ∈ SymbΓ(Dk) be a Up-eigensymbol with eigenvalue α ∈ L×,and let χ be a Dirichlet character of conductor pn with n ≥ 1. Let 0 ≤ j ≤ k. On Φ, thediagram

SymbΓ(Dk)Φ7→Φ0→∞|

Z×p //

ρ

D(Z×p )

µ7→∫Z×pχ(x)xj ·µ

SymbΓ(V∨k )

α−np ·Evχ,j // Qp

commutes.

Proof. Since we assume n ≥ 1, the function χ(x)xj is already supported on Z×p , and wehave ∫

Z×p

χ(x)xj · Φ0→∞ =

∫Zp

χ(x)xk · Φ0→∞.

As Φ is a Up-eigensymbol with non-zero eigenvalue, this is

= α−np

∫Zp

χ(x)xk · Φ|Unp 0→∞

= α−np∑

a (mod pn)

χ(a)

∫a+pnZp

xk · Φ|Unp 0→∞.

Now recall that

Unp =

pn−1∑b=0

Å1 b0 pn

ã.


We study this term by term. Recall Remark 14.10, where we saw that Φ|(

1 b0 pn

)has support

in b+ pnZp. Explicitly, we have∫a+pnZp

xj · Φ∣∣∣∣Å1 b

0 pn

ã0→∞

= Φ

∣∣∣∣Å1 b0 pn

ã0→∞

Åxj · 1a+pnZp(x)

ã= Φ

ßb

pn→∞

™Å(b+ pnx)j1a+pnZp(b+ pnx)

ã=

ßΦ|(

1 a0 pn

)0→∞(xj) : b = a

0 : b 6= a,

recalling that 1X denotes the indicator function of X. The double sum over a (mod pn) andb (mod pn) thus collapses to give∫

Z×p

χ(x)xj · Φ0→∞ = α−np∑

a (mod pn)

χ(a)Φ

∣∣∣∣Å1 a0 pn

ã(xj).

As we are working with 0 ≤ j ≤ k, we have xj ∈ Vk(L). Thus evaluation at xj factorsthrough ρ, and we may replace Φ with ρ(Φ). But this gives

= α−np∑

a (mod pn)

χ(a)ρ(Φ)

∣∣∣∣Å1 a0 pn

ã(xj)

= α−np Evχ,j(ρ(Φ)),

as required.

The case of n = 0 that is, the case of trivial character is slightly dierent, since unlikeabove, the function xj is not already supported on Z×p . We instead have the following.

Proposition 14.13. Let Φ ∈ SymbΓ(Dk) be a Up-eigensymbol with eigenvalue α ∈ L×,and let j ∈ Z (an arbitrary integer). Then∫

Z×p

xj · Φ0→∞ =

Å1− pj

α

ã∫Zp

xj · Φ0→∞.

In particular, if 0 ≤ j ≤ k and f is as in Denition 14.11, then

Lp(f, j + 1) = −Å

1− pj

αp

ã· j!

(2πi)j+1· L(f, j + 1).

Proof. We exploit the decomposition

Zp = pZp ∪ (1 + pZp) ∪ · · · ∪ (p− 1 + pZp)

= pZp ∪ Z×p .

From Remark 14.10, we know that if ψ is locally analytic on Zp, then(

1 a0 p

)is supported on

a+ pZp.

We now use Up. We have

Φ0→∞ = α−1UpΦ0→∞

= α−1

p−1∑a=0

Φ

∣∣∣∣Å1 a0 p

ã0→∞

= α−1Φ

∣∣∣∣Å1 00 p

ã0→∞+ α−1

p−1∑a=1

Φ

∣∣∣∣Å1 a0 p

ã0→∞ .

50 CHRIS WILLIAMS

The rst term is supported on pZp, and the second on Z×p . We deduce that

Φ0→∞|Z×p = α−1

p−1∑a=1

Φ

∣∣∣∣Å1 a0 p

ã0→∞

= Φ0→∞− α−1Φ

∣∣∣∣Å1 00 p

ã0→∞.

With this support condition in hand, we may now complete the calculation over Zp. At anyfunction ψ ∈ A(Zp, L), we compute that∫

Z×p

ψ(x) · Φ0→∞

=

∫Zp

ψ(x) ·ïΦ0→∞− α−1Φ

∣∣∣∣Å1 00 p

ã0→∞

ò=

∫Zp

Åψ(x)− ψ(px)

α

ã· Φ0→∞,

since(

1 00 p

)xes 0→∞ and sends x 7→ px. At ψ(x) = xj , this pulls out to be the required

factor (1− pj/α).

The nal statement follows immediately from combining this with Remark 12.21.

We nally summarise everything in the following theorem/diagram.

Φf- ((∈ SymbΓ(Dk(L))

⊂

0→∞, res. to Z×p // D(Z×p , L)

∫χ(x)xj

3 Lp(f)_

SymbΓ(Dk(L))

ρ

ϕf

_

OO

44∈ SymbΓ(V∨k (L))

α−np Evχ,j // Qp 3 (∗) · L(f, χ, j + 1).

Figure 14.2. (Commutative) diagram describing Theorem 14.14

Theorem 14.14. Let f ∈ Sk+2(Γ) be an eigenform, and suppose vp(α) < k + 1, whereUpf = αf . Let Lp(f) be the p-adic L-function of f as constructed in Denition 14.11. Letχ be a Dirichlet character of conductor pn, with n ≥ 0, and let 0 ≤ j ≤ k. Then∫

Z×p

χ(x)xj · Lp(f) = −α−np ·Å

1− χ(p)pj

αp

ã· G(χ) · j! · pnj

(2πi)j+1· L(f, χ, j + 1)

Ω±f.

Proof. First consider the case n ≥ 1, in which case χ(p) = 0 and (1 − χ(p)pj/αp) = 1.Since Φf is a Up-eigensymbol with non-zero eigenvalue, we may use Proposition 14.12 tocompute directly that ∫

Z×p

χ(x)xk · Lp(f) ..= Φf0→∞|Z×p (χ(x)xj)

= α−np Evχ,j(ρ(Φf ))

= α−np Evχ,j(ϕf ).


But in Corollary 11.26 we already saw that

Evχ,j(ϕf ) = −G(χ) · j! · pnj

(2πi)j+1· L(f, χ, j + 1)

Ω±f,

from which the result follows immediately.

In the case n = 0, the character χ is trivial, and (1− χ(p)pj/αp) = (1− pj/alphap). Weconclude from Proposition 14.13.

Remark 14.15. Our nal interpolation formula diers slightly from that in [PS11].We translate between the two:

we have a −1 scalar out the front; we have τ(χ) in the numerator, they have τ(χ−1) in the denominator; they have an additional (−1)j+1 in the denominator; they have an additional pn in the numerator.

These are partially explained by the identity τ(χ)τ(χ−1) = χ(−1)cond(χ) = χ(−1)pn. Afteraccounting for this, the two dier only by χ(−1)(−1)j , that is, by the Dirac distribution[−1]. Despite checking all the details, this author was unable to see where this dierencearises especially since they are ostensibly given by the same construction! but it is atleast consistent with what is possible.

In some other places in the literature, a formula is given which diers by only (−1)j .This formulation, and the one presented here, cannot both be true. For example, if ameasure takes the value +1 at all Dirichlet characters x 7→ χ(x), it must also take thevalue +1 at all functions x 7→ χ(x)x, which would not be true of the dierence betweenthese two formulations. This is reected in two contradictory general conjectures due toCoatesPerrin-Riou and FukayaKato, which dier by precisely this issue; the version herematches CoatesPerrin-Riou. I want to make no authoritative statement on which is correct,as though the calculations above have been carefully checked and rechecked, it is inherentlypossible that there remains a sign error somewhere (tedious exercise: conrm or deny this).There is a careful, detailed account of this problem, and a proposed ruling in favour ofCoatesPerrin-Riou, in [F18, IV.3]. (Of course, for elliptic curves only j = 0 arises, forwhich there is no problem).

15. Growth properties

Let f ∈ Sk+2(Γ) be a cuspidal eigenform with Upf = αpf and vp(α) < k+ 1. Then we'veconstructed a p-adic distribution Lp(f) interpolating the algebraic parts of certain specialL-values of f (Theorem 14.14). In this section, we look at the question:

Question 15.1. Is Lp(f) uniquely determined by this interpolation property?

In our setting, we prove that the answer is yes. This requires a deeper investigation intop-adic analysis. So far, we've considered the systems

A = A[0] ⊂ A[1] ⊂ A[2] ⊂ · · · ⊂ A ⊂ · · · C (Zp, L),

D = D[0] ⊃ D[1] ⊃ D[2] ⊃ · · · ⊃ D ⊃ · · · M (Zp, L)

in p-adic analysis. There remains a huge dierence between the space C (Zp, L) of arbi-trary continuous functions on Zp, and the space A(Zp, L) of locally analytic functions, andcorrespondingly between the spaces M (Zp, L) of measures and D of locally analytic distri-butions. In particular, we've seen that a measure has Amice transform in

∑n≥0 cn(µ)Tn ∈

O[[T ]] ⊗O L, and thus has bounded coecients, in the sense that |cn(µ)|p ≤ C for someconstant C.

52 CHRIS WILLIAMS

Exercise 15.2. Show that if µ ∈ D is locally analytic, then Aµ(T ) converges on theopen unit disc; that is,

Aµ(T ) =∑n≥0

cn(µ)zn

converges for all z ∈ Cp with vp(z) > 0.

Whilst obviously every power series with bounded coecients converges on the open unitdisc, the converse is far from true. For example, the power series

∑n≥1

zn

n has unboundedcoecients, but will converge on the open unit disc; and more generally, we could takecoecients cn with |cn| ∼ nh for any h ≥ 0, and the resulting power series will still convergeon the open unit disc.

In this section, we will:

For each h ≥ 0, dene a space Dh living between

D ⊃ D∞ ⊃ · · · ⊃ Dh ⊃ · · · ⊃ D0 = M (Zp, L),

cut out by having growth of order h. We give a criterion for detecting measures inside D; namely, we will have

measures = distributions with growth of order 0

= bounded distributions.

Remark 15.3. In the context of the remarks above, the space Dh will be equal to thespace of locally analytic distributions whose Amice transforms have coecients that growlike |cn| = O(nh). We will not prove this, however; see [Col10, Prop. II.3.1] for a proof (inthe language of valuations). This gives another way of seeing that measures are distributionswith growth of order 0.

15.1. Norms on distributions. Recall that ψ ∈ A[r] if for all b ∈ Zp, we may writethe restriction

ψ|b+prZp(x) =∑n≥0

an(b)(x− b)n

as a convergent power series for all x ∈ b+ prZp. Note that for this power series to convergeis equivalent to an(b)prn → 0 as n→∞; thus the following is well-dened.

Denition 15.4. Dene a norm || · ||r on A[r] by

||ψ||r = supb∈Zp

Åsupn

∣∣an(b) · prn∣∣p

ã.

An equivalent denition (see [Bel12, Prop. III.4.12]) is to pick any set B ⊂ Zp of repre-sentatives of Zp/p

rZp, and take

||ψ||r = supb∈B

Åsupn

∣∣an(b) · prn∣∣p

ã.

Remark 15.5 (The sets Br(Zp) and alternative descriptions)This denition is well-suited to our purposes, but there is a much more conceptual

denition which better explains why this does indeed dene a norm. Dene a subset

Br(Zp) ..=x ∈ Cp : ∃a ∈ Zp, |x− a| ≤ p−r

⊂ Cp.

For r = 0, this is simply the closed unit ball in Cp. For r = 1, this is the disjoint unionof the p closed balls of radius 1/p around the points 0, 1, ..., p − 1 in Zp. Note that each ofthese p balls contains exactly one of the open sets a + pZp in Zp. If ψ ∈ A[1], the powerseries dening ψ|a+pZp immediately extends to give a function on the entire correspondingclosed ball in Cp; and this gives an equality

A[1](Zp, L) = A(B1(Zp), L

)..=analytic functions on B1(Zp) dened over L

,


since such analytic functions on B1(Zp) are given by a separate power series (with coecientsin L) on each of the disjoint components.

More generally, Br(Zp) is the union of pr closed balls in Cp, and there is an equality

A[r](Zp, L) = A(Br(Zp), L

).

One then can prove that || · ||r is a sup norm in the sense that

||ψ||r = supx∈Br(Zp)

|ψ(x)|p.

This picture, which is the approach taken in [PS11], is described for p = 3 in Figure 15.1.

Figure 15.1. The sets Br(Z3) for small r

We get a corresponding dual norm on distributions.

Denition 15.6. Dene a norm

| · |r : D[r](L) −→ L

by

|µ|r ..= supψ∈A[r]

|µ(ψ)|p||ψ||r

.

Remark 15.7. Both A[r] and D[r] are L-Banach spaces with respect to these norms,and the inclusion maps D[r + 1] ⊂ D are compact (or completely continuous). Here, asin the usual functional analysis sense, this means that the image of every bounded set is arelatively compact set (that is, a set with compact closure). See [Col10] for further details.When considering generalisations of overconvergent modular symbols, and of Stevens' controltheorem, this property is extremely important (see, for example, [AS08, Urb11, Han17],which give very general formulations of this theorem; or [BSW19], where such a theorem isproved and used to construct p-adic L-functions in the more general setting of GL(2) overnumber elds).

54 CHRIS WILLIAMS

Since D is contained in D[r] for all r, we obtain an induced family of norms on D. WhilstD = lim←−D[r] is no longer a Banach space, it is a projective limit of Banach spaces withcompact transition maps. Such an object is called a compact Fréchet space.

Let µ ∈ D. SinceA[r] ⊂ A[r + 1],

we have an inequality

|µ|r = supψ∈A[r]

|µ(ψ)|p||ψ||r

≤ supψ∈A[r+1]

|µ(ψ)|p||ψ||r+1

= |µ|r+1.

Here we have implicitly used that if ψ ∈ A[r], then

||ψ||r+1 = supx∈Br+1(Zp)

|ψ(x)|p ≤ supx∈Br(Zp)

|ψ(x)|p = ||ψ||r,

so that the denominator cannot grow.

The growth condition we will impose is a restriction on how much bigger this can be, viaïgrowth order h

ò←→

ïph|µ|r ≥ |µ|r+1

ò.

Denition 15.8. Let µ ∈ D. We say that µ has growth of order h (or is h-admissible)if there exists a constant D such that

||µ||r ≤ Dprh

as r →∞. We write

Dh = Dh(Zp, L) = µ ∈ D(Zp, L) : µ has grwoth of order h ⊂ D.

The distributions that arise from overconvergent eigensymbols have growth equal to theslope at p.

Proposition 15.9. Suppose Φ ∈ SymbΓ(Dk) is an eigensymbol with UpΦ = αpΦ. Leth = vp(α). Then for every r, s ∈ P1(Q), we have

Φr → s ∈ Dh

is a distribution with growth of order h.

Proof. We give a very rough sketch of the proof here. Let µ = Φr → s be a value ofΦ. Recall that the action of the matrices in the Up operator dene mapsÅ

1 b0 p

ã: A[r + 1] −→ A[r].

So, we aim to perform a calculation akin to:

|µ|r+1 ∼ supψ∈A[r+1]

|µ(ψ)|

∼ supψ∈A[r+1]

|α−1p · (µ|Up)(ψ)|

≤ ph supψ∈A[r]

|µ(ψ)|

∼ ph|µ|r.

(Of course, the above is entirely nonsense: we cannot simply apply Up to the distribution µwithout using Φ itself, and there are also denominators to consider. The work in the realproof is in making this strategy make sense on a rigorous level.)


Since Lp(f) is built from Φf0→∞, we thus see that it has growth of order h = vp(αp),and that we have pinned down the p-adic L-function to an even smaller space of distributions:

D = D[0] ⊃ D[1] ⊃ · · · ⊃ D ⊃ Dh ⊃ · · · ⊃ M (Zp, L)

Lp(f).

∈

We conclude by giving the criterion for a distribution to be a measure.

Proposition 15.10. Let µ ∈ D0(Zp, L). Then µ is a measure.

Proof. Recall that a measure on Zp is equivalent to a bounded additive function on opencompact subsets of Zp [RJW17, Rem. 2.2], via the association µ 7→ µ(1X) =

∫Zp

1X · µ.

Let µ ∈ D(Zp, L), and suppose that is has growth of order 0. Then there exists a constantC such that

|µ|r = supψ∈A[r]

|µ(ψ)|p||ψ||r

≤ C

independently of r. Let X = a+ prZp be a basic open set in Zp; then its indicator function1X ∈ A[r], and clearly we have ||1X ||r = 1. We deduce that

|µ(1X)|p ≤ C · ||1X ||r = C

Thus µ(1X) is bounded, so that µ is a measure.

Corollary 15.11. Suppose f is an eigenform that is ordinary at p, that is, vp(αp) = 0.Then

Lp(f) ∈M (Zp, L) ∼= Λ(Zp)⊗O L

is a measure.

15.2. Uniqueness properties. One of the major reasons to introduce the abovegrowth conditions is the following uniqueness theorem, a distribution analogue of [RJW17,Lem. 3.8].

Theorem 15.12. Suppose µ ∈ D has growth of order h. Then µ is uniquely determinedby the values ∫

Zp

χ(x)xj · µ

for all χ of conductor ps and for all 0 ≤ j ≤ h.

Remark 15.13. Via some basic representation theory, one can rewrite the functions1a+psZp : a ∈ (Z/psZ)× as linear combinations of the characters χ : (Z/psZ)× → L×of conductor dviding ps. In particular, knowing the integrals in the theorem is completelyequivalent to knowing the values of the integrals∫

a+psZp

xj · µ

for 0 ≤ j ≤ h, for all a, and for all s ≥ 0. Thus to prove the theorem it suces to provethat µ is uniquely determined by these values; which in turn, is equivalent to knowing thevalues ∫

a+psZp

P (x) · µ

for all polynomials P of degree ≤ h, for all a, and for all s. This is the version of the theoremthat we prove.

56 CHRIS WILLIAMS

Proof. Let µ ∈ D, and let ψ ∈ A[r]. We want to exhibit a sequence (ψs) of locallypolynomial functions of degree ≤ h such that

µ(ψs) −→ µ(ψ) as s −→∞, (15.1)

which would mean the value of µ at the (arbitrary) function ψ is completely determined byits values at the functions ψs.

To construct such a sequence, we dene ψs by truncating the power series representationsdening ψ as an element of A[s]. We may consider

ψ =∑

b (mod pr)

ψ∣∣b+prZp

=∑

b (mod pr+1)

ψ∣∣b+pr+1Zp

...

=∑

b (mod ps)

ψ∣∣b+psZp

,

breaking ψ into pieces modulo pr, pr+1, and more generally modulo ps for any s ≥ r.Fix a compatible system of representatives of Z/ps as s varies; there is an obvious choice,

namely the integers 0, 1, ..., ps − 1. Then on each of the local pieces mod ps, we have a(necessarily unique) power series representation

ψ∣∣b+psZp

(x) =∑n≥0

an(b) · (x− b)n.

We truncate each term in this sum. Fix N = bhc the largest integer less than or equal tothe growth of µ, and dene

ψNs∣∣b+psZp

(x) =

N∑n=0

an(b)(x− b)n.

The term at s of our sequence will then be the sum of these approximations:

ψNs =∑

b (mod ps)

ψNs∣∣b+psZp

.

Thus we have dened a sequence

ψNr , ψNr+1, ...

of functions that are all locally polynomial degree ≤ N . (Note that after truncating, thefunction ψNs will in general only be locally analytic of order s, not of order r). We provethat this sequence satises the property (15.1).

Claim. We have ∣∣∣∣ψ − ψNs ∣∣∣∣s ≤ ÅprpsãN+1

||ψ||r.

Proof. Computing directly, we have∣∣∣∣ψ − ψNS ∣∣∣∣s = sup ps−1b=0

∣∣∣∣∣∣∣∣(ψ − ψNS )∣∣b+psZp

∣∣∣∣∣∣∣∣s

= sup ps−1b=0

∣∣∣∣∣∣∣∣∣∣∣∣ ∑n≥N+1

an(b)(x− b)n∣∣∣∣∣∣∣∣∣∣∣∣s

.


For each b, we have∣∣∣∣∣∣∣∣∣∣∣∣ ∑n≥N+1

an(b)(x− b)n∣∣∣∣∣∣∣∣∣∣∣∣s

= supn≥N+1

Å|an(b)| · p−ns

ã= supn≥N+1

ïÅpr

ps

ãn|an(b)| · p−nr

ò≤Åpr

ps

ãN+1

supn≥N+1

Å|an(b)| · p−nr

ã,

since N + 1 ≤ n. By denition of || · ||r, this is

≤Åpr

ps

ãN+1

||ψ||r,

completing the proof of the claim.

The claim implies that if we allow our norm function to change, any ψ can be arbitrarilyapproximated by polynomial functions of degree ≤ N . The growth condition on µ turns outto be exactly enough to force the sequence µ(ψNs ) to converge to µ(ψ). In particular, bydenition we have

|µ|s = supg∈A[s]

|µ(g)|||g||s

,

and in particular this must be

≥ |µ(ψ − ψNs )|||ψ − ψNs ||s

.

Rearranging, we have

|µ(ψ − ψNs )| ≤ |µ|s · ||ψ − ψNs ||s

≤ |µ|s ·Åpr

ps

ãN+1

· ||ψ||r (by the claim)

≤ Dpsh ·Åpr

ps

ãN+1

· ||ψ||r

for some constant D, since µ has growth of order h,

= D′ · (ps)h−N−1 (for some constant D′)

−→ 0 as s→∞,

since h < N + 1 = bhc+ 1. Thus µ(ψNs )→ µ(ψ), as required.

Remark 15.14. In [Col10], an alternative, but equivalent, approach is given. Colmezdenes the notion of being `continuous of order h' as an intrinisic notion on continuousfunctions. We may denote the space of such functions Ch(Zp, L). Then Dh is the dualof this space. This theorem is then equivalent to the statement that locally polynomialfunctions of degree ≤ h are dense in the space Ch. Note this also gives a converse toProposition 15.10, since the values of any measure are bounded, and hence any measureextends to a locally analytic distribution with growth of order 0.

Remark 15.15. This change of language makes leads to a natural converse of thetheorem. Let µ be an additive function on locally polynomial functions of degree ≤ h,satisfying the growth condition that∣∣∣∣∣

∫b+prZp

Åx− bpr

ãk· µ∣∣∣∣∣ ≤ Cprh,

for some constant C. Then µ extends to a unique distribution µ with growth of order h.See [Bel12, Thm. III.4.31] for a direct proof of this converse without introducing the spacesCh(Zp, L).

Finally, we arrive at the corollary we set out to prove.

58 CHRIS WILLIAMS

Corollary 15.16. If Upf = αpf with vp(αp) < k+ 1, then the p-adic L-function Lp(f)is uniquely determined by the interpolation property of Theorem 14.14.

Proof. We know from Proposition 15.9 that Lp(f) has growth order h = vp(αp). Thusby Theorem 15.12 it is determined uniquely by the values∫

Z×p

χ(x)xj · Lp(f)

for j ≤ h. But h < k + 1, and the interpolation formula gives these values for 0 ≤ j ≤ k,from which we conclude.


References

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60 CHRIS WILLIAMS

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Chris Williams, University of Warwick • E-mail : [email protected]

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