on the conjecture of birch and swinnerton-dyer for abelian ... · birch-swinnerton-dyer and the...

On the conjecture of Birch and Swinnerton-Dyer for Abelian schemes

over higher dimensional bases over finite fields

Timo Keller

August 6, 2014

Abstract

We formulate and prove an analogue of the conjecture of Birch and Swinnerton-Dyer for Abelian schemeswith everywhere good reduction over higher dimensional bases over finite fields.

Keywords: 𝐿-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture; Heights; Etalecohomology, higher regulators, zeta and 𝐿-functions; Brauer groups of schemes

MSC 2010: 11G40, 11G50, 19F27; 14F22

Contents

1 Preface 2

1.1 The main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Structure of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Preliminaries 7

2.1 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 The Brauer and the Tate-Shafarevich group 12

3.1 Higher direct images and the Brauer group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 The weak Neron model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 The Tate-Shafarevich group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4 Relation of the Brauer and Tate-Shafarevich group . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.5 Descent of finiteness of X under generically etale alterations . . . . . . . . . . . . . . . . . . . . 29

3.6 Isogeny invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.7 The generalised Cassels-Tate pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4 The special 𝐿-value in cohomological terms 32

4.1 The 𝐿-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2 Cohomological and height pairings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2.1 The pairing ⟨·, ·⟩ℓ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2.2 The pairing (·, ·)ℓ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.3 The case of a constant Abelian scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3.1 Heights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.3.2 The special 𝐿-value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

References 631

2 1 PREFACE

1 Preface

Let 𝐾 be a function field and 𝐴/𝐾 be an Abelian variety. By the theorem of Mordell-Weil, the group 𝐴(𝐾) isfinitely generated. Its torsion subgroup is easily computed and if also its rank 𝑟 ∈ N is known, one can calculategenerators of 𝐴(𝐾). Now, the Birch-Swinnerton-Dyer conjecture provides information about the rank byrelating it to the the 𝐿-function of the Abelian variety:

𝐿(𝐴/𝐾, 𝑠) =∏

𝑣 place

𝐿𝑣(𝐴/𝐾, 𝑞−𝑠)−1

with the Euler factors 𝐿𝑣(𝐴/𝐾,𝑇 ) ∈ Z[𝑇 ] given by certain polynomials depending on the number of rationalpoints of the reduction of 𝐴 at the places 𝑣. The 𝐿-function converges and can be continued to the whole complexplane, where it satisfies a functional equation relating 𝐿(𝐴/𝐾, 𝑠) with 𝐿(𝐴/𝐾, 2− 𝑠).

The weak Birch-Swinnerton-Dyer conjecture states that the rank 𝑟 is equal to the vanishing order of the𝐿-series at 𝑠 = 1:

ord𝑠=1

𝐿(𝐴/𝐾, 𝑠) = rk𝐴(𝐾)

The full Birch-Swinnerton-Dyer conjecture further describes the leading Taylor coefficient at 𝑠 = 1 in terms ofglobal data of 𝐴 (“𝐿-series encode local-global principles”). This is due to John Tate in [Tat66b]. Let 𝑋/F𝑞 bethe smooth projective geometrically connected model of 𝐾 and let A /𝑋 be the Neron model of 𝐴/𝐾. For theleading Taylor coefficient at 𝑠 = 1, one should have the formula

lim𝑠→1

𝐿(𝐴/𝐾, 𝑠)

(𝑠− 1)𝑟=|X(𝐴/𝐾)| ·𝑅 ·

∏𝑣 𝑐𝑣

|Tor𝐴(𝐾)| · |Tor𝐴∨(𝐾)|Here,

𝑅 = |det ℎ(·, ·)|is the regulator of the canonical height pairing ℎ : 𝐴(𝐾)×𝐴∨(𝐾)→ R. The factors

𝑐𝑣 =A (𝐾𝑣)/A

0(𝐾𝑣)

are the Tamagawa numbers (𝑐𝑣 = 1 if 𝐴 has good reduction at 𝑣, which is the case for almost all 𝑣), and finally

X(𝐴/𝐾) = ker

⎛⎝H1(𝐾,𝐴)→∏

𝑣 place

H1(𝐾𝑣, 𝐴)

⎞⎠the Tate-Shafarevich group, which is conjecturally finite. It classifies locally trivial 𝐴-torsors. A famous quote ofJohn Tate [Tat74], p. 198 (for the conjecture over Q) is:

“This remarkable conjecture relates the behavior of a function 𝐿 at a point where it is not at present known to bedefined1 to the order of a group X which is not known to be finite!”

From the finiteness of the Tate-Shafarevich group as well as from the equality rk𝐴(𝐾) = ord𝑠=1 𝐿(𝐴/𝐾, 𝑠)one would get algorithms for computing the Mordell-Weil group 𝐴(𝐾).

In his PhD thesis [Mil68], Milne proved the finiteness of the Tate-Shafarevich group and the Birch-Swinnerton-Dyer conjecture for constant Abelian varieties, i. e. those coming from base change from the finite constant field.Work of Peter Schneider [Sch82b] and Werner Bauer [Bau92], which was completed in the article [KT03] ofKazuya Kato and Fabien Trihan in 2003, proved that already the finiteness of one ℓ-primary component (ℓ prime,ℓ = char𝐾 allowed) of the Tate-Shafarevich group of an Abelian variety over a global function field impliesthe Birch-Swinnerton-Dyer conjecture. For these investigations, one considers the unique connected smoothprojective curve 𝐶 with function field 𝐾, as well as the Neron model A /𝐶 of 𝐴/𝐾.

The obvious generalisation of the weak Birch-Swinnerton-Dyer conjecture to higher dimensional functionfields 𝐾 = F𝑞(𝑋), namely that

ord𝑠=1

𝐿(𝐴/𝐾, 𝑠) = rk𝐴(𝐾), (1.0.1)

was already formulated by Tate in [Tat65], p. 104, albeit for the vanishing order at 𝑠 = dim𝑋, which is equivalentto (1.0.1) by the functional equation. For the leading coefficient, there was no conjecture up to now. Note thatthe vanishing order depends only on the generic fibre.

1by now proven for elliptic curves over Q

1 PREFACE 3

1.1 The main results

The aim of this paper is to formulate and prove an analogue of the conjecture of Birch and Swinnerton-Dyer forAbelian schemes with good reduction everywhere over higher dimensional bases over finite fields.

The Neron model

Given an Abelian variety 𝐴 over the generic point of a base scheme 𝑋, one would like to spread it out over thewhole of 𝑋 as an Abelian scheme for technical reasons. It turns out that this is not always possible, e. g. over theintegers SpecZ, there is no non-trivial Abelian scheme at all. But if one drops the condition that the spread outscheme is proper, there is such a model, called the Neron model, satisfying a universal property, called the Neronmapping property, if dim𝑋 = 1: For an Abelian scheme A /𝑋, there is an isomorphism

A∼−−→ 𝑔*𝑔

*A (1.1.1)

on the smooth site of 𝑋. Here 𝑔 : {𝜂} →˓ 𝑋 is the inclusion of the generic point. Intuitively, this means thatsmooth morphisms to the generic fibre can be spread out to the whole of A /𝑋.

In our situation where dim𝑋 ∈ N is arbitrary, we prove that if there is an Abelian scheme A /𝑋, then itsatisfies a weakened version of the universal property alluded to above, called the weak Neron mapping property,in the sense that the isomorphism (1.1.1) only holds on the etale site.

Theorem 1 (The weak Neron model). Let 𝑋 be a regular Noetherian, integral, separated scheme with 𝑔 : {𝜂} →˓ 𝑋the inclusion of the generic point. Let A /𝑋 be an Abelian scheme. Then

A∼−−→ 𝑔*𝑔

*A

as etale sheaves on 𝑋.

The Tate-Shafarevich and the Brauer group

In the following, let 𝑘 = F𝑞 be a finite field with 𝑞 = 𝑝𝑛 elements, ℓ = 𝑝 a prime, 𝑋/𝑘 a smooth projectivegeometrically connected variety, C /𝑋 a smooth projective relative curve admitting a section and 𝐵/𝑘 an Abelianvariety.

One important invariant of an Abelian scheme that turns up in the conjecture of Birch and Swinnerton-Dyeris the Tate-Shafarevich group. This group classifies (everywhere) locally trivial 𝐴-torsors.

Theorem 2 (The Tate-Shafarevich group). Define the Tate-Shafarevich group of an Abelian scheme A /𝑋 by

X(A /𝑋) := H1et(𝑋,A ).

Denote the function field of 𝑋 by 𝐾, the quotient field of the strict Henselisation of O𝑋,𝑥 by 𝐾𝑛𝑟𝑥 , the inclusion

of the generic point by 𝑗 : {𝜂} →˓ 𝑋 and 𝑗𝑥 : Spec(𝐾𝑛𝑟𝑥 ) →˓ Spec(O𝑠ℎ

𝑋,𝑥) →˓ 𝑋. Then we have

H1(𝑋,A )∼−−→ ker

(H1(𝐾, 𝑗*A )→

∏𝑥∈𝑋

H1(𝐾𝑛𝑟𝑥 , 𝑗*𝑥A )

).

One can replace the product over all points by(a) the closed points

H1(𝑋,A ) = ker

⎛⎝H1(𝐾, 𝑗*A )→⨁𝑥∈|𝑋|


⎞⎠or (b) the codimension-1 points if one disregards the 𝑝-torsion (𝑝 = char 𝑘) (for dim𝑋 ≤ 2, this also holds forthe 𝑝-torsion) and A = Pic0C/𝑋 :

H1(𝑋,A ) = ker

⎛⎝H1(𝐾, 𝑗*A )→⨁

𝑥∈𝑋(1)


⎞⎠ ,

One can also replace 𝐾𝑛𝑟𝑥 by the quotient field of the completion O𝑠ℎ

𝑋,𝑥 in the case of 𝑥 ∈ 𝑋(1).

4 1 PREFACE

Next, we partially generalise the relation between the Birch-Swinnerton-Dyer conjecture and the Artin-Tateconjecture to a higher dimensional basis. The classical Artin-Tate conjecture for a surface 𝑆 over 𝑘 is aboutthe Brauer group of 𝑆 and its invariants like the (rank of the) Neron-Severi group NS(𝑆) and its intersectionpairing.

Conjecture 1 (Artin-Tate conjecture). Let 𝑆/𝑘 be a smooth projective geometrically connected surface. Let𝑃𝑖(𝑆/𝑘, 𝑇 ) be the characteristic polynomial of the geometric Frobenius on H𝑖(𝑆,Qℓ), where the Frobenius acts viafunctoriality on the second factor of 𝑆 = 𝑆 ×𝑘 𝑘. Then the Brauer group Br(𝑆) is finite and

𝑃2(𝑆, 𝑞−𝑠) ∼ |Br(𝑆)| |det(𝐷𝑖.𝐷𝑗)|𝑞𝛼(𝑆) |Tor NS(𝑆)|2

(1− 𝑞1−𝑠)𝜌(𝑆) for 𝑠→ 1,

where

𝛼(𝑆) = 𝜒(𝑆,O𝑆)− 1 + dimPic0(𝑆),

NS(𝑆) is the Neron-Severi group of 𝑆, 𝜌(𝑆) = rk NS(𝑆) and (𝐷𝑖)1≤𝑖≤𝜌(𝑆) is a base for NS(𝑆) mod torsion. Thesymbol (𝐷𝑖.𝐷𝑗) denotes the total intersection multiplicity of 𝐷𝑖 and 𝐷𝑗.

Conjecture (d) in [Tat66b], p. 306–13, concerns a surface 𝑆 which is a relative curve C /𝑋 over a curve 𝑋over a finite field. It states the equivalence of the Birch-Swinnerton-Dyer conjecture for the Jacobian of thegeneric fibre of C /𝑋 and the Artin-Tate conjecture for C : The rank of the Mordell-Weil group should be relatedto the rank of the Neron-Severi group of the surface, the order of the Tate-Shafarevich group to the order of theBrauer group of C , and the height pairing to the intersection pairing on NS(C ). For the equivalence of the fullBirch-Swinnerton-Dyer and the Artin-Tate conjecture for the base a curve, see Gordon’s PhD thesis [Gor79].

Let 𝑋 be smooth projective over 𝑘 of arbitrary dimension. We prove the following partial generalisation,concerning the finiteness part of the conjecture.

Theorem 3 (The Artin-Tate and the Birch-Swinnerton-Dyer conjecture). Let 𝜋 : C → 𝑋 be a smooth projectiverelative curve over a regular variety 𝑋/𝑘. The finiteness of the (ℓ-torsion of the) Brauer group of C is equivalentto the finiteness of the (ℓ-torsion of the) Brauer group of the base 𝑋 and the finiteness of the (ℓ-torsion of the)Tate-Shafarevich group of PicC/𝑋 : One has an exact sequence

0→ 𝐾2 → Br(𝑋)𝜋*

→ Br(C )→X(PicC/𝑋/𝑋)→ 𝐾3 → 0,

where the groups 𝐾𝑖 are annihilated by 𝛿, the index of the generic fibre 𝐶/𝐾, e. g. 𝛿 = 1 if C /𝑋 has a section,and their prime-to-𝑝 part finite, and 𝐾𝑖 = 0 if 𝜋 has a section. Here, X(PicC/𝑋/𝑋) sits in a short exactsequence

0→ Z/𝑑→X(Pic0C/𝑋/𝑋)→X(PicC/𝑋/𝑋)→ 0,

where 𝑑 | 𝛿.

The conjecture of Birch and Swinnerton-Dyer

Now we come to our first statement on the conjecture of Birch and Swinnerton-Dyer for Abelian schemesA /𝑋 over higher dimensional schemes 𝑋/F𝑞. The proof we give is a generalisation of the methods of PeterSchneider [Sch82a] and [Sch82b] which assume dim𝑋 = 1. The theorem relates the vanishing order of a certain𝐿-function at 𝑠 = 1 to the Mordell-Weil rank, and the leading Taylor coefficient at 𝑠 = 1 to a product ofregulators of certain cohomological pairings, the order of the ℓ-primary component of the Tate-Shafarevich groupand the torsion subgroup, for each single prime ℓ = 𝑝. A main problem was to find the “correct” definitionof the 𝐿-function: One has to throw out the factors coming from dimension > 1 since these cause additionalcohomological terms in the special 𝐿-value which cannot be identified in terms of geometric invariants of theAbelian scheme.

Conjecture 2 (The conjecture of Birch and Swinnerton-Dyer for Abelian schemes over higher-dimensionalbases, cohomological version). Set �� = 𝑋 ×𝑘 𝑘 and let ℓ = char 𝑘 be prime. Define the 𝐿-function of the Abelianscheme 𝜋 : A → 𝑋 by

𝐿(A /𝑋, 𝑠) =𝑃1(A /𝑋, 𝑞−𝑠)

𝑃0(A /𝑋, 𝑞−𝑠)

1 PREFACE 5

where𝑃𝑖(A /𝑋, 𝑡) = det(1− Frob−1

𝑞 𝑡 | H𝑖(��,R1𝜋*Qℓ)).

Let 𝜌 be the vanishing order of 𝐿(A /𝑋, 𝑠) at 𝑠 = 1 and define the leading coefficient 𝑐 = 𝐿*(A /𝑋, 1) of𝐿(A /𝑋, 𝑠) at 𝑠 = 1 by

𝐿(A /𝑋, 𝑠) ∼ 𝑐 · (log 𝑞)𝜌(𝑠− 1)𝜌 for 𝑠→ 1.

Then X(A /𝑋)[ℓ∞] is finite, and one has the equality for the leading Taylor coefficient with the regulator𝑅ℓ(A /𝑋), the determinant of the cohomological pairing ⟨·, ·⟩ℓ, which is below shown to equal the Neron-Tatecanonical height pairing,

|𝑐|−1ℓ =

|X(A /𝑋)[ℓ∞]| ·𝑅ℓ(A /𝑋)

|Tor A (𝑋)[ℓ∞]| ·H2(��, 𝑇ℓA )Γ

where A (𝑋) = 𝐴(𝐾) with 𝐴 the generic fibre of A /𝑋 and 𝐾 = 𝑘(𝑋) the function field of 𝑋.

Our conditional result is:

Theorem 4 (The conjecture of Birch and Swinnerton-Dyer for Abelian schemes over higher-dimensional bases,cohomological version). In the situation of Conjecture 2, one has 𝜌 ≥ rkZ A (𝑋) and the following statementsare equivalent:

(a) 𝜌 = rkZ A (𝑋)(b) X(A /𝑋)[ℓ∞] is finiteIf these hold, Conjecture 2 is true for all ℓ = char 𝑘 and X(A /𝑋)[non-𝑝] is finite.

For example, the conjecture is true if 𝐿(A /𝑋, 1) = 0 since one then has 0 = 𝜌 ≥ rkZ A (𝑋) ≥ 0.We can improve this result by replacing the cohomological pairing with a height pairing, and with a trace

pairing in the case of constant Abelian schemes, i. e. A = 𝐵 ×𝑘 𝑋 for 𝐵/𝑘 an Abelian variety.

Theorem 5 (The cohomological, height and trace pairing). Let 𝑋/𝑘 be a smooth projective geometricallyconnected variety with Albanese 𝐴 such that Pic𝑋/𝑘 is reduced. Denote the constant Abelian scheme 𝐵 ×𝑘 𝑋/𝑋by A /𝑋. Then the trace pairing

Hom𝑘(𝐴,𝐵)×Hom𝑘(𝐵,𝐴)∘→ End(𝐴)

Tr→ Z

tensored with Zℓ equals the cohomological pairing

⟨·, ·⟩ℓ : H1(𝑋,𝑇ℓA )Tors ×H2𝑑−1(𝑋,𝑇ℓ(A∨)(𝑑− 1))Tors

∪→ H2𝑑(𝑋,Zℓ(𝑑))pr*1→ H2𝑑(��,Zℓ(𝑑)) = Zℓ.

If 𝑋/𝑘 is a curve, this equals the following height pairing

𝛾(𝛼) : 𝑋𝜙→ 𝐴

𝛼→ 𝐵,

𝛾′(𝛽) : 𝑋𝜙→ 𝐴

𝑐→ 𝐴∨ 𝛽∨

→ 𝐵∨,

(𝛾(𝛼), 𝛾′(𝛽))ℎ𝑡 = deg𝑋(−(𝛼𝜙, 𝛽∨𝑐𝜙)*P𝐵),

where 𝜙 : 𝑋 → 𝐴 is the Abel-Jacobi map associated to a rational point of 𝑋 and 𝑐 : 𝐴∼−−→ 𝐴 the canonical

principal polarisation associated to the theta divisor, and this is equal to the usual Neron-Tate canonical heightpairing.

For general 𝑋/𝑘, ⟨·, ·⟩ℓ equals the Neron-Tate canonical height pairing from section 4.2.1 and

(·, ·)ℓ : H2(𝑋,𝑇ℓA )Tors ×H2𝑑−1(𝑋,𝑇ℓ(A∨)(𝑑− 1))Tors

∪→ H2𝑑+1(𝑋,Zℓ(𝑑)) = Zℓ.

has determinant 1.

Finally, building upon work of Milne [Mil68], we give a full proof of the conjecture of Birch and Swinnerton-Dyer for constant Abelian schemes over higher dimensional bases with mild restrictions which works for allprimes, including the characteristic, at once. Again, one important step was to find the “correct” definition of the𝐿-function of a constant Abelian scheme. As in the first theorem on the Birch-Swinnerton-Dyer conjecture, onehas to throw out the factors coming from dimension > 1 of the base 𝑋.

6 1 PREFACE

Theorem 6 (The conjecture of Birch and Swinnerton-Dyer for constant Abelian schemes over higher-dimensionalbases, version with the height pairing). Assume �� = 𝑋 ×𝑘 𝑘 satisfies

(a) the Neron-Severi group of �� is torsion-free;

(b) the dimension of H1Zar(��,O��) as a vector space over 𝑘 equals the dimension of the Albanese of ��/𝑘.

Let A = 𝐵×𝑘𝑋. Define the 𝐿-function of the constant Abelian scheme A /𝑋 as the 𝐿-function of the motive

ℎ1(𝐵)⊗ ℎ≤1(𝑋) = ℎ1(𝐵)⊗ (ℎ0(𝑋)⊕ ℎ1(𝑋)) = ℎ1(𝐵)⊕ (ℎ1(𝐵)⊗ ℎ1(𝑋)),

namely

𝐿(𝐵 ×𝑘 𝑋/𝑋, 𝑠) =𝐿(ℎ1(𝐵)⊗ ℎ1(𝑋), 𝑠)

𝐿(ℎ1(𝐵), 𝑠)

with 𝐿(ℎ𝑖(𝑋)⊗ ℎ1(𝐵), 𝑡) = det(1− Frob−1𝑞 𝑡 | H𝑖(��,Qℓ)⊗ H1(��,Qℓ)). Let 𝑑 = dim𝐵 and 𝑔 = dim Alb(𝑋) and

𝑅(𝐵) the determinant of the above height pairing. Then one has A (𝑋) = 𝐴(𝐾) with 𝐴 the generic fibre of Aand 𝐾 = 𝑘(𝑋) the function field, and the following holds:

1. The Tate-Shafarevich group X(A /𝑋) is finite.

2. The vanishing order equals the Mordell-Weil rank 𝑟: ord𝑠=1

𝐿(A /𝑋, 𝑠) = rk𝐴(𝐾).

3. There is the equality for the leading coefficient

𝐿*(A /𝑋, 1) = 𝑞(𝑔−1)𝑑 · (log 𝑞)𝑟|X(A /𝑋)| ·𝑅(𝐵)

|Tor𝐴(𝐾))|.

For the question when (a) and (b) hold, see Theorem 4.3.10, Remark 4.3.11, Example 4.3.12 and Example 4.3.25.

For a constant Abelian variety A = 𝐵 ×𝑘 𝑋/𝑋, the two definitions of the 𝐿-function in Theorem 4 andTheorem 6 agree. For a motivation for the definitions of the 𝐿-functions, see Remark 4.3.31 below.

Combining the two results on the conjecture of Birch and Swinnerton-Dyer, one can identify the remainingtwo expressions in Theorem 4 under the assumptions of Theorem 6: One has

|det(·, ·)ℓ|−1ℓ = 1,

H2(��, 𝑇ℓA )Γ

= 1,

⟨·, ·⟩ℓ and (·, ·)ℓ are non-degenerate and X(A /𝑋)[ℓ∞] is finite.

1.2 Structure of the paper

In section 2, we collect some basic and technical facts.

Section 3.1 is concerned with the proof of a cohomological vanishing theorem R𝑞𝜋*G𝑚 = 0 for 𝑞 > 1. Insection 3.2, we introduce the notion of a weak Neron model and prove that an Abelian scheme is a weak Neronmodel of its generic fibre, see Theorem 1. The rest of section 3 is about the Tate-Shafarevich group: Section 3.3contains the definition of and theorems on the Tate-Shafarevich group in the higher dimensional basis case,see Theorem 2. In subsection 3.4, we relate the Tate-Shafarevich and the Brauer group, see Theorem 3. Weprove that the finiteness of the Tate-Shafarevich group descends under etale ℓ′-alterations in section 3.5 and theisogeny invariance of the finiteness of the Tate-Shafarevich groups under etale isogenies in section 3.6. In thefinal section 3.7, we construct the Cassels-Tate pairing.

Section 4 is about the Birch-Swinnerton-Dyer conjecture: In section 4.1, we define the 𝐿-function of an Abelianscheme and state a cohomological form of a Birch-Swinnerton-Dyer conjecture and give a criterion under whichconditions this theorem holds, see Theorem 4. In section 4.2, we study the height pairings: ⟨·, ·⟩ℓ in section 4.2.1and (·, ·)ℓ in section 4.2.2, see one part of Theorem 5. We specialise to constant Abelian schemes in section 4.3.The height pairing, see another part of Theorem 5, is treated in section 4.3.1, and the Birch-Swinnerton-Dyerconjecture for constant Abelian schemes, Theorem 6, in section 4.3.2.

2 PRELIMINARIES 7

1.3 Acknowledgements

Ich danke: meinem Doktorvater Uwe Jannsen fur jegliche Unterstutzung, die er mir hat zuteilwerden lassen; furviele hilfreiche Gesprache und Hinweise Brian Conrad, Patrick Forre, Walter Gubler, Armin Holschbach, PeterJossen, Moritz Kerz, Klaus Kunnemann, Niko Naumann, Maximilian Niklas, Tobias Sitte, Johannes Sprang,Jakob Stix, Tamas Szamuely, Georg Tamme und, von mathoverflow, abx, Angelo, anon, Martin Bright, KestutisCesnavicius, Torsten Ekedahl, Laurent Moret-Bailly, ulrich und xuhan; fur das Probelesen Patrick Forre, PeterJossen, Niko Naumann und Antonella Perucca; fur die angenehme Arbeitsatmosphare allen Mitgliedern derFakultat fur Mathematik Regensburg; fur die Unterstutzung zu Schulzeiten Gunter Malle und Arno Speicher;fur hilfreiche Ratschlage Jurgen Braun; der Studienstiftung des deutschen Volkes fur die finanzielle und ideelleForderung; schließlich meiner Familie dafur, dass ich mich zuhause immer wohlfuhlen konnte.

2 Preliminaries

Notation. Let 𝐴 be an Abelian group. Let Tor𝐴 be the torsion subgroup of 𝐴, 𝐴Tors = 𝐴/Tor𝐴. Let Div𝐴

be the maximal divisible subgroup of 𝐴 and 𝐴Div = 𝐴/Div𝐴. Denote the cokernel of 𝐴𝑛→ 𝐴 by 𝐴/𝑛 and its

kernel by 𝐴[𝑛], and the 𝑝-primary subgroup lim−→𝑛𝐴[𝑝𝑛] by 𝐴[𝑝∞].

Canonical isomorphisms are often denoted by “=”.If not stated otherwise, all cohomology groups are taken with respect to the etale topology.We denote Pontryagin duality, duals of 𝑅-modules or ℓ-adic sheaves and Abelian schemes and Cartier duals

by (−)∨. It should be clear from the context which one is meant.The Henselisation of a (local) ring 𝐴 is denoted by 𝐴ℎ and the strict Henselisation by 𝐴𝑠ℎ.The ℓ-adic valuation | · |ℓ is taken to be normalised by |ℓ|ℓ = ℓ−1.

2.1 Algebra

Lemma 2.1.1. Given a spectral sequence 𝐸𝑝,𝑞2 ⇒ 𝐸𝑛, one has an exact sequence

0→ 𝐸1,02 → 𝐸1 → 𝐸0,1

2 → 𝐸2,02 → ker(𝐸2 → 𝐸0,2

2 )→ 𝐸1,12 → 𝐸3,0

2 .

Proof. See [NSW00], p. 81, (2.1.3) Proposition.

Lemma 2.1.2. Let 𝐴,𝐵,𝐶 be Abelian groups and 𝑓 : 𝐴→ 𝐵, 𝑔 : 𝐵 → 𝐶 be homomorphisms. Then the sequence

0→ ker(𝑓)→ ker(𝑔𝑓)→ ker(𝑔)→ coker(𝑓)→ coker(𝑔𝑓)→ coker(𝑔)→ 0

is exact.

Proof. Apply the snake lemma to the commutative diagram with exact rows

𝐴𝑓//

𝑔𝑓

��

𝐵 //

𝑔

��

coker(𝑓) //

��

0

0 // 𝐶id // 𝐶 // 0.

Lemma 2.1.3. The tensor product of an Abelian torsion group 𝐴 with a divisible Abelian group 𝐵 is trivial.

Proof. Take an elementary tensor 𝑎⊗ 𝑏. There is some 𝑛 > 0 such that 𝑛𝑎 = 0, so, by divisibility of 𝐵 there issome 𝑏′ such that 𝑛𝑏′ = 𝑏, so 𝑎⊗ 𝑏 = 𝑎⊗ 𝑛𝑏′ = 𝑛𝑎⊗ 𝑏′ = 0⊗ 𝑏′ = 0.

Lemma 2.1.4. Let 𝐴 be an Abelian ℓ-primary torsion group such that 𝐴[ℓ] is finite. Then 𝐴 is a cofinitelygenerated Zℓ-module.

Proof. Equip 𝐴 with the discrete topology. Applying Pontryagin duality to 0→ 𝐴[ℓ]→ 𝐴ℓ→ 𝐴 gives us that

𝐴∨/ℓ is finite, hence by [NSW00], p. 179, (3.9.1) Proposition (𝐴∨ being profinite as a dual of a discrete torsiongroup), 𝐴∨ is a finitely generated Zℓ-module, hence 𝐴 a cofinitely generated Zℓ-module.

8 2 PRELIMINARIES

Definition 2.1.5. Let 𝐴 be an Abelian group and ℓ a prime number. Then the ℓ-adic Tate module 𝑇ℓ𝐴 of 𝐴is the projective limit

𝑇ℓ𝐴 = lim←−(. . .

ℓ→ 𝐴[ℓ𝑛+1]ℓ→ 𝐴[ℓ𝑛]

ℓ→ . . .ℓ→ 𝐴[ℓ]→ 0

).

The rationalised ℓ-adic Tate module is defined as 𝑉ℓ𝐴 = 𝑇ℓ𝐴⊗ZℓQℓ.

Lemma 2.1.6. One has 𝑇ℓ𝐴 = Hom(Qℓ/Zℓ, 𝐴).

Proof. One has Hom(Qℓ/Zℓ, 𝐴) = Hom(lim−→𝑛1ℓ𝑛Z/Z, 𝐴) = lim←−𝑛 Hom( 1

ℓ𝑛Z/Z, 𝐴) = lim←−𝑛𝐴[ℓ𝑛].

Lemma 2.1.7. Let 𝐴 be a finite Abelian group. Then 𝑇ℓ𝐴 is trivial.

Proof. There is an 𝑛0 such that 𝐴[ℓ𝑛] is stationary for 𝑛 ≥ 𝑛0, i. e. ℓ𝑛0𝐴 = 0, so there is no non-zero infinitesequence (. . . , 𝑎𝑛, 𝑎𝑛−1, . . . , 𝑎0) with ℓ𝑎𝑖+1 = 𝑎𝑖 since no non-zero element of 𝐴 is infinitely ℓ-divisible.

Lemma 2.1.8. Let 𝐴 be a non-finite cofinitely generated Zℓ-module. Then 𝑇ℓ𝐴 is a non-trivial Zℓ-module.

Proof. Since 𝐴 is a cofinitely generated Zℓ-module, 𝐴 ∼= 𝐵 ⊕ (Qℓ/Zℓ)𝑛 with 𝐵 finite, so 𝑇ℓ𝐴 ∼= Z𝑛ℓ . As 𝐴 is not

finite, 𝑛 > 0.

Lemma 2.1.9. Let 𝐴 be an Abelian group. Then 𝑇ℓ𝐴 is torsion free.

Proof. Let 𝑎 = (. . . , 𝑎𝑚, . . . , 𝑎1, 𝑎0) ∈ 𝑇ℓ𝐴 with ℓ𝑛𝑎 = 0. Then there is a 𝑛0 ∈ N minimal such that 𝑎𝑛0= 0.

Denote the order of 𝑎𝑛0by ℓ𝑚, 𝑚 > 0. If there is an 𝑖 > 0 such that ord(𝑎𝑛0+𝑖) < ℓ𝑚+𝑖, then 0 = ℓ𝑚+𝑖−1𝑎𝑛0+𝑖 =

ℓ𝑚+𝑖−2𝑎𝑛0+𝑖−1 = . . . = ℓ𝑚−1𝑎𝑛0, contradiction. Hence for 𝑖≫ 0, we have ℓ𝑛+1 | ord(𝑎𝑛0+𝑖) | ord(𝑎), contradiction

to ℓ𝑛𝑎 = 0.

Remark 2.1.10. Note that, in contrast, for an ℓ-adic sheaf F𝑛, lim←−𝑛 H𝑖(𝑋,F𝑛) need not be torsion-free.

Definition 2.1.11. For a profinite group 𝐺, a 𝐺-module 𝑀 is discrete iff

𝑀 = lim−→𝑈

𝑀𝑈

for 𝑈 running through the open normal subgroups of 𝐺.

Lemma 2.1.12. Let 𝐺 be a profinite group and 𝑀 a discrete 𝐺-module. Then H𝑞(𝐺,𝑀) is torsion for 𝑞 > 0.In particular, Galois cohomology is torsion in positive degrees.

Proof. We have an isomorphismlim−→𝑈

H𝑞(𝐺/𝑈,𝑀𝑈 )∼−−→ H𝑞(𝐺,𝑀),

the limit taken over the open normal subgroups 𝑈 of 𝐺. As H𝑞(𝐺/𝑈,𝑀𝑈 ) is torsion for 𝑞 > 0 because it is killedby |𝐺/𝑈 | <∞, the result follows.

Definition 2.1.13. A homomorphism of Zℓ-modules 𝑓 : 𝐴→ 𝐵 is called a quasi-isomorphism if ker(𝑓) andcoker(𝑓) are finite. In this case, define

𝑞(𝑓) =

|coker(𝑓)||ker(𝑓)|

ℓ

.

Lemma 2.1.14. In the situation of the previous definition, one has:

1. Assume 𝐴, 𝐵 are finitely generated Zℓ-modules of the same rank with bases (𝑎𝑖)𝑛𝑖=1 and (𝑏𝑖)

𝑛𝑖=1 and

𝑓(𝑎𝑖) =∑𝑗 𝑧𝑖𝑗𝑏𝑗 modulo torsion. Then 𝑓 is a quasi-isomorphism iff det(𝑧𝑖𝑗) = 0. In this case,

𝑞(𝑓) =

det(𝑧𝑖𝑗) ·

|Tor𝐵||Tor𝐴|

ℓ

.

2. Assume given 𝑓 : 𝐴→ 𝐵 and 𝑔 : 𝐵 → 𝐶. If two of 𝑓, 𝑔, 𝑔𝑓 are quasi-isomorphism, so is the third, and𝑞(𝑔𝑓) = 𝑞(𝑔) · 𝑞(𝑓).

3. For the Pontrjagin dual 𝑓∨ : 𝐵∨ → 𝐴∨, 𝑓 is a quasi-isomorphism iff 𝑓∨ is, and then 𝑞(𝑓) · 𝑞(𝑓∨) = 1.

2 PRELIMINARIES 9

4. Suppose 𝜗 is an endomorphism of a finitely generated Zℓ-module 𝐴. Let 𝑓 be the homomorphism ker(𝜗)→coker(𝜗) induced by the identity. Then 𝑓 is a quasi-isomorphism iff det(𝑇 − 𝜗Q) = 𝑇 𝜌𝑅(𝑇 ) with𝜌 = rkZℓ

(ker(𝜗)) and 𝑅(0) = 0. In this case, 𝑞(𝑓) = |𝑅(0)|ℓ.

Proof. See [Tat66b], p. 306-19–306-20, Lemma z.1–z.4.

Lemma 2.1.15. One has a canonical isomorphism

Z𝑝∼−−→ Hom(Z𝑝,Z𝑝), 𝑎 ↦→ (𝑥 ↦→ 𝑎𝑥)

of Abelian groups.

Proof. Injectivity is trivial. Surjectivity: We first prove that any homomorphism 𝑓 : Z𝑝 → Z𝑝 is continuous forthe 𝑝-adic topology. This is true since any such homomorphism maps 𝑝𝑛Z𝑝 to itself, so 𝑓−1(𝑝𝑛Z𝑝) ⊇ 𝑝𝑛Z𝑝 isopen. Hence 𝑓 is determined by 𝑓(1) ∈ Z𝑝.

2.2 Geometry

Theorem 2.2.1. Let 𝐴 be a Henselian local ring, 𝑆 = Spec(𝐴) with closed point 𝑠0, 𝑓 : 𝑋 → 𝑆 separated and offinite presentation, and 𝑋0 := 𝑓−1(𝑠0) of dimension ≤ 1. Then for every closed subscheme 𝑋 ′

0 of 𝑋 with thesame underlying space as 𝑋0 and of finite presentation over 𝑆, the canonical homomorphism Pic(𝑋)→ Pic(𝑋 ′

0)is surjective.

Proof. See [EGAIV4], p. 288, Corollaire (21.9.12).

Theorem 2.2.2. Let 𝐴 be an adic Noetherian ring, 𝑌 = Spec(𝐴) with I a defining ideal, 𝑌 ′ = 𝑉 (I ),𝑓 : 𝑋 → 𝑌 a separated morphism of finite type, 𝑋 ′ = 𝑓−1(𝑌 ′). Let 𝑌 = 𝑌/𝑌 ′ = Spf(𝐴), �� = 𝑋/𝑋′ the

completions of 𝑌 and 𝑋 along 𝑌 ′ and 𝑋 ′, 𝑓 : �� → 𝑌 the extension of 𝑓 to the completions. Then the functorF F/𝑋′ = F is an equivalence of categories of coherent O𝑋-modules with proper support on Spec(𝐴) to thecategory of coherent O��-modules with proper support on Spf(𝐴).

Proof. See [EGAIII1], p. 150, Theoreme (5.1.4).

Theorem 2.2.3 (Artin approximation). Let 𝑅 be a field, and let 𝐴 be the Henselisation of an 𝑅-algebra of finitetype at a prime ideal. Let m be a proper ideal of 𝐴. Let 𝐹 be a functor (𝐴−Alg)→ (Set) which is locally offinite presentation. Denote by 𝐴 the m-adic completion of 𝐴. Then 𝐹 (𝐴) = ∅ implies 𝐹 (𝐴) = ∅: Given 𝜉 ∈ 𝐹 (𝐴),for every integer 𝑐, there is a 𝜉 ∈ 𝐹 (𝐴) such that

𝜉 ≡ 𝜉 (mod m𝑐).

Proof. See [Art69], p. 26, Theorem (1.10) resp. Theorem (1.12).

Definition 2.2.4. Let 𝑋,𝑌 be schemes. A projective morphism 𝑋 → 𝑌 is a morphism that factors as aclosed immersion into a (possibly twisted) projective bundle 𝑋 →˓ P(E )→ 𝑌 .

Lemma 2.2.5. Let 𝑓 : 𝑋 → 𝑌 be a smooth projective morphism of locally Noetherian schemes. Then thefollowing are equivalent:

1. One has 𝑓*O𝑋 = O𝑌 (Zariski sheaves).

2. The fibres of 𝑓 are geometrically connected.

If these hold, G𝑚,𝑌∼−−→ 𝑓*G𝑚,𝑋 as Zariski (resp. etale, resp. fppf) sheaves.

Proof. Since 𝑓 is smooth, having geometrically integral fibres is equivalent to having geometrically connectedfibres. Hence:

1 =⇒ 2: See [Liu06], p. 200 f., Theorem 5.3.15/17.2 =⇒ 1: See [Liu06], p. 208, Exercise 5.3.12.If 2 holds, the last statement follows since the fibres of a base change of 𝑓 are also geometrically connected if

the fibres of 𝑓 are so.

10 2 PRELIMINARIES

Lemma 2.2.6. For a morphism of schemes 𝑓 : 𝑋 → 𝑌 and a sheaf F on 𝑋, the edge maps H𝑝(𝑌, 𝑓*F )→H𝑝(𝑋,F ) in the Leray spectral sequence are equal to 𝑓*.

Proof. Choose an injective resolution F → 𝐽∙ and an injective resolution 𝑓*F → 𝐼∙. Apply the exact functor𝑓* to the latter to obtain 𝑓*𝑓*F → 𝑓*𝐼∙, and we have the adjunction composed with the first injectiveresolution 𝑓*𝑓*F → F → 𝐽∙. Since 𝑓*𝐼∙ is exact and 𝐽∙ is injective, one gets a map 𝑓*𝐼∙ → 𝐽∙, and anadjoint map 𝐼∙ → 𝑓*𝐽

∙. Taking global sections H0(𝑌, 𝐼∙)→ H0(𝑌, 𝑓*𝐽∙) and cohomology yields the edge map

H𝑝(𝑌, 𝑓*F )→ H𝑝(𝑋,F ).

Lemma 2.2.7. Let 𝑆 = Spec(𝑅) be the spectrum of a strictly Henselian local ring. Then the global sectionsfunctor on the etale site for Abelian sheaves is exact, so all cohomology groups in positive dimension vanish.

Proof. See [Tam94], p. 124, Lemma (6.2.3).

Now we construct a Leray spectral sequence for etale cohomology with supports.

Theorem 2.2.8. If 𝑖 : 𝑍 →˓ 𝑌 is a closed immersion and 𝜋 : 𝑋 → 𝑌 is a morphism,

𝑍 ′ � � 𝑖′//

��

𝑋

𝜋

��

𝑍� � 𝑖 // 𝑌

there is a 𝐸2-spectral sequence for etale sheaves F

H𝑝𝑍(𝑌,R𝑞𝜋*F )⇒ H𝑝+𝑞

𝑍′ (𝑋,F ),

where 𝑖′ : 𝑍 ′ →˓ 𝑋 is the fibre product pr2 : 𝑍 ×𝑌 𝑋 →˓ 𝑋.

Proof. This is the Grothendieck spectral sequence for the composition of functors generalising the Leray spectralsequence [Mil80], p, 89, Theorem III.1.18 (a)

𝐹 :F ↦→ 𝜋*F

𝐺 :F ↦→ H0𝑍(𝑌,F ),

since

(𝐺𝐹 )(F ) = H0𝑍(𝑌, 𝜋*F )

= ker((𝜋*F )(𝑌 )→ (𝜋*F )(𝑌 ∖ 𝑍))

= ker(F (𝑋)→ F (𝜋−1(𝑌 ∖ 𝑍)))

= ker(F (𝑋)→ F (𝑋 ∖ 𝑍 ′))

= H0𝑍′(𝑋,F ).

We have to check if 𝜋*(−) maps injectives to H0𝑍(𝑌,−)-acyclics. Then [Mil80], p. 309, Theorem B.1 establishes

the existence of the spectral sequence.Injective sheaves I are flabby (defined in [Mil80], p. 87, Example III.1.9 (c)) and 𝜋* maps flabby sheaves

to flabby sheaves ([Mil80], p. 89, Lemma III.1.19). Therefore, it follows from the long exact localisationsequence [Mil80], p. 92, Proposition III.1.25

0→ H0𝑍(𝑌, 𝜋*I )→ H0(𝑌, 𝜋*I )→ H0(𝑌 ∖ 𝑍, 𝜋*I )

→ H1𝑍(𝑌, 𝜋*I )→ H1(𝑌, 𝜋*I )→ H1(𝑌 ∖ 𝑍, 𝜋*I )

→ H2𝑍(𝑌, 𝜋*I )→ H2(𝑌, 𝜋*I )→ H2(𝑌 ∖ 𝑍, 𝜋*I )→ . . .

and H𝑝(𝑌, 𝜋*I ) = 0 = H𝑝(𝑌 ∖ 𝑍, 𝜋*I ) for 𝑝 > 0 that H𝑞𝑍(𝑌, 𝜋*I ) = 0 for 𝑞 > 1. For H1

𝑍(𝑌, 𝜋*I ) = 0, itremains to show that H0(𝑌, 𝜋*I )→ H0(𝑌 ∖ 𝑍, 𝜋*I ) is surjective. For this, setting 𝑗 : 𝑈 = 𝑋 ∖ 𝑍 ′ →˓ 𝑋, applyHom(−,I ) to the exact sequence 0→ 𝑗!O𝑈 → O𝑋 (𝑈 = 𝑋 ∖ 𝑍 ′) and get

I (𝑋) = Hom(O𝑋 ,I )� Hom(𝑗!O𝑈 ,I ) = Hom(O𝑈 ,I |𝑈 ) = I (𝑈),

the arrow being surjective since I is injective.

2 PRELIMINARIES 11

Lemma 2.2.9. Let 𝐼 be a filtered category and (𝑖 ↦→ 𝑋𝑖) a contravariant functor from 𝐼 to schemes over 𝑋.Assume that all schemes are quasi-compact and that the transition maps 𝑋𝑖 ← 𝑋𝑗 are affine. Let 𝑋∞ = lim←−𝑋𝑖,and, for a sheaf F on 𝑋et, let F𝑖 and F∞ be its inverse images on 𝑋𝑖 and 𝑋∞ respectively. Then

lim−→H𝑝((𝑋𝑖)et,F𝑖)∼−−→ H𝑝((𝑋∞)et,F∞).

Assume the 𝑋𝑖 ⊆ 𝑋 are open, the transition morphisms are affine and all schemes are quasi-compact. Let𝑍 →˓ 𝑋 be a closed subscheme. Then

lim−→H𝑝𝑍∩𝑋𝑖

((𝑋𝑖)et,F𝑖)∼−−→ H𝑝

𝑍∩𝑋∞((𝑋∞)et,F∞).

Proof. See [Mil80], p. 88, Lemma III.1.16 for the first statement. The second one follows from the first, the longexact localisation sequence (note that the morphisms (𝑋 ∖ 𝑍) ∩𝑋𝑖 ← (𝑋 ∖ 𝑍) ∩𝑋𝑗 are affine as well since theyare base changes of affine morphisms) and the five lemma.

Now we construct a Mayer-Vietoris sequence for cohomology with supports.

Theorem 2.2.10. Let 𝑌1 and 𝑌1 be closed subschemes of 𝑋 and F a sheaf on 𝑋. Then there is a long exactsequence of cohomology with supports

. . .→ H𝑖𝑌1∩𝑌2

(𝑋,F )→ H𝑖𝑌1

(𝑋,F )⊕H𝑖𝑌2

(𝑋,F )→ H𝑖𝑌1∪𝑌2

(𝑋,F )→ . . .

Proof. Let I be an injective sheaf on 𝑋. Consider the diagram

0

��

0

��

0

��

0 // Γ𝑌1∩𝑌2(𝑋,I )

��

// Γ𝑌1(𝑋,I )⊕ Γ𝑌2

(𝑋,I )

��

// Γ𝑌1∪𝑌2(𝑋,I )

��

// 0

0 // Γ(𝑋,I )

��

// Γ(𝑋,I )⊕ Γ(𝑋,I )

��

// Γ(𝑋,I )

��

// 0

0 // Γ(𝑋 ∖ (𝑌1 ∩ 𝑌2),I )

��

// Γ(𝑋 ∖ 𝑌1,I )⊕ Γ(𝑋 ∖ 𝑌2,I )

��

// Γ(𝑋 ∖ (𝑌1 ∪ 𝑌2),I )

��

// 0

0 0 0

The maps are induced by the restrictions, the two maps into the direct sum have the opposite sign and the mapout of the direct sum is induced by the summation.

Since I is injective, by the same argument as in the proof of Theorem 2.2.8 the columns are exact. Thesecond row is trivially exact and the third row is exact since I is a sheaf and I is injective. Hence by the snakelemma, the first row is exact.

Applying this to an injective resolution 0→ F → I ∙, we get an exact sequence of complexes

0→ Γ𝑌1∩𝑌2(𝑋,I ∙)→ Γ𝑌1(𝑋,I ∙)⊕ Γ𝑌2(𝑋,I ∙)→ Γ𝑌1∪𝑌2(𝑋,I ∙)→ 0

and from this the long exact sequence in the usual way.

Lemma 2.2.11. Let 𝑓 : 𝑋 → 𝑆 be a morphism of schemes. Then 𝑓 is locally of finite presentation iff

Mor𝑆(lim𝑖∈𝐼

𝑇𝑖, 𝑋) = lim−→𝑖∈𝐼 Mor𝑆(𝑇𝑖, 𝑋)

for any directed partially ordered set 𝐼, and any inverse system (𝑇𝑖, 𝑓𝑖𝑖′) of 𝑆-schemes over 𝐼 with each 𝑇𝑖 affine.

Proof. See [EGAIV3], p. 52, Proposition 8.14.2.

Theorem 2.2.12 (Lang-Steinberg). Let 𝑋0/𝑘 be a scheme such that 𝑋0 ×𝑘 𝑘 is an Abelian variety. Then 𝑋0

has a 𝑘-rational point.

12 3 THE BRAUER AND THE TATE-SHAFAREVICH GROUP

Proof. See [Mum70], p. 205, Theorem 3.

Theorem 2.2.13 (Zariski-Nagata purity). Let 𝑋 be a locally notherian regular scheme, 𝑈 →˓ 𝑋 open withclosed complement 𝑍 of codimension ≥ 2. Then the functor 𝑋 ′ ↦→ 𝑋 ′ ×𝑋 𝑈 of the category of etale coverings of𝑋 to the category of etale coverings of 𝑈 is an equivalence of categories.

Proof. See [SGA1], Exp. X, Corollaire 3.3.

Theorem 2.2.14 (Barsotti-Weil formula). Let A /𝑋 be a projective Abelian scheme over a locally Noetherianscheme 𝑋. Then there are isomorphisms of Abelian groups resp. of fpqc sheaves

Ext1𝑋𝑓𝑝𝑝𝑓(A ,G𝑚)

∼−−→ A ∨(𝑋),

E xt1𝑋𝑓𝑝𝑝𝑓(A ,G𝑚)

∼−−→ A ∨

given by(1→ G𝑚 → 𝐺→ A → 0) ↦→ [𝐺] ∈ H1(A ,G𝑚) ⊆ A ∨(𝑋),

i. e. by regarding 𝐺 as a G𝑚-torsor.

Proof. See [Oor66], III.18 – 1, Theorem (18.1).

3 The Brauer and the Tate-Shafarevich group

All cohomology groups are with respect to the etale topology unless stated otherwise.

3.1 Higher direct images and the Brauer group

Lemma 3.1.1. Let 𝑋 be a scheme and ℓ a prime invertible on 𝑋. Then there are exact sequences

0→ H𝑖−1(𝑋,G𝑚)⊗Z Qℓ/Zℓ → H𝑖(𝑋,𝜇ℓ∞)→ H𝑖(𝑋,G𝑚)[ℓ∞]→ 0

for each 𝑖 ≥ 1.

Proof. This follows from the long exact sequence induced by the Kummer sequence (which is exact by theinvertibility of ℓ)

1→ 𝜇ℓ𝑛 → G𝑚ℓ𝑛→ G𝑚 → 1

and passage to the colimit.

Definition 3.1.2. A variety over a field 𝑘 is a separated scheme of finite type over 𝑘.

Recall the definition [Mil80], IV.2, p. 140 ff. of the Brauer group Br(𝑋) of a scheme 𝑋 as the group ofequivalence classes of Azumaya algebras on 𝑋.

Definition 3.1.3. Br′(𝑋) := Tor H2(𝑋,G𝑚) is called the cohomological Brauer group.

Theorem 3.1.4. Br′(𝑋) = H2(𝑋,G𝑚) if 𝑋 is a regular integral quasi-compact scheme.

Proof. See [Mil80], p. 106 f., Example 2.22: We have an injection H2(𝑋,G𝑚) →˓ H2(𝐾,G𝑚) and the latter istorsion as Galois cohomology by Lemma 2.1.12.

Theorem 3.1.5. There is an injection Br(𝑋) →˓ Br′(𝑋), where Br(𝑋) is the Brauer group of 𝑋.

Proof. See [Mil80], p. 142, Theorem 2.5.

Theorem 3.1.6. Let 𝑋 be a scheme endowed with an ample invertible sheaf. Then Br(𝑋) = Br′(𝑋).

Proof. See [dJ].

Corollary 3.1.7. Let 𝑋/𝑘 be a smooth projective geometrically connected variety. Then Br(𝑋) = Br′(𝑋) =H2(𝑋,G𝑚).

3 THE BRAUER AND THE TATE-SHAFAREVICH GROUP 13

Proof. The first equality follows from Theorem 3.1.6 since 𝑋/𝑘 is projective, and the second equality followsfrom Theorem 3.1.4.

Theorem 3.1.8. Let 𝑋 be a smooth projective geometrically connected variety over a finite field 𝑘 = F𝑞, 𝑞 = 𝑝𝑛.(a) H𝑖(𝑋,G𝑚) is torsion for 𝑖 = 1, finite for 𝑖 = 1, 2, 3 and trivial for 𝑖 > 2 dim(𝑋) + 1.(b) For ℓ = 𝑝 and 𝑖 = 2, 3, one has H𝑖(𝑋,G𝑚)[ℓ∞] = (Qℓ/Zℓ)

𝜌𝑖,ℓ ⊕𝐶𝑖,ℓ, where 𝐶𝑖,ℓ is finite and trivial for allbut finitely many ℓ, and 𝜌𝑖,ℓ a non-negative integer.

Proof. See [Lic83], p. 180, Proposition 2.1 a)–c), f).

Corollary 3.1.9. Let 𝑋 be a smooth projective geometrically connected variety over a finite field 𝑘 = F𝑞, 𝑞 = 𝑝𝑛.Let ℓ = 𝑝 be prime. Then one has

H𝑖(𝑋,𝜇ℓ∞)∼−−→ H𝑖(𝑋,G𝑚)[ℓ∞]

for 𝑖 = 2.

Proof. By Lemma 2.1.3, this follows from Lemma 3.1.1 and Theorem 3.1.8 since H𝑖−1(𝑋,G𝑚)⊗Z Qℓ/Zℓ is thetensor product of a torsion group (for 𝑖 = 2) with a divisible group (Lemma 2.1.3).

Lemma 3.1.10. Let 𝑋 be a quasi-compact scheme, quasi-projective over an affine scheme. Assume 𝐴 ∈H1(𝑋,PGL𝑛) is an Azumaya algebra trivialised by 𝐴 ∼= End(𝑉 ) with 𝑉 ∈ H1(𝑋,GL𝑛) a locally free sheaf of rank𝑛. Then every other such 𝑉 ′ differs from 𝑉 by tensoring with an invertible sheaf.

Proof. Consider for 𝑛 ∈ N the central extension of etale sheaves on 𝑋 (see [Mil80], p. 146)

1→ G𝑚 → GL𝑛 → PGL𝑛 → 1.

By [Mil80], p. 143, Step 3, this induces a long exact sequence in (Cech) cohomology of pointed sets

Pic(𝑋) = H1(𝑋,G𝑚)𝑔→ H1(𝑋,GL𝑛)

ℎ→ H1(𝑋,PGL𝑛)𝑓→ H2(𝑋,G𝑚).

Note that by assumption and [Mil80], p. 104, Theorem III.2.17, H1(𝑋,G𝑚) = H1(𝑋,G𝑚) = Pic(𝑋) andH2(𝑋,G𝑚) = H2(𝑋,G𝑚). Further, Br(𝑋) = Br′(𝑋) since a scheme quasi-compact and quasi-projective overan affine scheme has an ample line bundle ([Liu06], p. 171, Corollary 5.1.36), so Theorem 3.1.6 applies andBr′(𝑋) →˓ H2(𝑋,G𝑚). Since 𝐴 is an Azumaya algebra, 𝑓(𝐴) = [𝐴] ∈ Br(𝑋) →˓ H2(𝑋,G𝑚). Therefore 𝑓 factorsthrough Br(𝑋) →˓ H2(𝑋,G𝑚).

Assume the Azumaya algebra 𝐴 ∈ H1(𝑋,PGL𝑛) lies in the kernel of 𝑓 , i. e. there is a 𝑉 such that 𝐴 ∼= End(𝑉 ).Then it comes from 𝑉 ∈ H1(𝑋,GL𝑛) by [Mil80], p. 143, Step 2 (ℎ is the morphism 𝑉 ↦→ End(𝑉 )). So, since G𝑚

is central in GL𝑛, by the analogue of [Ser02], p. 54, Proposition 42 for etale Cech cohomology, if 𝑉 ′ ∈ H1(𝑋,GL𝑛)also satisfies 𝐴 ∼= End(𝑉 ′), they differ by an invertible sheaf.

Lemma 3.1.11. Let 𝑓 : C → 𝑋 be a projective smooth morphism with 𝑋 the spectrum of a Henselisation ofa variety at a regular prime ideal. Suppose the transition maps of (Pic(C𝑛))𝑛∈N are surjective (in fact, theMittag-Leffler condition would suffice). Then the canonical homomorphism

Br(C )→ lim←−𝑛∈N

Br(C𝑛)

is injective.

Proof. Let 𝐴 be an Azumaya algebra over C which lies in the kernel of the map in this lemma, i. e. such that forevery 𝑛 ∈ N there is an isomorphism

𝑢𝑛 : 𝐴𝑛∼= End(𝑉 𝑛) (3.1.1)

with 𝑉 𝑛 a locally free OC𝑛-module. Such a 𝑉 𝑛 is uniquely determined by 𝐴𝑛 modulo tensoring with an invertible

sheaf 𝐿𝑛 by Lemma 3.1.10.Because of surjectivity of the transition maps of (Pic(C𝑛))𝑛∈N, one can choose the 𝑉 𝑛, 𝑢𝑛 such that the 𝑉 𝑛

and 𝑢𝑛 form a projective system:

𝑉 𝑛 = 𝑉 𝑛+1 ⊗OC𝑛+1OC𝑛 (3.1.2)


and the isomorphisms (3.1.1) also form a projective system: Construct the 𝑉 𝑛, 𝑢𝑛 inductively. Take 𝑉 0 suchthat

𝐴⊗OC OC0∼= 𝐴0

∼= End(𝑉 0).

One has𝐴𝑛 = 𝐴⊗OC OC𝑛

and by Lemma 3.1.10, there is an invertible sheaf L𝑛 ∈ Pic(C𝑛) such that

𝑉 𝑛+1 ⊗OC𝑛+1OC𝑛

∼−−→ 𝑉 𝑛 ⊗OC𝑛L𝑛.

By assumption, there is an invertible sheaf L𝑛+1 ∈ Pic(C𝑛+1) such that L𝑛+1 ⊗OC𝑛+1OC𝑛

∼= L𝑛, so redefine

𝑉 𝑛+1 as 𝑉 𝑛+1 ⊗OC𝑛+1L −1𝑛+1. Then (3.1.2) is satisfied.

Let �� be the completion of 𝑋, and denote by C , 𝐴, . . . the base change of C , 𝐴, . . . by �� → 𝑋.Recall that an adic Noetherian ring 𝐴 with defining ideal I is a Noetherian ring with a basis of neighbourhoods

of zero of the form I 𝑛, 𝑛 > 0 such that 𝐴 is complete and Hausdorff in this topology. For such a ring 𝐴, there isthe formal spectrum Spf(𝐴) with underlying space Spec(𝐴/I ).

According to Theorem 2.2.2, to give a projective system (𝑉 𝑛, 𝑢𝑛)𝑛∈N on (C𝑛)𝑛∈N as in (3.1.1) and (3.1.2) is

equivalent to giving a locally free module 𝑉 on C and an isomorphism

�� : 𝐴∼−−→ End(𝑉 ). (3.1.3)

If 𝑋 = ��, we are done: 𝐴 = 𝐴 is trivial.In the general case, one has to pay attention to the fact that one does not know if with the preceding

construction 𝑉 comes from a locally free module 𝑉 on C . However, there is a locally free module E on C suchthat there exists an epimorphism

E → 𝑉 .

Indeed, choosing a projective immersion for C (by projectivity of C /Spec(𝐴)) with an ample invertible sheafOC (1), it suffices to take a direct sum of sheaves of the form OC (−𝑁), 𝑁 ≫ 0. Now, for 𝑁 ≫ 0, there is an

epimorphism O⊕𝑘C� 𝑉 (𝑁) for a suitable 𝑘 ∈ N, so twisting with OC (−𝑁) gives

OC (−𝑁)⊕𝑘 � 𝑉

(“there are enough vector bundles”). Set E = OC (−𝑁)⊕𝑘.Now consider, for schemes 𝑋 ′ over 𝑋, the contravariant functor 𝐹 : (Sch/𝑋)∘ → (Set) given by 𝐹 (𝑋 ′) = the set

of pairs (𝑉 ′, 𝜙′), where 𝑉 ′ is a quotient of a locally free module E ′ = E ⊗𝑋𝑋 ′ and 𝜙′ : 𝐴′ = 𝐴⊗𝑋𝑋 ′ ∼−−→ End(𝑉 ′).Since 𝑓 is projective and flat, by [SGA4.3], p. 133 f., Lemme XIII 1.3, one sees that the functor 𝐹 is

representable by a scheme, also denoted 𝐹 , locally of finite type over 𝑋, hence locally of finite presentationsince our schemes are Noetherian (what matters is the functor being locally of finite presentation, not itsrepresentability). By assumption of Lemma 3.1.11 and (3.1.3), (𝑉 , ��) is an element from 𝐹 (��). By Artinapproximation Theorem 2.2.3, 𝐹 (��) = ∅ implies 𝐹 (𝑋) = ∅:

This proves that 𝐴 is isomorphic to an algebra of the form End(𝑉 ) with 𝑉 locally free over C , so it is trivialas an element of Br(C ).

The following is a generalisation of [Gro68], pp. 98–104, Theoreme (3.1) from the case of 𝑋/𝑌 with dim𝑋 = 2,dim𝑌 = 1 to 𝑋/𝑌 with relative dimension 1. One can remove the assumption dim𝑋 = 1 if one uses Artin’sapproximation theorem Theorem 2.2.3 instead of Greenberg’s theorem on p. 104, l. 4 and l. −2, and replaces“proper” by “projective” and does some other minor modifications; also note that in our situation the Brauergroup coincides with the cohomological Brauer group by Theorem 3.1.8 and Theorem 3.1.6.

Theorem 3.1.12. Let 𝑓 : C → 𝑋 be a smooth projective morphism with fibres of dimension ≤ 1, C and 𝑋regular and 𝑋 the spectrum of a Henselisation of a variety at a prime ideal with closed point 𝑥, and C0 →˓ C thesubscheme 𝑓−1(𝑥). Then the canonical homomorphism

H2(C ,G𝑚)→ H2(C0,G𝑚)

induced by the closed immersion C0 →˓ C is bijective.


Proof. Let 𝑋 = Spec(𝐴), 𝑋𝑛 = Spec(𝐴/m𝑛+1), C𝑛 = C ×𝑋 𝑋𝑛.Note that for C and 𝑋, Br, Br′ and H2(−,G𝑚) are equal since there is an ample sheaf (Theorem 3.1.6) and

by regularity (Theorem 3.1.4).There are exact sequences for every 𝑛

0→ F → G𝑚,C𝑛+1 → G𝑚,C𝑛 → 1 (3.1.4)

with F a coherent sheaf on C0: Zariski-locally on the source, C → 𝑋 is of the form Spec(𝐵)→ Spec(𝐴) andhence C𝑛 → 𝑋𝑛 of the form Spec(𝐵/m𝑛+1)→ Spec(𝐴/m𝑛+1). There is an exact sequence

1→ (1 + m𝑛/m𝑛+1)→ (𝐵/m𝑛+1)× → (𝐵/m𝑛)× → 1.

The latter map is surjective since m𝑛/m𝑛+1 ⊂ 𝐵/m𝑛+1 is nilpotent (deformation of units: Let 𝑓 : 𝐵 → 𝐴 bea surjective ring homomorphism with nilpotent kernel. If 𝑓(𝑏) is a unit, so is 𝑏: this is because a unit plus anilpotent element is a unit: Let 𝑓(𝑏)𝑐 = 1𝐴. Then there is a 𝑐 ∈ 𝐵 such that 𝑏𝑐− 1𝐵 ∈ ker(𝑓), so 𝑏𝑐 is a unit,so 𝑏 is invertible in 𝐵). By the logarithm, (1 + m𝑛/m𝑛+1)

∼−−→ m𝑛/m𝑛+1 is a coherent sheaf on Spec(𝐵). Thesequences for a Zariski-covering of C0 glue to an exact sequence of sheaves on C0 (3.1.4), equivalently, on C𝑛 forany 𝑛 since these have the same underlying topological space.

Therefore, the associated long exact sequence to (3.1.4) yields

H2(C0,F )→ H2(C0,G𝑚,C𝑛+1)→ H2(C0,G𝑚,C𝑛

)→ H3(C0,F ).

Now, H𝑝et(C0,F ) = H𝑝Zar(C0,F ) since F is coherent by [SGA4.2], VII 4.3. Thus, since dim C0 ≤ 1, H2(C0,F ) =H3(C0,F ) = 0. Thus we get an isomorphism

H2(C0,G𝑚,C𝑛+1)

∼−−→ H2(C0,G𝑚,C𝑛)

Next note that C0 →˓ C𝑛 is a closed immersion defined by a nilpotent ideal sheaf, so there is an equivalence ofcategories of etale C0-sheaves and etale C𝑛-sheaves by [Mil80], p. 30, Theorem I.3.23, so we get

H2(C𝑛+1,G𝑚)∼−−→ H2(C𝑛,G𝑚).

Taking torsion, it follows that Br′(C𝑛+1)∼−−→ Br′(C𝑛), and then Theorem 3.1.6 yields that the Br(C𝑛+1)→ Br(C𝑛)

are isomorphisms (in fact, injectivity suffices for the following). Therefore the injectivity of Br(C )→ Br(C0)follows from Lemma 3.1.11. One can apply this in our situation since the transition maps Pic(C𝑛+1)→ Pic(C𝑛)are surjective by Theorem 2.2.1.

The surjectivity in Theorem 3.1.12 is shown analogously: Take an element of Br(C0), represented by anAzumaya algebra 𝐴0. As Br(C𝑛)

∼−−→ Br(C0), see above, there is a compatible system of Azumaya algebras 𝐴𝑛on C𝑛. Therefore, as above, there is an Azumaya algebra 𝐴 on C . Choose a locally free module E on C suchthat there is an epimorphism E � 𝐴 and consider the functor 𝐹 : (Sch/𝑋)∘ → (Set) defined by 𝐹 (𝑋 ′) = the setof pairs (𝐵′, 𝑝′) where 𝐵′ is a locally free module of E ⊗𝑋 𝑋 ′ and 𝑝′ a multiplication law on 𝐵′ which makes itinto an Azumaya algebra. Then 𝐹 is representable by a scheme locally of finite type over 𝑋 (loc. cit.), andthe point in 𝐹 (𝑥) (recall that 𝑥 is the closed point of 𝑋) defined by 𝐴0 gives us a point in 𝐹 (��), so by Artinapproximation Theorem 2.2.3 it comes from a point in 𝐹 (𝑋), which proves surjectivity.

Lemma 3.1.13. Let 𝑋 be a scheme, ℓ a prime number and 𝑛 an integer such that cdℓ(𝑋) ≤ 𝑛. ThenH𝑖(𝑋,G𝑚)[ℓ∞] = 0 if 𝑖 > 𝑛+ 1, resp. 𝑖 > 𝑛 if ℓ is invertible on 𝑋.

Proof. If ℓ is invertible on 𝑋, the Kummer sequence

1→ 𝜇ℓ𝑟 → G𝑚ℓ𝑟→ G𝑚 → 1

induces a long exact sequence in cohomology, part of which is

0 = H𝑖+1(𝑋,𝜇ℓ𝑟 )→ H𝑖+1(𝑋,G𝑚)ℓ𝑟→ H𝑖+1(𝑋,G𝑚)→ H𝑖+2(𝑋,𝜇ℓ𝑟 ) = 0,

for 𝑖 > 𝑛, i. e. multiplication by ℓ𝑟 induces an isomorphism on H𝑖+1(𝑋,G𝑚) for 𝑖 > 𝑛. If 𝑐 ∈ H𝑖+1(𝑋,G𝑚)[ℓ𝑟] ⊆H𝑖+1(𝑋,G𝑚)[ℓ∞], then 0 = ℓ𝑟𝑐, therefore 𝑐 = 0 by the injectivity of ℓ𝑟, so H𝑖(𝑋,G𝑚)[ℓ∞] = 0 for 𝑖 > 𝑛.


If ℓ = 𝑝, one has an exact sequence

1→ ker(ℓ𝑟)→ G𝑚ℓ𝑟→ G𝑚 → coker(ℓ𝑟)→ 1

of etale sheaves which splits up into

1 // ker(ℓ𝑟) // G𝑚ℓ𝑟 //

## ##

G𝑚// coker(ℓ𝑟) // 1

im(ℓ𝑟)-

;;

By the same argument as in the case ℓ invertible on 𝑋, one finds H𝑖(𝑋,G𝑚)∼−−→ H𝑖(𝑋, im(ℓ𝑟)) for 𝑖 > 𝑛, and,

since ℓ𝑟 coker(ℓ𝑟) = 0, H𝑖(𝑋, im(ℓ𝑟))∼−−→ H𝑖(𝑋,G𝑚) for 𝑖 > 𝑛+ 1. So, altogether

ℓ𝑟 : H𝑖(𝑋,G𝑚)∼−−→ H𝑖(𝑋, im(ℓ𝑟))

∼−−→ H𝑖(𝑋,G𝑚)

is injective for 𝑖 > 𝑛+ 1, and therefore H𝑖(𝑋,G𝑚)[ℓ∞] = 0 for 𝑖 > 𝑛+ 1.

Lemma 3.1.14. Let 𝐶/𝐾 be a projective regular curve over a separably closed field. Then Br(𝐶) = Br′(𝐶) =H2(𝐶,G𝑚) = 0.

Proof. One has Br(𝐶) = Br′(𝐶) = 0 by [Gro68], p. 132, Corollaire (5.8) since 𝐶 is a proper curve over a separablyclosed field. Moreover, Theorem 3.1.4 implies Br′(𝐶) = H2(𝐶,G𝑚).

As a corollary of Theorem 3.1.12, we get

Theorem 3.1.15. Let 𝜋 : C → 𝑋 be projective and smooth with C and 𝑋 regular, all fibres of dimension 1 and𝑋 be a variety. Then

R𝑞𝜋*G𝑚 = 0 for 𝑞 > 1.

Proof. By [SGA4.2], VIII 5.2, resp. [Mil80], p. 88, III.1.15 one can assume 𝑋 strictly local and we must proveH𝑖(C ,G𝑚) = 0 for 𝑖 > 1. By the proper base change theorem [Mil80], p. 224, Corollary VI.2.7, one has fortorsion sheaves F on C with restriction F0 to the closed fibre C0 restriction isomorphisms

H𝑖(C ,F )→ H𝑖(C0,F0).

Since dim C0 = 1, the latter term vanishes for 𝑖 > 2 and for 𝑖 > 1 if F is 𝑝-torsion, where 𝑝 is the residue fieldcharacteristic. Therefore

cd(C ) ≤ 2, and cd𝑝(C ) ≤ 1.

The relation H𝑖(C ,G𝑚) = 0 for 𝑖 > 2 follows from the fact that these groups are torsion by Theorem 3.1.8 (a)and from Lemma 3.1.13.

It remains to treat the case 𝑖 = 2, i. e. to prove

H2(C ,G𝑚) = 0.

If ℓ is invertible on 𝑋, thenH2(C ,G𝑚)[ℓ∞] = 0

follows as in the case 𝑖 > 2. From the Kummer sequence Lemma 3.1.1 and the inclusion C0 →˓ C , one gets acommutative diagram with exact rows

0 // Pic(C )⊗Qℓ/Zℓ //

��

H2(C , 𝜇ℓ∞) //

∼=��

H2(C ,G𝑚)[ℓ∞]

��

// 0

0 // Pic(C0)⊗Qℓ/Zℓ // H2(C0, 𝜇ℓ∞) // H2(C0,G𝑚)[ℓ∞] // 0,

and the middle vertical arrow is bijective by proper base change [Mil80], p. 224, Corollary VI.2.7, and the firstvertical arrow is surjective by Theorem 2.2.1 and the right exactness of the tensor product. Hence, by the fivelemma, the right vertical morphism ist bijective.


Thus, by Lemma 3.1.14, the diagram gives us H2(C ,G𝑚)[ℓ∞] = 0For ℓ = 𝑝, one uses Theorem 3.1.12, which gives us

H2(C ,G𝑚)∼−−→ H2(C0,G𝑚),

and H2(C0,G𝑚) = 0 by Lemma 3.1.14.

Remark 3.1.16. Note that the difficult Theorem 3.1.12 is only needed for the 𝑝-torsion in Theorem 3.1.15.

We now draw some consequences from Theorem 3.1.15:

Corollary 3.1.17. In the situation of Theorem 3.1.15, assume we have locally Noetherian separated schemeswith geometrically reduced and connected fibres. Then one has the long exact sequence

0→H1(𝑋,G𝑚)𝜋*

→ H1(C ,G𝑚)→ H0(𝑋,R1𝜋*G𝑚)→ . . .

H𝑞(𝑋,G𝑚)𝜋*

→ H𝑞(C ,G𝑚)→ H𝑞−1(𝑋,R1𝜋*G𝑚)→ . . .

Proof. This follows from the Leray spectral sequence and Theorem 3.1.15 combined with [Wei97], p. 124,Exercise 5.2.2 (spectral sequence with two rows; here for a cohomological spectral sequence). One has 𝜋*G𝑚 = G𝑚

by Lemma 2.2.5. The edge maps H𝑝(𝑋,G𝑚) = H𝑝(𝑋, 𝜋*G𝑚)→ H𝑝(C ,G𝑚) are identified as 𝜋* by Lemma 2.2.6.

Lemma 3.1.18. Let 𝜋 : C → 𝑋 be a proper relative curve with 𝑋 integral and let 𝐷 be an irreducible Weildivisor in the generic fibre 𝐶 of 𝜋. Then 𝜋�� : �� → 𝑋 is a finite morphism.

Proof. Take 𝐷 an irreducible Weil divisor in 𝐶, i. e. a closed point of 𝐶. Then ��/𝑋 is finite of degree deg(𝐷):��/𝑋 is of finite type, generically finite and dominant (since 𝐷 lies over the generic point of 𝑋) and �� (as aclosure of an irreducible set) and 𝑋 are irreducible, C and 𝑋 are integral, hence by [Har83], p. 91, Exercise II.3.7there is a dense open subset 𝑈 ⊆ 𝑋 such that ��|𝑈 → 𝑈 is finite. Since 𝜋|�� : �� → 𝑋 is proper (since it factors

as a composition of a closed immersion and a proper morphism �� →˓ C𝜋→ 𝑋), it suffices to show that it is

quasi-finite. If there is an 𝑥 ∈ 𝑋 such that 𝜋|−1��

(𝑥) is not finite, we must have 𝜋|−1��

(𝑥) = C𝑥, i. e. the whole curve

as a fibre, since the fibres of 𝜋 are irreducible. But then �� would have more than one irreducible component, acontradiction.

Corollary 3.1.19. If in the situation of Corollary 3.1.17, 𝜋 has a section 𝑠 : 𝑋 → C , one has split short exactsequences

0 H𝑖(𝑋,G𝑚) H𝑖(C ,G𝑚) H𝑖−1(𝑋,R1𝜋*G𝑚) 0𝜋*

𝑠*

for 𝑖 ≥ 1.In the general case, denote by 𝐶/𝐾 the generic fibre of C /𝑋, and assume that for every Weil divisor 𝐷 in 𝐶,

�� ⊆ C has everywhere the same dimension as 𝑋 with no embedded components, and denote by 𝛿 the greatestcommon divisor of the degrees of Weil divisors on 𝐶/𝐾, i. e. the index of 𝐶/𝐾. Then one has an exact sequence

0→ 𝐾2 → H2(𝑋,G𝑚)𝜋*

→ H2(C ,G𝑚)→ H1(𝑋,R1𝜋*G𝑚)→ 𝐾3 → 0,

where 𝐾𝑖 = ker(H𝑖(𝑋,G𝑚)𝜋*

→ H𝑖(C ,G𝑚)) are Abelian groups annihilated by 𝛿 whose prime-to-𝑝 torsion is finite.

Proof. The first assertion is obvious from the previous Corollary 3.1.17 and the existence of a section.For the second claim, take an irreducible Weil divisor 𝐷 in 𝐶. Then 𝜋�� : �� → 𝑋 is a finite morphism

by Lemma 3.1.18. We have the commutative diagram

�� 𝑖 //

𝜋|��

C

𝜋

��

𝑋.


By the Leray spectral sequence, we have that H𝑖(��,G𝑚) = H𝑖(𝑋, 𝜋|��,*G𝑚) as 𝜋|�� is finite, hence exact for theetale topology, see [Mil80], p. 72, Corollary II.3.6. If 𝜋|�� is also flat, by finite locally freeness we have a normmap 𝜋|��,*G𝑚 → G𝑚 whose composite with the inclusion G𝑚 → 𝜋|��,*G𝑚 is the 𝛿-th power map. If not, thereis still a norm map since 𝑓 is flat in codimension 1 since �� has everywhere the same dimension as 𝑋 with noembedded components, so one can take the norm there, which then will land in G𝑚 as 𝑋 is normal.

We have 𝜋|��* ∘ 𝑖* ∘𝜋* = 𝜋|��* ∘𝜋|*�� = deg(𝐷), so ker(𝜋* : H*(𝑋,G𝑚)→ H*(C ,G𝑚)) is annihilated by deg(𝐷)for all 𝐷, hence by the index 𝛿. Now the finiteness of the prime-to-𝑝 part of the 𝐾𝑖 follows from Theorem 3.1.8.

In the following, assume 𝜋 is smooth (automatically projective since C ,𝑋 are projective over F𝑞), withall geometric fibres integral and of dimension 1, and that it has a section 𝑠 : 𝑋 → C . Assume further that𝜋*OC = O𝑋 holds universally and 𝜋 is cohomologically flat in dimension 0, e. g. if 𝜋 is a flat proper morphism oflocally Noetherian separated schemes with geometrically connected fibres (Lemma 2.2.5).

We recall some definitions from [FGI+05], p. 252, Definition 9.2.2.

Definition 3.1.20. The relative Picard functor Pic𝑋/𝑆 on the category of (locally Noetherian) 𝑆-schemesis defined by

Pic𝑋/𝑆(𝑇 ) := Pic(𝑋 ×𝑆 𝑇 )/ pr*2 Pic(𝑇 ).

Its associated sheaves in the Zariski, etale and fppf topology are denoted by

Pic𝑋/𝑆,Zar, Pic𝑋/𝑆,et, Pic𝑋/𝑆,fppf .

Now we come to the representability of the relative Picard functor by a group scheme, whose connectedcomponent of unity is an Abelian scheme.

Theorem 3.1.21. PicC/𝑋 is represented by a separated smooth 𝑋-scheme PicC/𝑋 locally of finite type. Pic0C/𝑋is represented by an Abelian 𝑋-scheme Pic0C/𝑋 . For every 𝑇/𝑋,

0→ Pic(𝑇 )→ Pic(C ×𝑋 𝑇 )→ PicC/𝑋(𝑇 )→ 0

is exact.

Proof. Since 𝜋 has a section 𝑠 : 𝑋 → C and 𝜋*OC = O𝑋 holds universally, by [FGI+05], p. 253, Theorem 9.2.5

PicC/𝑋∼−−→ PicC/𝑋,Zar

∼−−→ PicC/𝑋,et∼−−→ PicC/𝑋,fppf .

Since 𝜋 is projective and flat with geometrically integral fibres, by [FGI+05] p. 263, Theorem 9.4.8, PicC/𝑋

exists, is separated and locally of finite type over 𝑋 and represents PicC/𝑋,et. Since 𝑋 is Noetherian andC /𝑋 projective, by loc. cit., PicC/𝑋 is a disjoint union of open subschemes, each an increasing union of openquasi-projective 𝑋-schemes.

By [BLR90], p. 259 f., Proposition 4, Pic0C/𝑋/𝑋 is an Abelian scheme (this uses that C /𝑋 is a relative curveor an Abelian scheme).

The last assertion follows from [BLR90], p. 204, Proposition 4.

Theorem 3.1.22. Let 𝜋 : 𝑋 → 𝑆 be a smooth proper morphism. Then R1𝜋*G𝑚 = Pic𝑋/𝑆, the higher directimage taken with respect to the fppf or etale topology.

Proof. See [BLR90], p. 202 f.

Theorem 3.1.23. Let A /𝑋 be an Abelian scheme over a locally Noetherian, integral, geometrically unibranch(e. g., normal) base 𝑋. Then A /𝑋 is projective.

Proof. See [Ray70], p. 161, Theoreme XI 1.4.


3.2 The weak Neron model

Lemma 3.2.1. Let 𝐴 be a regular local ring. Then

𝐴ℎ ⊗𝐴 Quot(𝐴) = Quot(𝐴ℎ),

𝐴𝑠ℎ ⊗𝐴 Quot(𝐴) = Quot(𝐴𝑠ℎ).

Proof. By [Mil80], p. 38, Remark 4.11, 𝐴𝑠ℎ is the localisation at a maximal ideal lying over the maximal idealm of 𝐴 of the integral closure of 𝐴 in (Quot(𝐴)sep)𝐼 with 𝐼 ⊆ Gal(Quot(𝐴)sep/Quot(𝐴)) the inertia subgroup.Pick an element 𝑎 ∈ 𝐴𝑠ℎ. It is a root of a monic polynomial 𝑓(𝑇 ) ∈ 𝐴[𝑇 ], which we can assume to be monicirreducible since 𝐴 is a regular local ring and hence factorial by [Mat86], p 163, Theorem 20.3, and hence 𝐴[𝑇 ] isfactorial by [Mat86], p. 168, Exercise 20.2. Since 𝐴 is factorial and 𝑓 ∈ 𝐴[𝑇 ] is irreducible, by the lemma ofGauß [Bos03], p. 64, Korollar 6, 𝑓 is also irreducible over Quot(𝐴). Hence 𝐴[𝑎]⊗𝐴 Quot(𝐴) = Quot(𝐴)[𝑇 ]/(𝑓(𝑇 ))is a field, and 𝐴𝑠ℎ ⊗𝐴 Quot(𝐴) is a directed colimit of fields since tensor products commute with colimits, andhence itself a field, namely Quot(𝐴𝑠ℎ). Now note that localisation does not change the quotient field.

The proof for 𝐴ℎ is the same: Just replace the inertia group by the decomposition group.

Theorem 3.2.2 (The weak Neron model). Let 𝑆 be a regular, Noetherian, integral, separated scheme with𝑔 : {𝜂} →˓ 𝑆 the inclusion of the generic point. Let 𝑋/𝑆 be a smooth projective variety with geometrically integralfibres that admits a section such that its Picard functor is representable (e. g., 𝑋/𝑆 a smooth projective curveadmitting a section or an Abelian scheme). Then

Pic𝑋/𝑆∼−−→ 𝑔*𝑔

*Pic𝑋/𝑆

as etale sheaves on 𝑆. Let A /𝑆 be an Abelian scheme. Then

A∼−−→ 𝑔*𝑔

*A

as etale sheaves on 𝑆.

We call this the weak Neron mapping property and A a weak Neron model of its generic fibre,while a Neron model of 𝐴/𝐾 in the usual sense is a model A /𝑆 which satisfies the Neron mapping propertyA

∼−−→ 𝑔*𝑔*A for the smooth topology.

The main idea for injectivity is to use the separatedness of our schemes, and the main idea for surjectivity isthat Weil divisors spread out and that the Picard group equals the Weil divisor class group by regularity.

Proof. Let 𝑓 : Pic𝑋/𝑆 → 𝑔*𝑔*Pic𝑋/𝑆 be the natural map of etale sheaves induced by adjointness. Let 𝑠→ 𝑆 be

a geometric point. We have to show that (coker(𝑓))𝑠 = 0 = (ker(𝑓))𝑠.Taking stalks and using Lemma 2.2.11, we get the following commutative diagram:

Pic𝑋/𝑆(O𝑠ℎ𝑆,𝑠)

𝑓// Pic𝑋/𝑆(Quot(O𝑠ℎ

𝑆,𝑠))// // (coker(𝑓))𝑠

Pic(𝑋 ×𝑆 O𝑠ℎ𝑆,𝑠)

OOOO

// Pic(𝑋 ×𝑆 Quot(O𝑠ℎ𝑆,𝑠))

OOOO

Div(𝑋 ×𝑆 O𝑠ℎ𝑆,𝑠)

OOOO

// Div(𝑋 ×𝑆 Quot(O𝑠ℎ𝑆,𝑠))

OOOO

Note that O𝑠ℎ𝑆,𝑠 is a domain since it is regular as a strict Henselisation of a regular local ring by [Fu11],

p. 111, Proposition 2.8.18. By Lemma 2.2.11, one has (Pic𝑋/𝑆)𝑠 = Pic𝑋/𝑆(O𝑠ℎ𝑆,𝑠) and (𝑔*𝑔

*Pic𝑋/𝑆)𝑠 =

Pic𝑋/𝑆(O𝑠ℎ𝑆,𝑠⊗O𝑆,𝑠

𝐾(𝑆)). But for a regular local ring 𝐴, one has 𝐴𝑠ℎ⊗𝐴 Quot(𝐴) = Quot(𝐴𝑠ℎ) by Lemma 3.2.1.By [Har83], p. 145, Corollary II.6.16, one has a surjection from Div to Pic since 𝑆 is Noetherian, integral,

separated and locally factorial. By Theorem 3.1.21, the upper vertical arrows are surjective (under the assumptionthat 𝑋/𝑆 has a section)

But here, the lower horizontal map is surjective: A preimage under 𝜄 : 𝑋 ×𝑆 Quot(O𝑠ℎ𝑆,𝑠)→ 𝑋 ×𝑆 O𝑠ℎ

𝑆,𝑠 of

𝐷 ∈ Div(𝑋 ×𝑆 Quot(O𝑠ℎ𝑆,𝑠)) is �� ∈ Div(𝑋 ×𝑆 O𝑠ℎ

𝑆,𝑠), the closure taken in 𝑋 ×𝑆 O𝑠ℎ𝑆,𝑠. In fact, note that 𝐷 is


closed in 𝑋 ×𝑆 Quot(O𝑠ℎ𝑆,𝑠) since it is a divisor; the closure of an irreducible subset is irreducible again, and the

codimension is also 1 since the codimension is the dimension of the local ring at the generic point 𝜂𝐷 of 𝐷, andthe local ring of 𝜂𝐷 in 𝑋 ×𝑆 Quot(O𝑠ℎ

𝑆,𝑠) is the same as the local ring of 𝜂𝐷 in 𝑋 ×𝑆 O𝑠ℎ𝑆,𝑠 as it is the colimit of

the global sections taken for all open neighbourhoods of 𝜂𝐷. Hence (coker(𝑓))𝑠 = 0.For (ker(𝑓))𝑠 = 0, consider the diagram

Spec Quot(O𝑠ℎ𝑆,𝑠)

jJ

ww

� _

��

// Pic𝑋/𝑆

��

Spec O

''

33

Spec O𝑠ℎ𝑆,𝑠

??

// 𝑆.

We want to show that a lift Spec O𝑠ℎ𝑆,𝑠 → Pic𝑋/𝑆 of Spec Quot(O𝑠ℎ

𝑆,𝑠) → Pic𝑋/𝑆 is unique. As Pic𝑋/𝑆/𝑆 is

separated, this is true for all valuation rings O ⊂ Quot(O𝑠ℎ𝑆,𝑠) by the valuative criterion of separatedness [EGAII],

p. 142, Proposition (7.2.3). But by [Mat86], p. 72, Theorem 10.2, every local ring (O𝑠ℎ𝑆,𝑠)p is dominated by

a valuation ring O of Quot(O𝑠ℎ𝑆,𝑠). It follows from the valuative criterion for separatedness that the lift is

topologically unique. Assume 𝜙,𝜙′ are two lifts. Now cover Pic𝑋/𝑆/𝑆 by open affines 𝑈𝑖 = Spec𝐴𝑖 and theirpreimages 𝜙−1(𝑈𝑖) = 𝜙′−1(𝑈𝑖) by standard open affines {𝐷(𝑓𝑖𝑗)}𝑗 .

𝐴𝑖

uu

𝜙#

��

𝜙′#

��

O𝑓𝑖𝑗

(O𝑠ℎ𝑆,𝑠)𝑓𝑖𝑗R2

dd

It follows that 𝜙 = 𝜙′.

Lemma 3.2.3. Let 𝑆 be a locally Noetherian scheme, 𝑓 : 𝑋 → 𝑆 be a proper flat morphism and E a locally freesheaf on 𝑋. Then the Euler characteristic

𝜒E (𝑠) =∑𝑖≥0

(−1)𝑖 dim H𝑖(𝑋𝑠,E𝑠)

is locally constant on 𝑆.

Proof. See [EGAIII2], p. 76 f., Theoreme (7.9.4).

For the statement about Pic0𝑋/𝑆 for curves: Restricting the isomorphism to Pic0 gives a well-definedhomomorphism since a line bundle being of degree 0 can be checked on geometric fibres by [Conb], p. 3,Proposition 4.1 and [Conb], p. 1, Theorem 2.2. Injectivity is immediate. Surjectivity follows since if L is apreimage in Pic of a line bundle lying in Pic0, it must already lie in Pic0 since, again, being of degree 0 can bechecked fibrewise.

Now we prove the last statement of the theorem not only for curves, but also for Abelian schemes, so one candeduce the statement for Abelian varieties by noting that A = (A ∨)∨ = Pic0A ∨/𝑆 .

We want to show that

Pic0𝑋/𝑆(𝑆′)→ Pic0𝑋/𝑆(𝑆′𝜂) (3.2.1)

is bijective for any etale 𝑆-scheme 𝑆′.2

2The following argument has been communicated to me by Brian Conrad.


First note that such an 𝑆′ is regular, so its connected components are integral, and 𝑆′𝜂 is the disjoint union of

the generic points of the connected components of 𝑆′, so we can replace 𝑋 → 𝑆 with the restrictions of the basechange 𝑋 ′ → 𝑆′ over each connected component of 𝑆′ separately to reduce to checking for 𝑆′ = 𝑆.

For any section 𝑆 → Pic𝑋/𝑆 , since 𝑆 is connected and Pic𝜏𝑋/𝑆 is open and closed in Pic𝑋/𝑆 by [SGA6],p. 647 f., exp. XIII, Theoreme 4.7, the preimage of Pic𝜏𝑋/𝑆 under the section is open and closed in 𝑆, henceempty or 𝑆. Thus, if even a single point of 𝑆 is carried into Pic𝜏𝑋/𝑆 under the section, then the whole of 𝑆 is.More generally, when using 𝑆′-valued points of Pic𝑋/𝑆 for any etale 𝑆-scheme 𝑆′, such a point lands in Pic𝜏𝑋/𝑆if and only if some point in each connected component of 𝑆′ does, such as the generic point of each connectedcomponent 𝑆′

𝜂.

This proves the statement in (3.2.1) for Pic0𝑋/𝑆 replaced by Pic𝜏𝑋/𝑆 , and hence for Pic0𝑋/𝑆 in cases where itcoincides with Pic𝜏𝑋/𝑆 (i. e., when the geometric fibers have component group for Pic𝑋/𝑆—i. e., Neron-Severigroup—that is torsion-free, e. g. for Abelian schemes or curves, see Example 4.3.25 below).

3.3 The Tate-Shafarevich group

Proposition 3.3.1. Let 𝑋 be integral and A /𝑋 be an Abelian scheme. Then H𝑖(𝑋,A ) is torsion for 𝑖 > 0.

Proof. Consider the Leray spectral sequence for the inclusion 𝑔 : {𝜂} →˓ 𝑋 of the generic point

H𝑝(𝑋,R𝑞𝑔*𝑔*A )⇒ H𝑝+𝑞(𝜂, 𝑔*A ).

Calculation modulo the Serre subcategory of torsion sheaves, and exploiting the fact that Galois cohomologygroups are torsion in dimension > 0 by Lemma 2.1.12, and therefore also the higher direct images R𝑞𝑔*𝑔

*A aretorsion sheaves by [Mil80], p. 88, Proposition III.1.13 (the higher direct images are the sheaf associated with thepresheaf “cohomology of the preimages”), the spectral sequence degenerates giving

H𝑝(𝑋, 𝑔*𝑔*A ) = 𝐸𝑝,02 = 𝐸𝑝 = H𝑝(𝜂, 𝑔*A ) = 0

for 𝑝 > 0. Because of the weak Neron mapping property we have H𝑝(𝑋,A )∼−−→ H𝑝(𝑋, 𝑔*𝑔

*A ), which finishesthe proof.

This gives a necessary condition for the existence of a weak Neron model:

Remark 3.3.2. In [MB], Moret-Bailly constructs, using [Ray70], XIII, 𝐴-torsors of infinite order over the affineline with two points identified, and even over normal two-dimensional schemes.

Definition 3.3.3. Define the Tate-Shafarevich group of an Abelian scheme A /𝑋 by

X(A /𝑋) = H1et(𝑋,A ).

Theorem 3.3.4. Let A /𝑋 be an Abelian scheme. For 𝑥 ∈ 𝑋, denote the function field of 𝑋 by 𝐾, the quotientfield of the strict Henselisation of O𝑋,𝑥 by 𝐾𝑛𝑟

𝑥 , the inclusion of the generic point by 𝑗 : {𝜂} →˓ 𝑋 and let𝑗𝑥 : Spec(𝐾𝑛𝑟

𝑥 ) →˓ Spec(O𝑠ℎ𝑋,𝑥) →˓ 𝑋 be the composition. Then we have

H1(𝑋,A )∼−−→ ker

(H1(𝐾, 𝑗*A )→

∏𝑥∈𝑋


).

Proof. By Lemma 2.1.1, the Leray spectral sequence H𝑝(𝑋,R𝑞𝑗*(𝑗*A )))⇒ H𝑝+𝑞(𝐾, 𝑗*A ) yields the exactnessof 0→ 𝐸1,0

2 → 𝐸1 → 𝐸0,12 , i. e.

H1(𝑋, 𝑗*𝑗*A ) = ker

(H1(𝐾, 𝑗*A )→ H0(𝑋,R1𝑗*(𝑗*A ))

).

SinceH0(𝑋,R1𝑗*(𝑗*A ))→

∏𝑥∈𝑋

R1𝑗*(𝑗*A )��

is injective ([Mil80], p. 60, Proposition II.2.10: If a section of an etale sheaf is non-zero, there is a geometricpoint for which the stalk of the section is non-zero) and

R1𝑗*(𝑗*A )�� = H1(𝐾𝑛𝑟𝑥 , 𝑗*𝑥A )

by Lemma 3.2.1, the theorem follows from Lemma 2.1.2 and the weak Neron mapping property H1(𝑋,A )∼−−→

H1(𝑋, 𝑗*𝑗*A ).


Remark 3.3.5. In order to apply Lemma 3.3.19 below, we assume in this section that all schemes lie overSpec 𝑘 with 𝑘 = F𝑞 a finite field.

Theorem 3.3.6 (The Tate-Shafarevich group). In the situation of Theorem 3.3.4, one can replace the productover all points by

(a) the codimension-1 points if one disregards the 𝑝-torsion (𝑝 = char 𝑘) (for dim𝑋 ≤ 2, this also holds forthe 𝑝-torsion), assuming 𝑋/𝑘 smooth projective and let A /𝑋 be an Abelian scheme such that the vanishingtheorem Lemma 3.3.15 below is satisfied:

H1(𝑋,A ) = ker

⎛⎝H1(𝐾, 𝑗*A )→⨁

𝑥∈𝑋(1)


⎞⎠ ,

or (b) the closed points

H1(𝑋,A ) = ker

⎛⎝H1(𝐾, 𝑗*A )→⨁𝑥∈|𝑋|


⎞⎠One can also replace 𝐾𝑛𝑟

𝑥 by the quotient field of the completion O𝑠ℎ𝑋,𝑥 in the case of 𝑥 ∈ 𝑋(1).

We first establish some vanishing results for etale cohomology with supports.

Lemma 3.3.7. In the situation of the previous lemma, one has

H𝑖𝑍(𝑋,G𝑚) = 0 for 𝑖 ≤ 2,

and for 𝑖 = 3 at least away from 𝑝.

Proof. See [Gro68], p. 133 ff.: Using the local-to-global spectral sequence ([Gro68], p. 133, (6.2))

𝐸𝑝,𝑞2 = H𝑝(𝑋,H 𝑞𝑍 (G𝑚))⇒ H𝑝+𝑞

𝑍 (𝑋,G𝑚)

and [Gro68], p. 133–135

H 0𝑍 (G𝑚) = 0 [Gro68], p. 133, (6.3)

H 1𝑍 (G𝑚) = 0 [Gro68], p. 133, (6.4) since the codimension of 𝑍 in 𝑋 is = 1

H 2𝑍 (G𝑚) = 0 [Gro68], p. 134, (6.5)

H 3𝑍 (G𝑚)(𝑝

′) = 0 [Gro68], p. 134 f., Thm. (6.1),

(even H 3𝑍 (G𝑚) = 0 for dim𝑋 = 2; if not, we have to calculate modulo suitable Serre subcategories in the

following), we have 𝐸𝑝,𝑞2 = 0 for 𝑞 ≤ 3, and hence the result 𝐸𝑛 = 0 for 0 ≤ 𝑛 ≤ 3 follows from the exactsequences

0→ 𝐸𝑛,02 → 𝐸𝑛 → 𝐸0,𝑛2

for 𝑛 = 1, 2, 3.

Lemma 3.3.8. If 𝑓 : 𝑋 → 𝑌 is a flat morphism of locally Noetherian schemes and 𝑍 →˓ 𝑌 is a closed immersionof codimension ≥ 𝑐, then also the base change 𝑍 ′ := 𝑍 ×𝑌 𝑋 →˓ 𝑋 is a closed immersion of codimension ≥ 𝑐.

Proof. Without loss of generality, let 𝑍 ′ = {𝑧′} be irreducible. Since all involved schemes are locally Noetherianand 𝑓 is flat, by [GW10], p. 464, Corollary 14.95 (𝑓 : 𝑋 → 𝑌 , 𝑦 = 𝑓(𝑥), then the codimension of {𝑥} is≥ the codimension of {𝑓(𝑥)}), we have codim𝑋 𝑍

′ ≥ codim𝑌 𝑓(𝑧′). But 𝑓(𝑧′) ⊆ 𝑍 = 𝑍, so codim𝑌 𝑓(𝑧′) ≥codim𝑌 𝑍 ≥ 𝑐, hence the result.

Lemma 3.3.9. Let 𝑋 be a normal scheme and C /𝑋 a smooth proper relative curve. Then there is an exactsequence

0→ Pic0C/𝑋 → PicC/𝑋 → Z→ 0.


Proof. Without loss of generality, assume 𝑋 connected (as all schemes are of finite type over a field, so thereare only finitely many connected components all of which are open: Every connected component is closed, andthey are finite in number, so they are open). Let 𝑔 = 1 − 𝜒OC be the genus of C /𝑋 (well-defined becauseof Lemma 3.2.3). Consider

deg : PicC/𝑋 → Z,

Pic(C ×𝑋 𝑌 )/pr*2 Pic(𝑌 ) ∋ L ↦→ 𝜒L (𝑦)− (1− 𝑔), 𝑦 ∈ 𝑌 ;

this is a well-defined morphism of Abelian sheaves on the small etale site of 𝑋 because of Lemma 3.2.3.

Now the statement follows from [Conb], p. 3 f., Proposition 4.1 and Theorem 4.4.

Lemma 3.3.10. Let 𝑋/𝑘 be a smooth variety and 𝜋 : C → 𝑋 a smooth proper relative curve which admits asection 𝑠 : 𝑋 → C . Let 𝑍 →˓ 𝑋 be a reduced closed subscheme of codimension ≥ 2. Assume dim𝑋 ≤ 2. Then

H𝑖𝑍(𝑋,PicC/𝑋) = 0 for 𝑖 ≤ 2.

For dim𝑋 > 2, this holds at least up to 𝑝-torsion.

Proof. By Theorem 3.1.15 and the Leray spectral sequence with supports Theorem 2.2.8, we get a long exactsequence

0→ 𝐸1,02 → 𝐸1 → 𝐸0,1

2 → 𝐸2,02 → 𝐸2 → 𝐸1,1

2 → 𝐸3,0 → 𝐸3 → 𝐸2,12 → 𝐸4,0 → 𝐸4 (3.3.1)

by [Wei97], p. 124, Exercise 5.2.2 (spectral sequence with two rows; here for a cohomological spectral sequence).But 𝐸𝑖,02 = H𝑖

𝑍(𝑋,G𝑚) = 0 for 𝑖 ≤ 3 by Lemma 3.3.7 (for 𝑖 = 3 at least away from 𝑝). Therefore the long exactsequence (3.3.1) yields isomorphisms

𝐸𝑖 → 𝐸𝑖−1,12

for 𝑖 ≤ 2, but 𝐸𝑖 = H𝑖𝑍′(C ,G𝑚) = 0 for 𝑖 ≤ 3, again by Lemma 3.3.7 and Lemma 3.3.8, hence H𝑖𝑍(𝑋,R1𝜋*G𝑚) =

𝐸𝑖−1,12 = 0 for 𝑖 ≤ 2 from (3.3.1). For the vanishing of 𝐸2,1

2 note that

0 = 𝐸3 → 𝐸2,12 → 𝐸4,0 → 𝐸4

is exact by (3.3.1), but by Lemma 2.2.6 the latter map is 𝜋* : H4𝑍(𝑋,G𝑚) →˓ H4

𝑍′(C ,G𝑚), which is injective as𝜋 admits a section.

Lemma 3.3.11. Let 𝑋 be geometrically unibranch (e. g. normal) and 𝑖 : 𝑈 →˓ 𝑋 dominant. Then for anyconstant sheaf 𝐴, one has 𝐴

∼−−→ 𝑖*𝑖*𝐴 as etale sheaves.

Proof. See [SGA4.3], p. 25 f., IX Lemme 2.14.1.

Lemma 3.3.12. Let 𝑋 be a connected normal Noetherian scheme with generic point 𝜂. Then H𝑝(𝑋,Q) = 0 forall 𝑝 > 0 and H1(𝑋,Z) = 0.

Proof. Denote the inclusion of the generic point by 𝑔 : {𝜂} →˓ 𝑋. Since 𝑋 is connected normal, by Lemma 3.3.11

𝐴∼−−→ 𝑔*𝑔

*𝐴.

As R𝑞𝑔*(𝑔*Q) = 0 for 𝑞 > 0 (since Galois cohomology is torsion and Q is uniquely divisible; then use [Mil80],p. 88, Proposition III.1.13), the Leray spectral sequence

H𝑝(𝑋,R𝑞𝑔*𝑔*Q)⇒ H𝑝+𝑞(𝜂,Q)

degenerates to H𝑝(𝑋,Q) = H𝑝(𝜂,Q), which is trivial as, again, Galois cohomology is torsion and Q is uniquelydivisible.

Similarly, as R1𝑔*(𝑔*Z) = 0 (since Z carries the trivial Galois action and homomorphisms from profinitegroups to discrete groups have finite image, and Z has no nontrivial finite subgroups; again, use [Mil80], p. 88,Proposition III.1.13), the Leray spectral sequence gives H1(𝑋,Z) = H1(𝜂,Z) = 0.


Remark 3.3.13. If 𝑋 is not normal, H𝑖(𝑋,Z) = 0 in general:3 Consider 𝑋 = Spec(𝑘[𝑋, 𝑌 ]/(𝑌 2 − (𝑋3 +𝑋2))),the normalisation morphism 𝜋 : A1

𝑘 → 𝑋 and the inclusion 𝑖 : {𝑥} →˓ 𝑋 of 𝑥 = (0, 0). There is a short exactsequence of etale sheaves on 𝑋

0→ Z𝑋 → 𝜋*ZA1𝑘→ 𝑖*Z{𝑥} → 0.

Taking the long exact cohomology sequence yields H1(𝑋,Z𝑋) = Z.

Proposition 3.3.14. Let 𝑋 be a connected Noetherian scheme and �� a geometric point. Then

H1(𝑋,Q/Z) = Hom𝑐𝑜𝑛𝑡(𝜋et1 (𝑋, ��),Q/Z)

andH1(𝑋,Zℓ) = Hom𝑐𝑜𝑛𝑡(𝜋

et1 (𝑋, ��),Zℓ).

Proof. This follows from [Fu11], p. 245, Proposition 5.7.20 (for a connected Noetherian scheme 𝑋 and a finitegroup 𝐺, one has H1(𝑋,𝐺) = Hom𝑐𝑜𝑛𝑡(𝜋

et1 (𝑋, ��), 𝐺)) by passing to the colimit over 𝐺𝑛 = 1

𝑛Z/Z, or passing tothe limit over 𝐺𝑛 = Z/ℓ𝑛, respectively.

Lemma 3.3.15. Let 𝑋/𝑘 be a smooth variety and C /𝑋 a smooth proper relative curve. Assume dim𝑋 ≤ 2.Let 𝑍 →˓ 𝑋 be a reduced closed subscheme of codimension ≥ 2. Then

H𝑖𝑍(𝑋,Pic0C/𝑋) = 0 for 𝑖 ≤ 2.

If dim𝑋 > 2, this holds at least up to 𝑝-torsion.

Proof. Taking the long exact sequence associated to the short exact sequence of Lemma 3.3.9 with respect toH𝑖𝑍(𝑋,−), by Lemma 3.3.10, it suffices to show that H𝑖

𝑍(𝑋,Z) = 0 for 𝑖 = 0, 1, 2.For this, consider the long exact sequence

. . .→ H𝑖𝑍(𝑋,Z)→ H𝑖(𝑋,Z)→ H𝑖(𝑋 ∖ 𝑍,Z)→ . . .

It suffices to show that H𝑖(𝑋,Z)→ H𝑖(𝑋 ∖ 𝑍,Z) is an isomorphism for 𝑖 = 0, 1 and an injection for 𝑖 = 2.For 𝑖 = 0, this map is Z

∼−−→ Z.For 𝑖 = 1, both groups are equal to 0 by Lemma 3.3.12.For 𝑖 = 2, this map is H2(𝑋,Z)→ H2(𝑋 ∖ 𝑍,Z). Consider the long exact sequence associated to

0→ Z→ Q→ Q/Z→ 0.

In the following, we omit the base point. Now, 𝑋 being normal because smooth over a field, H2(𝑋,Z) =H1(𝑋,Q/Z) = Hom𝑐𝑜𝑛𝑡(𝜋

et1 (𝑋),Q/Z). (For the first equality, use the long exact sequence associated to 0→ Z→

Q→ Q/Z→ 0 and H𝑝(𝑋,Q) = 0 for 𝑝 > 0 by Lemma 3.3.12. For the second equality use Proposition 3.3.14.)Since 𝜋et1 (𝑋 ∖𝑍)→ 𝜋et1 (𝑋) is an isomorphism because 𝑍 is of codimension ≥ 2 (by Theorem 2.2.13), H2(𝑋,Z)→H2(𝑋 ∖ 𝑍,Z) is an isomorphism:

H2(𝑋,Z)∼= //

��

Hom𝑐𝑜𝑛𝑡(𝜋et1 (𝑋),Q/Z)

∼=��

H2(𝑋 ∖ 𝑍,Z)∼= // Hom𝑐𝑜𝑛𝑡(𝜋

et1 (𝑋 ∖ 𝑍),Q/Z)

Lemma 3.3.16. We havelim−→𝑈

H𝑝(𝑈,F |𝑈 )∼−−→ H𝑝(𝐾,F𝜂),

the colimit with respect to the restriction maps, where 𝑈 runs through the non-empty standard affine opensubschemes 𝐷(𝑓𝑖) of an non-empty affine open subscheme Spec(𝐴) ⊆ 𝑋.

Proof. Corollary to Lemma 2.2.9.

3http://mathoverflow.net/questions/84414/etale-cohomology-with-coefficients-in-the-integers

http://mathoverflow.net/questions/84414/etale-cohomology-with-coefficients-in-the-integers


Lemma 3.3.17 (Excision of codimension ≥ 2 subschemes). One can excise subschemes 𝑍 →˓ 𝑌 of codimension≥ 2 in 𝑋:

ker(H1(𝑈,A )→ H2

𝑌 (𝑋,A ))

= ker(

H1(𝑈,A )→ H2𝑌 ∖𝑍(𝑋 ∖ 𝑍,A |𝑋∖𝑍)

). (3.3.2)

Proof. From the long exact localisation sequence for cohomology with supports [Mil80], p. 92, Remark III.1.26,and from Lemma 3.3.15 one gets the injectivity

0→ H2𝑌 (𝑋,A ) →˓ H2

𝑌 ∖𝑍(𝑋 ∖ 𝑍,A |𝑋∖𝑍), (3.3.3)

hence by Lemma 2.1.2 the claim.

Lemma 3.3.18. Let 𝑌 →˓ 𝑋 be a closed subscheme with open complement 𝑈 = 𝑋 ∖ 𝑌 and with all of itsirreducible components of codimension 1 in 𝑋. Denote the finitely many irreducible components of 𝑌 by (𝑌𝑖)

𝑛𝑖=1.

Then


𝑌 (𝑋,A ))

= ker

(H1(𝑈,A )→

𝑛⨁𝑖=1

H2𝑌𝑖∖𝑍𝑖

(𝑋 ∖ 𝑍𝑖,A )

)(3.3.4)

with certain closed subschemes 𝑍𝑖 →˓ 𝑌𝑖.

Proof. Excise the intersections 𝑌𝑖 ∩𝑌𝑗 for 𝑖 = 𝑗 (they are of codimension ≥ 2 in 𝑋 since our schemes are catenaryas they are varieties by [Liu06], p. 338, Corollary 8.2.16 and

2 = 1 + 1 ≤ codim(𝑌𝑖 ∩ 𝑌𝑗 →˓ 𝑌 ) + codim(𝑌 →˓ 𝑋) = codim(𝑌𝑖 ∩ 𝑌𝑗 →˓ 𝑋)).

Now, by a repeated application of the Mayer-Vietoris sequence with supports Theorem 2.2.10, one gets theclaim.

Lemma 3.3.19. Let 𝑋 be a regular variety over a finite field and 𝑥 ∈ 𝑋(1). Then there is an open affinesubscheme Spec𝐴 = 𝑋0 ⊆ 𝑋 containing 𝑥 such that 𝐴 is a unique factorisation domain.

Proof. The class group Cl(𝑋) = Pic(𝑋) (using that 𝑋 is regular) is finitely generated since NS(𝑋) is so by [Mil80],p. 215, Theorem IV.3.25 and Pic0(𝑋) is the group of rational points of the Picard scheme and the ground field isfinite. Excise the finite set of generators of the Picard group using [Har83], p. 133, Proposition II.6.5. (If {𝑥} isone of the generators, replace it by a moving lemma.) Then the rest of the variety has trivial class group. Nowtake an open affine subset of the remaining scheme containing 𝑥. It is normal and has trivial class group, soby [Har83], p. 131, Proposition II.6.2 it is a unique factorisation domain.

Lemma 3.3.20. One has


𝑌 (𝑋,A ))

= ker

(H1(𝑈,A )→

𝑛⨁𝑖=1

H2{𝑥𝑖}(𝑋 ∖ 𝑍𝑖,A )

)(3.3.5)

with the 𝑥𝑖 ∈ 𝑋 ∖ 𝑍𝑖 closed points and generic points of the 𝑌𝑖 and 𝑍𝑖 →˓ 𝑋 certain subschemes.

Proof. This follows basically by excising (using (3.3.2)) everything except the generic points of the 𝑌𝑖 in (3.3.4).The only technical difficulty is that for applying Lemma 2.2.9 one has to make sure that the transition maps areaffine.

Fix an irreducible component 𝑌𝑖 of 𝑌 and call it 𝑌 with 𝑥 its generic point. We have to construct an injection

H2𝑌 ∖𝑍(𝑋 ∖ 𝑍,A ) →˓ H2

{𝑥}(𝑋 ∖ 𝑍,A ). (3.3.6)

Using Lemma 3.3.19, choose an affine open 𝑋0 := Spec𝐴 ⊂ 𝑋 ∖ 𝑍 containing the point 𝑥 such that 𝐴 is aunique factorisation domain. One has H𝑞

𝑌 (𝑋,A ) →˓ H𝑞𝑌 ∩𝑋0

(𝑋0,A ): 𝑋 ∖𝑋0 is a closed subscheme 𝑉 →˓ 𝑋 suchthat 𝑉 ∩ 𝑌 →˓ 𝑌 is of codimension ≥ 1 and hence (since varieties are catenary by [Liu06], p. 338, Corollary 2.16)𝑉 ∩𝑌 →˓ 𝑋 of codimension ≥ 2, so we conclude by excision (3.3.2). We construct a sequence (𝑋𝑖)

∞𝑖=0 of standard

affine open subsets 𝑋𝑖 = 𝐷(𝑓𝑖) ⊂ 𝑋0, 𝑋𝑖+1 ⊂ 𝑋𝑖, all of them containing the point 𝑥 such that

H𝑞𝑌 ∩𝑋𝑖

(𝑋𝑖,A ) →˓ H𝑞𝑌 ∩𝑋𝑖+1

(𝑋𝑖+1,A ), and thus by (3.3.2)


𝑌 (𝑋,A ))

= ker(H1(𝑈,A )→ H2

𝑌 ∩𝑋𝑖(𝑋𝑖,A )

),

and such that lim←−𝑗 𝑋𝑖 ∩ 𝑌 = {𝑥}. Then (3.3.6) follows from Lemma 2.2.9.


Construction of the (𝑋𝑖)∞𝑖=0. Since 𝑋 is countable, one can choose an enumeration (𝑍𝑖)

∞𝑖=1 of the closed

integral subschemes of codimension 1 of 𝑋0 not equal to 𝑋0 ∩ 𝑌 .

Given 𝑋𝑖 = 𝐷(𝑓𝑖), take 𝑓 ∈ Spec𝐴𝑓𝑖 such that 𝑉 (𝑓) = 𝑍𝑖+1. (By the converse of Krull’s Hauptideal-satz [Eis95], p. 233, Corollary 10.5, since 𝑍𝑖+1 is of codimension 1 in 𝑋0, there is an 𝑓 ∈ 𝐴 such that 𝑍𝑖+1 ⊆ 𝑉 (𝑓)is minimal. Since 𝑋0 is a unique factorisation domain, one can assume 𝑓 prime, hence 𝑍𝑖+1 = 𝑉 (𝑓) bycodimension reasons.) Then 𝑉 (𝑓) ∩ (𝑋(1) ∩ 𝑌 ) = ∅ and 𝑉 (𝑓) ∩ 𝑌 ( 𝑌 has codimension ≥ 1 in 𝑌 , hence (again,varieties being catenary) 𝑉 (𝑓) ∩ 𝑌 has codimension ≥ 2 in 𝑋. Therefore one can apply (3.3.3) to yield aninjection

H2𝑌 ∩𝑋𝑖

(𝑋𝑖,A ) →˓ H2(𝑌 ∩𝑋𝑖)∖𝑉 (𝑓)(𝑋𝑖 ∖ (𝑌 ∩ 𝑉 (𝑓)),A ).

Now, by excision ([Mil80], p. 92, Proposition III.1.27) of 𝑉 (𝑓), one has

H2(𝑌 ∩𝑋𝑖)∖𝑉 (𝑓)(𝑋𝑖 ∖ (𝑌 ∩ 𝑉 (𝑓)),A )

∼−−→ H2(𝑌 ∩𝑋𝑖)∖𝑉 (𝑓)(𝑋𝑖 ∖ 𝑉 (𝑓),A ).

Set 𝑋𝑖+1 = 𝑋𝑖 ∖ 𝑉 (𝑓) = 𝐷(𝑓𝑖𝑓), 𝑓𝑖+1 = 𝑓𝑖𝑓 .

Apply this to the direct summands in (3.3.4).

Proof of Theorem 3.3.6. Now for the proof of Theorem 3.3.6, at least for prime-to-𝑝 torsion:

First note that the map H1(𝑋,A ) → ker(. . .) in Theorem 3.3.6 is well-defined since if 𝑥 ∈ H1(𝑋,A ) isrestricted to H1(𝐾,A ) via 𝑔 : {𝜂} →˓ 𝑋, its pullback to 𝐾𝑛𝑟

𝑥 = Spec(Quot(O𝑠ℎ𝑋,𝑥)) factors as

H1(𝑋,A )→ H1(Spec(O𝑠ℎ𝑋,𝑥),A )→ H1(Spec(Quot(O𝑠ℎ

𝑋,𝑥)),A ),

but the etale site of Spec(O𝑠ℎ𝑋,𝑥) is trivial by Lemma 2.2.7.

Let ∅ = 𝑈 →˓ 𝑋 be open with reduced closed complement 𝑌 →˓ 𝑋. Then one has the long exact localisationsequence [Mil80], p. 92, Proposition III.1.25

0→ H0𝑌 (𝑋,A )→ H0(𝑋,A )→ H0(𝑈,A )→ H1

𝑌 (𝑋,A )→ H1(𝑋,A )→ H1(𝑈,A )→ . . . .

Because of the injectivity of H0(𝑋,A ) →˓ H0(𝑈,A ) (If two sections of A /𝑋 coincide on 𝑈 open dense, theyagree on 𝑋 since A is separated and 𝑋 reduced), the exactness of the sequence yields H0

𝑌 (𝑋,A ) = 0, and hencea short exact sequence

0→ H1𝑌 (𝑋,A )/A (𝑈)→ H1(𝑋,A )→ ker

(H1(𝑈,A )→ H2

𝑌 (𝑋,A ))→ 0. (3.3.7)

Conclusion of the proof I, surjectivity. Applying excision in the form of [Mil80], p. 93, Corollary III.1.28to (3.3.5) (using that the 𝑥𝑖 ∈ 𝑋 ∖ 𝑍𝑖 are closed points), one gets


𝑌 (𝑋,A ))

= ker

(H1(𝑈,A )

(𝑟𝑖)→𝑛⨁𝑖=1

H2{𝑥𝑖}(𝑋ℎ

𝑥𝑖,A )

).

Now (𝑟𝑖) factors as

𝑟𝑖 : H1(𝑈,A )𝑗*𝑥𝑖→ H1(𝐾ℎ

𝑥𝑖,A )

𝛿𝑖→ H2{𝑥𝑖}(𝑋ℎ

𝑥𝑖,A ),

where the latter map is the boundary map of the localisation sequence associated to the discrete valuation ringOℎ𝑋,𝑥𝑖

(𝑋 is normal as it is smooth over a field, 𝑥𝑖 is a codimension-1 point, and the Henselisation of a normalring is normal again by [Fu11], p. 106, Proposition 2.8.10, and normal rings are (R1))

𝑋ℎ𝑥𝑖

= Spec(Quot(Oℎ𝑋,𝑥𝑖

)) ∪ {𝑥𝑖}

(𝑥𝑖 is the closed point [Henselisation preserves residue fields], and Spec(𝐾ℎ𝑥𝑖

) is the generic point of 𝑋ℎ𝑥𝑖

). Onehas H1(𝑋ℎ

𝑥𝑖,A ) = H1(𝜅(𝑥𝑖),A ) and the inflation-restriction exact sequence

0→ H1(𝑋ℎ𝑥𝑖,A )

inf→ H1(𝐾ℎ𝑥 ,A )

res→ H1(𝐾𝑛𝑟𝑥 ,A ).


Since O𝑠ℎ𝑋,𝑥𝑖

is a discrete valuation ring (as above; the strict Henselisation of a normal ring is normal againby [Fu11], p. 111, Proposition 2.8.18), the valuative criterion of properness [EGAII], p. 144 f., Theoreme (7.3.8)gives us A (𝐾𝑛𝑟

𝑥𝑖) = A (𝑋𝑠ℎ

𝑥𝑖). Hence one can write 𝑗*𝑖 = inf as the inflation

H1(𝜅(𝑥𝑖),A (𝑋𝑠ℎ𝑥𝑖

)) →˓ H1(𝐾ℎ𝑥𝑖,A ),

so the cokernel of 𝑗*𝑥𝑖injects into H1(𝐾𝑛𝑟

𝑥𝑖,A ) and in H2

{𝑥𝑖}(𝑋ℎ𝑥𝑖,A ), so we get

ker

(H1(𝑈,A )→

𝑛⨁𝑖=1

coker(𝑗𝑥𝑖)

)∼−−→ ker

(H1(𝑈,A )→

𝑛⨁𝑖=1

H2{𝑥𝑖}(𝑋ℎ

𝑥𝑖,A )

)

ker

(H1(𝑈,A )→

𝑛⨁𝑖=1

coker(𝑗𝑥𝑖)

)∼−−→ ker

(H1(𝑈,A )→

𝑛⨁𝑖=1

H1(𝐾𝑛𝑟𝑥𝑖,A )

)

and hence


𝑌 (𝑋,A ))

= ker

(H1(𝑈,A )

(𝑟𝑖)→𝑛⨁𝑖=1

H1(𝐾𝑛𝑟𝑥𝑖,A )

).

Conclusion of the proof II, surjectivity. Taking the limit over all 𝑌 (choose an enumeration (𝑌𝑖)∞𝑖=1 of

the integral closed subschemes of codimension 1 of 𝑋 [𝑋 is countable]) yields by Lemma 3.3.16 and the fact that(Ab) satisfies (AB5)

ker

(H1(𝐾,A )→ lim−→

𝑌

H2𝑌 (𝑋,A )

)= ker

⎛⎝H1(𝐾,A )→⨁

𝑥∈𝑋(1)

H1(𝐾𝑛𝑟𝑥 ,A )

⎞⎠ .

Now (3.3.7) gives us

0→ lim−→H1𝑍(𝑋,A )/A (𝑈)→ H1(𝑋,A )→ ker

⎛⎝H1(𝐾,A )→⨁

𝑥∈𝑋(1)


⎞⎠→ 0.

Injectivity. But the latter, surjective, map factors as (with the isomorphism from Theorem 3.3.4)

H1(𝑋,A )∼−−→ ker

(H1(𝐾,A )→

∏𝑥∈𝑋


)

→˓ ker

⎛⎝H1(𝐾,A )→⨁

𝑥∈𝑋(1)


⎞⎠ ,

so the claim follows.Now one has to check that the isomorphism is in fact the one induced by the natural maps. We skip this.

Remark 3.3.21. For generalising the previous Theorem 3.3.6 from Jacobians to arbitrary Abelian schemes, itwould be desirable to have a generalisation of the crucial Lemma 3.3.15, which depends on Theorem 3.1.15.

Closed points suffice in Theorem 3.3.6. In the proof of Theorem 3.3.4, even

H0(𝑋,R1𝑗*(𝑗*A ))→∏𝑥∈|𝑋|

R1𝑗*(𝑗*A )��

is injective by [Mil80], p. 65, Remark II.2.17 (b): If a section of an etale sheaf is non-zero, there is a closedgeometric point for which the stalk of the section is non-zero. This is because 𝑋/𝑘 is a variety and henceJacobson.

One can replace the strict Henselisation O𝑠ℎ𝑋,𝑥 of O𝑋,𝑥 by its completion O𝑠ℎ

𝑋,𝑥 (respectively by their quotient

fields) in the case of 𝑥 ∈ 𝑋(1):


Lemma 3.3.22. Let (𝐴,m) be a discrete valuation ring of a variety. Let 𝑍 be a smooth proper scheme of finitetype over 𝐴𝑠ℎ. The following are equivalent:

1. 𝑍 has a point over Quot(𝐴𝑠ℎ).

2. 𝑍 has a point over 𝐴𝑠ℎ.

3. 𝑍 has points over 𝐴𝑠ℎ/m𝑛𝐴𝑠ℎ = 𝐴𝑠ℎ/m𝑛𝐴𝑠ℎ for all 𝑛≫ 0.

4. 𝑍 has a point over 𝐴𝑠ℎ.

5. 𝑍 has a point over Quot(𝐴𝑠ℎ).

In 3, the equality 𝐴𝑠ℎ/m𝑛𝐴𝑠ℎ = 𝐴𝑠ℎ/m𝑛𝐴𝑠ℎ holds by [Eis95], p. 183, Theorem 7.1.

Proof. One has 1 ⇐⇒ 2 and 4 ⇐⇒ 5 by the valuative criterion for properness [EGAII], p. 144 f.,Theoreme (7.3.8), note that we are in codimension 1.

2 =⇒ 3 and 4 =⇒ 3 are trivial.3 =⇒ 4 is trivial since smooth implies formally smooth [EGAIV4], Definition (17.1.1).4 =⇒ 2 follows from Artin approximation, see Theorem 2.2.3, applied to the functor 𝑇 ↦→ 𝑍(𝑇 ) locally of

finite presentation.

3.4 Relation of the Brauer and Tate-Shafarevich group

From the above, one gets the “finiteness part” of the relation of the (generalised) Artin-Tate and the Birch-Swinnerton-Dyer conjecture:

Theorem 3.4.1 (The Artin-Tate and the Birch-Swinnerton-Dyer conjecture). Let 𝜋 : C → 𝑋 be projective andsmooth with C and 𝑋 regular, all fibres of dimension 1 and 𝑋 be a variety. Then one has an exact sequence

0→ 𝐾2 → Br(𝑋)𝜋*

→ Br(C )→X(PicC/𝑋/𝑋)→ 𝐾3 → 0

in which the groups 𝐾𝑖 annihilated by 𝛿, the index of the generic fibre 𝐶/𝐾, e. g. 𝛿 = 1 if C /𝑋 has a section,and their prime-to-𝑝 parts are finite, and 𝐾𝑖 = 0 if 𝜋 has a section. Here, X(PicC/𝑋/𝑋) sits in a short exactsequence

0→ Z/𝑑→X(Pic0C/𝑋/𝑋)→X(PicC/𝑋/𝑋)→ 0,

where 𝑑 | 𝛿.

Hence the finiteness of the (ℓ-torsion of the) Brauer group of C is equivalent to the finiteness of the (ℓ-torsionof the) Brauer group of the base 𝑋 and the finiteness of the (ℓ-torsion of the) Tate-Shafarevich group of PicC/𝑋 .

Proof. Combining Corollary 3.1.19 (here the theory of the Picard functor and the exactness of the sequence stillworks [we claim this here without proof] if we do not have a section, but etale locally a section; and the latteris the case since a smooth morphism factors locally into an etale morphism followed by an affine projection)with Theorem 3.1.22 and Theorem 3.3.6 yields the exact sequence

0→ 𝐾2 → Br(𝑋)𝜋*

→ Br(C )→ H1(𝑋,PicC/𝑋)→ 𝐾3 → 0

with the prime-to-𝑝 part of the 𝐾𝑖 finite, and 𝐾𝑖 = 0 if 𝜋 has a section.Now the long exact sequence associated to the short exact sequence in Lemma 3.3.9 yields the exact sequence

H0(𝑋,PicC/𝑋)→ H0(𝑋,Z)→ H1(𝑋,Pic0C/𝑋)→ H1(𝑋,PicC/𝑋)→ H1(𝑋,Z) = 0,

and H1(𝑋,Z) = 0 using Lemma 3.3.12. Now, choose a Weil divisor 𝐷 on the generic fibre 𝐶/𝐾 of C /𝑋 withdegree 𝛿 the index of 𝐶/𝐾. By Lemma 3.1.18 and [Conb], p. 3, Proposition 4.1, �� is a Weil divisor on C ofdegree 𝛿, and its image under H0(𝑋,PicC/𝑋)→ H0(𝑋,Z) = Z (here we use that 𝑋 is connected) is 𝛿. Hencecoker(H0(𝑋,PicC/𝑋)→ H0(𝑋,Z) = Z) is a quotient of Z/𝛿.


3.5 Descent of finiteness of X under generically etale alterations

Theorem 3.5.1. Let 𝑓 : 𝑋 ′ → 𝑋 be a proper, surjective, generically etale ℓ′-morphism of regular schemes (anℓ′-alteration). Let ℓ be some prime invertible on 𝑋. If A is an Abelian scheme on 𝑋 such that the ℓ-torsion of theTate-Shafarevich group X(A ′/𝑋 ′) of A ′ := 𝑓*A = A ×𝑋 𝑋 ′ is finite, then the ℓ-torsion of the Tate-Shafarevichgroup X(A /𝑋) is finite.

Proof. Without loss of generality, assume 𝑋 integral with generic point 𝜂. Define 𝑋 ′𝜂 by the commutativity of

the cartesian diagram

𝑋 ′𝜂� � 𝑔

′//

𝑓

��

𝑋 ′

𝑓

��

{𝜂} �� 𝑔

// 𝑋.

(3.5.1)

Step 1: H1(𝑋, 𝑓*A ′)[ℓ∞] is finite. This follows from the low terms exact sequence

0→ H1(𝑋, 𝑓*A′)→ H1(𝑋 ′,A ′)

associated to the Leray spectral sequence H𝑝(𝑋,R𝑞𝑓*A ′)⇒ H𝑝+𝑞(𝑋 ′,A ′) and the finiteness of H1(𝑋 ′,A ′)[ℓ∞] =X(A ′/𝑋 ′)[ℓ∞].

Step 2: The theorem holds if there is a trace morphism. Assume there is a trace morphism𝑓*𝑓

*A → A such that the composition with the adjunction morphism

A → 𝑓*𝑓*A → A

is multiplication with deg 𝑓 and deg 𝑓 is invertible on 𝑋. Let 𝐴 = H1(𝑋,A )[ℓ∞] and 𝐵 = H1(𝑋, 𝑓*A ′)[ℓ∞] anddenote the induced morphisms on cohomology by 𝑔 : 𝐴→ 𝐵 und ℎ : 𝐵 → 𝐴. By Lemma 4.1.21 below, 𝐴 iscofinitely generated. Since 𝐵 is a finite ℓ-group by Step 1, there is an 𝑁 ∈ N such that ℓ𝑁 ·𝐵 = 0. Then onehas ℓ𝑁𝑔(𝐴) = 0, thus ℓ𝑁 (ℎ ∘ 𝑔) = ℓ𝑁 [deg 𝑓 ] = 0 as an endomorphism of 𝐴. As 𝐴 is cofinitely generated andℓ𝑁 · deg 𝑓 = 0, the finiteness of 𝐴 follows.

Step 3: Construction of the trace morphism for 𝑋 = Spec 𝑘 a field. Since 𝑓 is etale, by [Fu11],p. 205, Proposition 5.5.1 (i), 𝑓! is left adjoint to 𝑓* and by loc. cit. (iv), one has

(𝑓!F )�� =⨁

𝑥′∈𝑋′⊗O𝑋𝑘(𝑥)

F��′ .

Since 𝑓 is etale, proper and of finite type, by [Fu11], p. 207, Proposition 5.5.2, one has 𝑓! = 𝑓*. Hence, there isthe adjunction morphism 𝑓*𝑓

*A = 𝑓!𝑓*A → A , and we have to prove that

A�� → (𝑓*𝑓*A )�� = (𝑓!𝑓

*A )�� → A��

is multiplication by deg 𝑓 . Since one can assume �� = Spec 𝑘 = 𝑋 with 𝑘 separably closed is strictly Henselian,𝑋 ′ = ⨿deg 𝑓

𝑖=1 Spec 𝑘. Hence the morphisms are

A�� → (𝑓*𝑓*A )��, 𝑠 ↦→ (𝑠)deg 𝑓𝑖=1 ,

(𝑓!𝑓*A )�� → A��, (𝑠𝑖)

deg 𝑓𝑖=1 ↦→

∑deg 𝑓𝑖=1 𝑠𝑖,

and the claim is obvious.Step 4: Construction of the trace morphism for 𝑋 arbitrary regular. One has to construct a

commutative diagram

A∼= //

��

𝑔*𝑔*A

��

𝑓*𝑓*A //

��

𝑔*𝑔*(𝑓*𝑓

*A )

��

A∼= // 𝑔*𝑔

*A

(3.5.2)


such that the composition of the vertical maps equals multiplication by deg 𝑓 .The upper square comes from the functoriality of the adjunction morphism associated to the natural

transformation of functors id→ 𝑔*𝑔*. The crucial point is to construct the lower right morphism 𝑔*𝑔

*(𝑓*𝑓*A )→

𝑔*𝑔*A corresponding to the trace morphism (it is not directly possible to apply 𝑔*𝑔

* to a map 𝑓*𝑓*A → A

since it is not clear how to define the latter). This map arises as follows: Since the domain of 𝑔 is a field, byStep 3, there is a trace morphism

𝑓*𝑓*(𝑔*A ) = 𝑓!𝑓

*(𝑔*A )→ 𝑔*A .

By the commutativity of the diagram (3.5.1), one has 𝑔′*𝑓*A = 𝑓*𝑔*A , giving us an arrow

𝑓*𝑔′*(𝑓*A )→ 𝑔*A .

By proper base change [Den88], p. 231, Theorem 1.1 (The assumptions are: Noetherian, 𝑓 proper, 𝑋 ′ excellent,𝑔′ normal, i. e. flat with geometrically normal fibres: 𝑔′ is flat as it is the base change of the flat morphism𝑔 : {𝜂} →˓ 𝑋; flat since 𝑔 is the inclusion of O𝑋,𝜂), 𝑔*𝑓* = 𝑓*𝑔

′*, so

𝑔*𝑓*(𝑓*A )→ 𝑔*A .

Finally, applying 𝑔*, we get our trace morphism on the right hand side.By Step 3, it is obvious that the composition of the vertical arrows on the right hand side is multiplication by

deg 𝑓 .Now the lower left arrow 𝑓*𝑓

*A → A can be defined by the commutativity of the lower square of (3.5.2),noting that A

∼−−→ 𝑔*𝑔*A is invertible by Theorem 3.2.2. It follows from the right hand side that the composition

of the vertical arrows on the left hand side is multiplication by deg 𝑓 .

(Note that this works also for more general sheaves than A .)

Theorem 3.5.2. In the situation of Corollary 3.1.19, if Br(C )[ℓ∞] is finite for ℓ invertible on 𝑋, Br(𝑋)[ℓ∞] isfinite.

Proof. Obvious from Corollary 3.1.19.

3.6 Isogeny invariance

Theorem 3.6.1. Let 𝑋/𝑘 be proper, A and A ′ Abelian schemes over 𝑋 and 𝑓 : A ′ → A an etale isogeny. Letℓ = char 𝑘 be a prime. Then X(A /𝑋)[ℓ∞] is finite if and only if X(A ′/𝑋)[ℓ∞] is finite.

Proof. The long exact cohomology sequence associated to the short exact sequence of etale sheaves (𝑓 is etale)0→ ker(𝑓)→ A ′ → A → 0

H1(𝑋, ker(𝑓))→ H1(𝑋,A ′)→ H1(𝑋,A )→ H2(𝑋, ker(𝑓))

implies the result since by [Mil80], p. 224 f., Corollary VI.2.8 the H𝑖(𝑋, ker(𝑓)) are finite because of ker(𝑓) beingconstructible.

3.7 The generalised Cassels-Tate pairing

Let 𝑘 = F𝑞 be a finite field with absolute Galois group Γ = 𝐺𝑘 and 𝑋/𝑘 a smooth projective geometricallyconnected variety of dimension 𝑑. Let A /𝑋 be an Abelian scheme and ℓ = 𝑝 = char 𝑘 be prime.

Reduction to the curve case. Let 𝑌 →˓ 𝑋 be an ample smooth geometrically connected hypersurface section(this exists by [Poo05], Proposition 2.7) with (necessarily) affine complement 𝑈 →˓ 𝑋. Base changing to 𝑘 andwriting �� = 𝑋 ×𝑘 𝑘 etc., one has by [Mil80], p. 94, Remark III.1.30 a long exact sequence

. . .→ H𝑖𝑐(�� ,A [ℓ𝑛])→ H𝑖(��,A [ℓ𝑛])→ H𝑖(𝑌 ,A [ℓ𝑛])→ H𝑖+1

𝑐 (�� ,A [ℓ𝑛])→ . . . (3.7.1)

(Note that H𝑖𝑐(��,F ) = H𝑖(��,F ) since �� is proper, and likewise for 𝑌 .)

Since A [ℓ𝑛]/𝑋 is etale, Poincare duality [Mil80], p. 276, Corollary VI.11.2 gives us

H𝑖𝑐(�� ,A [ℓ𝑛]) = H2𝑑−𝑖(�� , (A [ℓ𝑛])∨(𝑑)).


(Note that the varieties live over a separably closed field.) By the affine Lefschetz theorem [Mil80], p. 253,Theorem VI.7.2, one has H2𝑑−𝑖(�� , (A [ℓ𝑛])∨(𝑑)) = 0 for 2𝑑− 𝑖 > 𝑑, i. e. for 𝑖 < 𝑑. Analogously, H𝑖+1

𝑐 (�� ,A [ℓ𝑛]) = 0for 𝑖+ 1 < 𝑑. Plugging this into (3.7.1), one gets an isomorphism

H𝑖(��,A [ℓ𝑛])∼−−→ H𝑖(𝑌 ,A [ℓ𝑛])

for 𝑖+ 1 < 𝑑. Inductively, it follows that the cohomology groups in dimension 0 and 1 are isomorphic to thecohomology groups of a surface 𝑆.

If 𝑌 = 𝐶 →˓ 𝑋 is a curve, one gets at least

H0(��,A [ℓ𝑛])∼−−→ H0(𝐶,A [ℓ𝑛]), (3.7.2)

H1(��,A [ℓ𝑛]) →˓ H𝑖(𝐶,A [ℓ𝑛]). (3.7.3)

There are short exact sequences

0→ H𝑖−1(��,A [ℓ𝑛])Γ → H𝑖(𝑋,A [ℓ𝑛])→ H𝑖(��,A [ℓ𝑛])Γ → 0.

Using (3.7.2), one gets a commutative diagram

H0(𝑋,A [ℓ𝑛])∼= //

��

H0(��,A [ℓ𝑛])Γ

∼=��

H0(𝐶,A [ℓ𝑛])∼= // H0(𝐶,A [ℓ𝑛])Γ,

and hence an isomorphism H0(𝑋,A [ℓ𝑛])∼−−→ H0(𝐶,A [ℓ𝑛]).

This impliesTor A (𝑋)[ℓ∞] = H0(𝑋,A [ℓ∞]) = H0(𝐶,A [ℓ∞]) = Tor A (𝐶)[ℓ∞]. (3.7.4)

Using (3.7.2) and (3.7.3), one gets a commutative diagram

0 // H0(��,A [ℓ𝑛])Γ

∼=��

// H1(𝑋,A [ℓ𝑛])� _

��

// H1(��,A [ℓ𝑛])Γ� _

��

// 0

0 // H0(𝐶,A [ℓ𝑛])Γ // H1(𝐶,A [ℓ𝑛]) // H1(𝐶,A [ℓ𝑛])Γ // 0.

and hence, by the snake lemma, an injection H1(𝑋,A [ℓ𝑛]) →˓ H1(𝐶,A [ℓ𝑛]).

Remark 3.7.1. By the snake lemma, this injection is an isomorphism iff H1(��,A [ℓ𝑛])Γ →˓ H1(𝐶,A [ℓ𝑛])Γ

is surjective, e. g. dim𝐶 ≥ 2. Under this condition, one gets (by passing to the inverse limit lim←−𝑛 and

tensoring with Qℓ) the equality of the 𝐿-functions 𝐿(A /𝑋, 𝑠) = 𝐿(A /𝐶, 𝑠), and hence, under assumption of theBirch-Swinnerton-Dyer conjecture (X(A /𝐶)[ℓ∞] finite), the equality of the ranks rkZ A (𝑋) = rkZ A (𝐶).

We want the map H1(𝑋,A )[ℓ𝑛] → H1(𝐶,A )[ℓ𝑛] of the ℓ𝑛-torsion of the Tate-Shafarevich groups to beinjective in order for the generalised Cassels-Tate pairing to be non-degenerate (at least on the non-divisiblepart).

0 // A (𝑋)/ℓ𝑛� _

��

// H1(𝑋,A [ℓ𝑛])� _

��

// H1(𝑋,A )[ℓ𝑛]

��

// 0

0 // A (𝐶)/ℓ𝑛 // H1(𝐶,A [ℓ𝑛]) // H1(𝐶,A )[ℓ𝑛] // 0

By the snake lemma, the left vertical map is an injection and H1(𝑋,A )[ℓ𝑛] → H1(𝐶,A )[ℓ𝑛] is injective ifthis injection A (𝑋)/ℓ𝑛 →˓ A (𝐶)/ℓ𝑛 is surjective. This is e. g. the case if rk A (𝐶) = 0 since Tor A (𝑋)[ℓ𝑛] =Tor A (𝐶)[ℓ𝑛] by (3.7.4) and A (𝑋)/ℓ𝑛 →˓ A (𝐶)/ℓ𝑛 is injective. For example, one has rk A (𝐶) = 0 if A /𝐶 isconstant and 𝐶 ∼= P1

𝑘, see Example 4.3.2 below.

Remark 3.7.2. Note that for dim𝑌 ≥ 2 and 𝑌 →˓ 𝑋 excised by a sequence of ample hypersurface sections,rk A (𝑋) = rk A (𝑌 ) by Remark 3.7.1.

32 4 THE SPECIAL 𝐿-VALUE IN COHOMOLOGICAL TERMS

The curve case. By [Mil86b], p. 176, (a), one has for 𝐶/𝑘 a smooth proper geometrically connected curveover the finite ground field 𝑘

H3(𝐶,G𝑚) = Q/Z.

By the local-to-global spectral sequence H𝑝(𝑋,E xt𝑞𝑋(A ,G𝑚))⇒ Ext𝑝+𝑞𝑋 (A ,G𝑚), and using H om𝑋(A ,G𝑚) =0 and the Barsotti-Weil formula Theorem 2.2.14 E xt1𝑋(A ,G𝑚) = A ∨, we get edge morphisms

H𝑟(𝑋,A ∨)→ Ext𝑟+1𝑋 (A ,G𝑚),

which are injective for 𝑟 = 1. Composing this for 𝑟 = 1 with the Yoneda pairing for 𝑟 = 2

Ext𝑟𝐶(A ,G𝑚)×H3−𝑟(𝐶,A )→ H3(𝐶,G𝑚)∼−−→ Q/Z,

induces a pairing

H1(𝐶,A ∨)×H1(𝐶,A )∪→ H3(𝐶,G𝑚) = Q/Z.

This is the Cassels-Tate pairing, see [Mil86b], p. 199–203.

Remark 3.7.3. Note that we cannot directly define a pairing with values in H2𝑑+1(𝑋,G𝑚) since this group isfinite for dim𝑋 = 𝑑 > 1 by Theorem 3.1.8 (a).

4 The special 𝐿-value in cohomological terms

4.1 The 𝐿-function

Let 𝑘 = F𝑞 be a finite field with 𝑞 = 𝑝𝑛 elements and let ℓ = 𝑝 be a prime. For a variety 𝑋/𝑘 denote by �� itsbase change to a separable closure 𝑘 of 𝑘.

Denote by Frob𝑞 the arithmetic Frobenius, by Γ the absolute Galois group of our finite base field 𝑘 = F𝑞, by𝑛 the dimension of the base scheme 𝑋 and by �� the base change 𝑋 ×𝑘 𝑘.

Let 𝑋/𝑘 be a smooth projective geometrically connected variety.

Definition 4.1.1. Let 𝜋 : A → 𝑋 be an Abelian scheme of (relative) dimension 𝑑. Then its 𝐿-function isdefined as

𝐿(A /𝑋, 𝑠) =∏𝑥∈|𝑋|

det(1− 𝑞−𝑠 deg(𝑥) Frob−1

𝑥 | (R1𝜋*Qℓ)��)−1

.

Here, Frob−1𝑥 is the geometric Frobenius and (R1𝜋*Qℓ)�� = H1(A 𝑥,Qℓ) by proper base change.

Proposition 4.1.2. One has

𝐿(A /𝑋, 𝑠) =

2𝑑∏𝑖=0

det(1− 𝑞−𝑠 Frob−1

𝑞 | H𝑖(��,R1𝜋*Qℓ))(−1)𝑖+1

,

where the Frobenius acts via functoriality on the second factor of �� = 𝑋 ×𝑘 𝑘.

Proof. This follows from the Lefschetz trace formula [KW01], p. 7, Theorem 1.1.

Definition 4.1.3. Let𝑃𝑖(A /𝑋, 𝑠) = det

(1− 𝑞−𝑠 Frob−1

𝑞 | H𝑖(��,R1𝜋*Qℓ)).

and define the relative 𝐿-function of an Abelian scheme A /𝑋 by

𝐿(A /𝑋, 𝑠) =𝑃1(A /𝑋, 𝑞−𝑠)

𝑃0(A /𝑋, 𝑞−𝑠).

For our purposes, it is better to consider the following 𝐿-function:

Definition 4.1.4. Let𝐿𝑖(A /𝑋, 𝑠) = det(1− 𝑞−𝑠 Frob−1

𝑞 | H𝑖(��, 𝑉ℓA )).

4 THE SPECIAL 𝐿-VALUE IN COHOMOLOGICAL TERMS 33

Remark 4.1.5. Note that

𝑃𝑖(A /𝑋, 𝑠) = det(1− 𝑞−𝑠 Frob−1𝑞 | H𝑖(��,R1𝜋*Qℓ))

= det(1− 𝑞−𝑠 Frob−1𝑞 | H𝑖(��, 𝑉ℓ(A

∨))(−1)) by (4.1.1) below

= det(1− 𝑞−𝑠𝑞 Frob−1𝑞 | H𝑖(��, 𝑉ℓA )) by Lemma 4.3.21 below

= 𝐿𝑖(A /𝑋, 𝑠− 1),

so the vanishing order of 𝑃𝑖(A /𝑋, 𝑠) at 𝑠 = 1 is equal to the vanishing order of 𝐿𝑖(A /𝑋, 𝑠) at 𝑠 = 0, and therespective leading coefficients agree.

Definition 4.1.6. A Qℓ[Γ]-module is said to be pure of weight 𝑛 if all eigenvalues 𝛼 of the geometric Frobeniusautomorphism are algebraic integers which have absolute value 𝑞𝑛/2 under all embeddings 𝜄 : Q(𝛼) →˓ C.

We often use the yoga of weights:

Theorem 4.1.7. Let 𝑓 : 𝑋 → 𝑌 be a smooth proper morphism of schemes of finite type over F𝑞 and F asmooth sheaf pure of weight 𝑛. Then R𝑖𝑓*F is a smooth sheaf pure of weight 𝑛+ 𝑖 for any 𝑖.

Proof. Apply Poincare duality to [Del80], p. 138, Theoreme 1.

Lemma 4.1.8. If 𝑉 is a Qℓ[Γ]-module pure of weight 𝑛, its 𝑖-th Tate twist 𝑉 (𝑖) is pure of weight 𝑛− 2𝑖.

Lemma 4.1.9. Let 𝑉 , 𝑊 be Qℓ[Γ]-modules pure of weight 𝑚 and 𝑛, respectively.(a) 𝑉 ⊗Qℓ

𝑊 is a Qℓ[Γ]-module pure of weight 𝑚+ 𝑛.(b) HomQℓ

(𝑉,𝑊 ) is a Qℓ[Γ]-module pure of weight 𝑛−𝑚.

Lemma 4.1.10. If 𝑉 is a Qℓ[Γ]-module pure of weight = 0, 𝑉Γ = 𝑉 Γ = 0. More generally, if 𝑉 and 𝑊 areQℓ[Γ]-modules pure of weights 𝑚 = 𝑛, every Γ-morphism 𝑉 →𝑊 is zero.

The following is a generalisation of [Sch82a], p. 134–138 and [Sch82b], p. 496–498.For the definition of an ℓ-adic sheaf see [FK88], p. 122, Definition 12.6.

Lemma 4.1.11. Let (𝐺𝑛)𝑛∈N be a Barsotti-Tate group consisting of finite etale group schemes. Then it is anℓ-adic sheaf.

Proof. By [Tat67], p. 161, (2)

0→ ker[ℓ]→ 𝐺𝑛+1[ℓ]→ 𝐺𝑛 → 0

is exact. But ker[ℓ] = ℓ𝑛𝐺𝑛+1. Furthermore, 𝐺𝑛 = 0 for 𝑛 < 0 and ℓ𝑛+1𝐺𝑛 = 0 by [Tat67], p. 161, (ii). Finally,the 𝐺𝑛 are constructible since they are finite etale group schemes.

Corollary 4.1.12. For ℓ invertible on 𝑋, 𝑇ℓA = (A [ℓ𝑛])𝑛∈N is an ℓ-adic sheaf.

Proof. This follows from Lemma 4.1.11 since ℓ is invertible on 𝑋, so A [ℓ𝑛]/𝑋 is finite etale.

Theorem 4.1.13. Let 𝐾 be an arbitrary field, ℓ = char𝐾 be prime and 𝐴/𝐾 an Abelian variety. Then we havean isomorphism of (ℓ-adic discrete) 𝐺𝐾-modules, equivalently, by [Mil80], p. 53, Theorem II.1.9, of (ℓ-adic)etale sheaves on Spec𝐾,

𝑇ℓ(𝐴) = H1(𝐴,Zℓ)∨.

In particular, 𝑇ℓ(𝐴) has weight −1.

Proof. Consider the Kummer sequence


on 𝐴. Taking etale cohomology, one gets an exact sequence of 𝐺𝐾-modules

0→ G𝑚(𝐴)/ℓ𝑛 → H1(𝐴,𝜇ℓ𝑛)→ H1(𝐴,G𝑚)[ℓ𝑛]→ 0.


Since Γ(𝐴,O𝐴) = �� is separably closed and ℓ = char𝐾, G𝑚(𝐴) is ℓ-divisible (one can extract ℓ-th roots), andhence

H1(𝐴,𝜇ℓ𝑛)∼−−→ H1(𝐴,G𝑚)[ℓ𝑛] = Pic(𝐴)[ℓ𝑛] = Pic0(𝐴)[ℓ𝑛],

the latter equality since NS(𝐴) is torsion-free by [Mum70], p. 178, Corollary 2. Taking Tate modules lim←−𝑛 yields

H1(𝐴,Zℓ(1))∼−−→ 𝑇ℓ Pic0(𝐴), (4.1.1)

so (the first equality coming from the perfect Weil pairing (4.3.4))

Hom(𝑇ℓ𝐴,Zℓ(1)) = 𝑇ℓ(𝐴∨) = H1(𝐴,Zℓ(1)),

so(𝑇ℓ𝐴)∨ = Hom(𝑇ℓ𝐴,Zℓ) = H1(𝐴,Zℓ),

so𝑇ℓ𝐴 = H1(𝐴,Zℓ)

∨.

Alternatively, 𝜋1(𝐴, 0) =∏ℓ 𝑇ℓ(𝐴) by [Mum70], p. 171, and H1(𝐴,Zℓ) = Hom(𝜋1(𝐴, 0),Zℓ) by Proposi-

tion 3.3.14.

Remark 4.1.14. Note that both 𝑇ℓ(−) and H1(−,Zℓ)∨ are covariant functors.

Lemma 4.1.15. Isogenous Abelian varieties over a finite field 𝑘 have the same number of points.

Proof. Let 𝑓 : 𝐴→ 𝐵 be an 𝑘-isogeny. Take Galois invariants of

0→ ker 𝑓(𝑘)→ 𝐴(𝑘)→ 𝐵(𝑘)→ 0

and using Lang-Steinberg Theorem 2.2.12 in the form H1(𝑘,𝐴) = 0 = H1(𝑘,𝐵) and the Herbrand quotientℎ(ker 𝑓(𝑘)) = 1 (since ker 𝑓(𝑘) is finite) yields |𝐴(𝑘)| = |𝐵(𝑘)|.

Alternatively, use that 𝐴(F𝑞𝑛) = ker(1− Frob𝑛𝑞 ) and 𝑓(1− Frob𝑛𝑞 ) = (1− Frob𝑛𝑞 )𝑓 and deg 𝑓 > 0 is finite,and take degrees.

Lemma 4.1.16. Let (F𝑛)𝑛∈N, F = lim←−𝑛 F𝑛 be an ℓ-adic sheaf. For every 𝑖, there is a short exact sequence

0→ H𝑖−1(��,F )Γ → H𝑖(𝑋,F )→ H𝑖(��,F )Γ → 0. (4.1.2)

The following argument is a generalisation of [Mil88], p. 78, Lemma 3.4.

Proof. Since Γ has cohomological dimension 1, we get from the Hochschild-Serre spectral sequence for ��/𝑋 [Mil80],p. 106, Remark III.2.21 (b) short exact sequences for every 𝑛 and 𝑖

0→ H𝑖−1(��,F𝑛)Γ → H𝑖(𝑋,F𝑛)→ H𝑖(��,F𝑛)Γ → 0.

Since all involved groups are finite (because the two outer groups are finite by [Mil80], p. 224, Corollary VI.2.8since ��/𝑘 is proper and F𝑛 is constructible by definition of an ℓ-adic sheaf), the system satisfies the Mittag-Lefflercondition, so taking the projective limit yields an exact sequence

0→ lim←−𝑛

(H𝑖−1(��,F𝑛)Γ)→ H𝑖(𝑋,F )→ lim←−𝑛

(H𝑖(��,F𝑛)Γ)→ 0.

Write 𝑀 [𝑛] for H𝑖(��,F𝑛). Breaking the exact sequence

0→𝑀 [𝑛]Γ →𝑀 [𝑛]Frob−1→ 𝑀 [𝑛]→𝑀 [𝑛]Γ → 0

into two short exact sequences and applying lim←−𝑛, one obtains, setting 𝑄[𝑛] = (Frob−1)𝑀 [𝑛], two exact sequences

0→ lim←−𝑛

(𝑀 [𝑛]Γ)→ lim←−𝑛

(𝑀 [𝑛])Frob−1→ lim←−

𝑛

(𝑄𝑛)→ lim←−𝑛

1(𝑀 [𝑛]Γ) (4.1.3)

0→ lim←−𝑛

(𝑄[𝑛])→ lim←−𝑛

(𝑀 [𝑛])→ lim←−𝑛


1(𝑄[𝑛]) (4.1.4)


Since the 𝑀 [𝑛], and hence the 𝑀 [𝑛]Γ are finite (argument as above), they form a Mittag-Leffler system, andhence one gets from (4.1.3) an exact sequence

0→ lim←−𝑛



𝑛

(𝑄𝑛)→ 0.

Similarly, the 𝑄[𝑛] ⊆𝑀 [𝑛] are finite, and hence

0→ lim←−𝑛

(𝑄[𝑛])→ lim←−𝑛


(𝑀 [𝑛]Γ)→ 0

is exact from (4.1.4). Combining the above two short exact sequences, one gets the exactness of

0→ lim←−𝑛



𝑛


(𝑀 [𝑛]Γ)→ 0,

which shows that for all 𝑖

lim←−𝑛

(H𝑖(��,F𝑛)Γ) = lim←−𝑛

(𝑀 [𝑛]Γ) = ker(lim←−𝑛


𝑛

(𝑀 [𝑛])) = H𝑖(��,F )Γ

lim←−𝑛

(H𝑖(��,F𝑛)Γ) = lim←−𝑛

(𝑀 [𝑛]Γ) = coker(lim←−𝑛


𝑛

(𝑀 [𝑛])) = H𝑖(��,F )Γ,

which is what we wanted.

This implies

H2𝑑(��, 𝑇ℓA )Γ∼−−→ H2𝑑+1(𝑋,𝑇ℓA ) (4.1.5)

since H2𝑑+1(��, 𝑇ℓA ) = 0 for dimension reasons. Because of H𝑖(��, 𝑇ℓA ) = 0 for 𝑖 > 2𝑑, it follows thatH𝑖(𝑋,𝑇ℓA ) = 0 for 𝑖 > 2𝑑+ 1. Furthermore, one has

Zℓ = (Zℓ)Γ = H2𝑑(��,Zℓ(𝑑))Γ∼−−→ H2𝑑+1(𝑋,Zℓ(𝑑)),

the second equality by Poincare duality [Mil80], p. 276, Theorem VI.11.1 (a).

Lemma 4.1.17. Let Frob be a topological generator of Γ and 𝑀 be a finitely generated Zℓ-module with continuousΓ-action. Then the following are equivalent:

1. det(1− Frob |𝑀 ⊗ZℓQℓ) = 0.

2. H0(Γ,𝑀) = 𝑀Γ is finite.

3. H1(Γ,𝑀) is finite.

If one of these holds, we have H1(Γ,𝑀) = 𝑀Γ and

|det(1− Frob |𝑀 ⊗ZℓQℓ)|ℓ =

H0(Γ,𝑀)

|H1(Γ,𝑀)|

=

𝑀Γ

|𝑀Γ|

=|ker (1− Frob)||coker (1− Frob)|

= 𝑞(1− Frob).

Proof. See [BN78], p. 42, Lemma (3.2). If 𝑀 is torsion, H1(Γ,𝑀) = 𝑀Γ by [NSW00], p. 69, (1.6.13) Proposition (i).For the last equality, see Lemma 2.1.14.

Since 𝑇ℓA = H1(𝐴,Zℓ)∨ (see Theorem 4.1.13) has weight −1, H𝑖(��, 𝑇ℓA ) has weight 𝑖− 1. So the conditions

of Lemma 4.1.17 are fulfilled for the Γ-module 𝑀 = H𝑖(��, 𝑇ℓA ) and 𝑖 = 1. Therefore infinite groups in the shortexact sequences in (4.1.2) can only occur in the following two sequences:

0 // H1(��, 𝑇ℓA )Γ𝛽// H2(𝑋,𝑇ℓA ) // H2(��, 𝑇ℓA )Γ // 0

0 // H0(��, 𝑇ℓA )Γ // H1(𝑋,𝑇ℓA )𝛼 // H1(��, 𝑇ℓA )Γ

𝑓

kk

// 0

(4.1.6)


Here, 𝑓 is induced by the identity on H1(��, 𝑇ℓA ). Since H2(��, 𝑇ℓA )Γ and H0(��, 𝑇ℓA )Γ are finite (havingweight 2− 1 = 0 and 0− 1 = 0), 𝛼 and 𝛽 are quasi-isomorphisms, i. e. they have finite kernel and cokernel.

Recall that𝐿1(A /𝑋, 𝑡) = det(1− Frob−1

𝑞 𝑡 | H1(��, 𝑉ℓA )).

Define ��1(A /𝑋, 𝑡) and 𝜌 by

𝐿1(A /𝑋, 𝑡) = (𝑡− 1)𝜌 · ��1(A /𝑋, 𝑡),

𝜌 = ord𝑡=1

𝐿1(A /𝑋, 𝑡) ∈ N.

By writing Frob−1𝑞 in Jordan normal form, one sees that 𝜌 is equal to

dimQℓ

⋃𝑛≥1

ker(1− Frob−1𝑞 )𝑛 ≥ dimQℓ

ker(1− Frob−1𝑞 ) = dimQℓ

H1(��, 𝑉ℓA )Γ,

i. e.𝜌 ≥ dimQℓ

H1(��, 𝑉ℓA )Γ,

and that equality holds iff the operation of the Frobenius on H1(��, 𝑉ℓA ) is semi-simple at 1, i. e.

dimQℓ

⋃𝑛≥1

ker(1− Frob−1𝑞 )𝑛 = dimQℓ

ker(1− Frob−1𝑞 ),

i. e. the generalised eigenspace at 1 equals the eigenspace, which is equivalent to 𝑓Qℓin (4.1.6) being an

isomorphism, i. e. 𝑓 being a quasi-isomorphism. From (4.1.6), since H0(��, 𝑇ℓA )Γ is finite, one sees that

dimQℓH1(��, 𝑉ℓA )Γ = rkZℓ

H1(��, 𝑇ℓA )Γ = rkZℓH1(𝑋,𝑇ℓA ).

Corollary 4.1.18. If 𝑖 = 1, one has

|𝐿𝑖(A /𝑋, 𝑞−1)|ℓ =

H𝑖(��, 𝑇ℓA )Γ

H𝑖(��, 𝑇ℓA )Γ

.Proof. This follows from Lemma 4.1.17 1 if 𝑖 = 1 since H𝑖(��, 𝑉ℓA ) has weight 𝑖− 1.

Lemma 4.1.19. 𝜌 = rkZℓH1(𝑋,𝑇ℓA ) iff 𝑓 is a quasi-isomorphism. In this case,

|��1(A /𝑋, 𝑞−1)|−1ℓ = 𝑞(𝑓) =

| coker 𝑓 || ker 𝑓 |

and

|��1(A /𝑋, 𝑞−1)|−1ℓ = 𝑞((𝛽𝑓𝛼)Tors) ·

H0(��, 𝑇ℓA )Γ

H2(��, 𝑇ℓA )Γ

· Tor H2(𝑋,𝑇ℓA )

|Tor H1(𝑋,𝑇ℓA )|.

The idea is that for infinite cohomology groups H1(��, 𝑇ℓA ), one should insert a regulator term 𝑞(𝑓) or𝑞((𝛽𝑓𝛼)Tors).

Proof. From the discussion above, the first statement follows. Assuming this, one has

|��1(A /𝑋, 𝑞−1)| = |(Frob𝑞 −1)H𝑖(��, 𝑇ℓA )Γ||(Frob𝑞 −1)H𝑖(��, 𝑇ℓA )Γ|

=|(Frob𝑞 −1)H𝑖(��, 𝑇ℓA )Γ|

|(Frob𝑞 −1)H𝑖(��, 𝑇ℓA ) : (Frob𝑞 −1)2H𝑖(��, 𝑇ℓA )|

=| ker 𝑓 || coker 𝑓 |

= 𝑞(𝑓)−1.

For the second equation,

𝑞(𝑓) =𝑞(Tor(𝛽𝑓))

𝑞(𝛽)· 𝑞((𝛽𝑓)Tors) by Lemma 2.1.14 2

=1

H2(��, 𝑇ℓA )Γ · Tor H2(𝑋,𝑇ℓA )Γ

Tor H1(��, 𝑇ℓA )Γ

· 𝑞((𝛽𝑓)Tors)

= 𝑞((𝛽𝑓𝛼)Tors) ·H0(��, 𝑇ℓA )Γ

H2(��, 𝑇ℓA )Γ


|Tor H1(𝑋,𝑇ℓA )|since coker(𝛼) = 0.


Lemma 4.1.20. Let ℓ = 𝑝 be invertible on 𝑋. Then the sequence

0→ A (𝑋)⊗Qℓ/Zℓ → H1(𝑋,A [ℓ∞])→ H1(𝑋,A )[ℓ∞]→ 0

is exact.

Proof. The short exact sequence of etale sheaves

0→ A [ℓ𝑛]→ A → A → 0

induces0→ A (𝑋)/ℓ𝑛 → H1(𝑋,A [ℓ𝑛])→ H1(𝑋,A )[ℓ𝑛]→ 0. (4.1.7)

Passing to the colimit lim−→𝑛yields the result.

Lemma 4.1.21. Let ℓ be invertible on 𝑋. Then the Zℓ-corank of X(A /𝑋)[ℓ∞] is finite.

Proof. From (4.1.7), one sees that H1(𝑋,A )[ℓ] is finite as it is a quotient of H1(𝑋,A [ℓ]) and A [ℓ]/𝑋 isconstructible. Hence X(A /𝑋)[ℓ∞] is cofinitely generated by Lemma 2.1.4.

Lemma 4.1.22. Let 𝑋/𝑘 be proper, ℓ be invertible on 𝑋. There is a long exact sequence

. . .→ H𝑖(𝑋,𝑇ℓA (𝑛))→ H𝑖(𝑋,𝑇ℓA (𝑛))⊗ZℓQℓ → H𝑖(𝑋,A [ℓ∞](𝑛))→ . . .

which induces isomorphismsH𝑖−1(𝑋,A [ℓ∞](𝑛))Div → Tor H𝑖(𝑋,𝑇ℓA (𝑛))

and short exact sequences

0→ H𝑖(𝑋,𝑇ℓA (𝑛))→ H𝑖(𝑋,𝑇ℓA (𝑛))⊗ZℓQℓ → Div H𝑖(𝑋,A [ℓ∞](𝑛))→ 0.

Proof. Consider for 𝑚,𝑚′ ∈ N the short exact sequence of etale sheaves

0→ A [𝑚] →˓ A [𝑚𝑚′]·𝑚→ A [𝑚′]→ 0.

Twisting with Zℓ(𝑛) gives the exact sequence

0→ A [𝑚](𝑛) →˓ A [𝑚𝑚′](𝑛)·𝑚→ A [𝑚′](𝑛)→ 0.

Setting 𝑚 = ℓ𝜇,𝑚′ = ℓ𝜈 , the associated long exact sequence is

. . .→ H𝑖(𝑋,A [ℓ𝜇](𝑛))→ H𝑖(𝑋,A [ℓ𝜇+𝜈 ](𝑛))→ H𝑖(𝑋,A [ℓ𝜈 ](𝑛))→ . . . .

Passing to the projective limit lim←−𝜇 and then to the inductive limit lim−→𝜈yields the desired long exact sequence

since all involved cohomology groups are finite by [Mil80], p. 224, Corollary VI.2.8 since 𝑋/𝑘 is proper and oursheaves are constructible.

For the second statement, consider the exact sequence

H𝑖−1(𝑋,𝑇ℓA (𝑛))⊗ZℓQℓ

𝑓→ H𝑖−1(𝑋,A [ℓ∞](𝑛))𝑑→ H𝑖(𝑋,𝑇ℓA (𝑛))

𝑔→ H𝑖(𝑋,𝑇ℓA (𝑛))⊗ZℓQℓ.

Since H𝑖(𝑋,𝑇ℓA (𝑛)) is a finitely generated Zℓ-module (since (A [ℓ𝑛])𝑛∈N is an ℓ-adic sheaf) and 𝑔 is inducedby the identity, we have ker 𝑔 = Tor H𝑖(𝑋,𝑇ℓA (𝑛)). Since H𝑖−1(𝑋,𝑇ℓA (𝑛)) is a finitely generated Zℓ-moduleand H𝑖−1(𝑋,A [ℓ∞](𝑛)) is a cofinitely generated ℓ-torsion module, we have im 𝑓 = Div H𝑖−1(𝑋,A [ℓ∞](𝑛)). Theclaim follows from the exactness of the sequence.

For the third statement, consider the exact sequence

H𝑖(𝑋,𝑇ℓA (𝑛))𝑔→ H𝑖(𝑋,𝑇ℓA (𝑛))⊗Zℓ

Qℓ𝑓→ H𝑖(𝑋,A [ℓ∞](𝑛)).

Since H𝑖(𝑋, 𝑇ℓA (𝑛)) is a finitely generated Zℓ-module (since (A [ℓ𝑛](𝑛))𝑛∈N is an ℓ-adic sheaf) and 𝑔 is inducedby the identity, we have ker 𝑔 = Tor H𝑖(𝑋,𝑇ℓA (𝑛)). Since H𝑖(𝑋,𝑇ℓA (𝑛)) is a finitely generated Zℓ-module andH𝑖(𝑋,A [ℓ∞](𝑛)) is a cofinitely generated ℓ-torsion module, we have im 𝑓 = Div H𝑖(𝑋,A [ℓ∞](𝑛)). The claimfollows from the exactness of the sequence.


Note that one can easily generalise this to ℓ-adic sheaves.

Lemma 4.1.23. Assume ℓ is invertible on 𝑋. Then one has the following identities for the etale cohomologygroups of 𝑋:

H𝑖(𝑋,𝑇ℓA ) = 0 for 𝑖 = 1, 2, . . . , 2𝑑+ 1 (4.1.8)

Tor H1(𝑋,𝑇ℓA ) = H0(𝑋,A [ℓ∞])Div = Tor H0(𝑋,A )[ℓ∞] (4.1.9)

Tor H2(𝑋,𝑇ℓA ) = H1(𝑋,A [ℓ∞])Div (4.1.10)

H1(𝑋,A [ℓ∞])Div = X(A /𝑋)[ℓ∞] if X(A /𝑋)[ℓ∞] is finite (4.1.11)

Proof. (4.1.8): For 𝑖 > 2𝑑+ 1 this follows from (4.1.2), and it holds for 𝑖 = 0 since H0(𝑋,A [ℓ𝑛]) ⊆ Tor A (𝑋) isfinite (since A (𝑋) is a finitely generated Abelian group by the Mordell-Weil theorem) hence its Tate-module istrivial by Lemma 2.1.7.

(4.1.9) and (4.1.10): From Lemma 4.1.22, we getTor H𝑖(𝑋,𝑇ℓA )

=H𝑖−1(𝑋,A [ℓ∞])Div

The desired equalities follow by plugging in 𝑖 = 1, 2.

Further, one has H0(𝑋,A [ℓ∞])Div = Tor H0(𝑋,A )[ℓ∞] in (4.1.9) because H0(𝑋,A [ℓ∞]) is cofinitely generatedby the Mordell-Weil theorem.

Finally, (4.1.11) holds since by Lemma 4.1.20, H1(𝑋,A [ℓ∞])Div = H1(𝑋,A )[ℓ∞] if the latter is finite, andthis equals X(A /𝑋)[ℓ∞].

Now we have two pairings

⟨·, ·⟩ℓ : H1(𝑋,𝑇ℓA )Tors ×H2𝑑−1(𝑋,𝑇ℓ(A∨)(𝑑− 1))Tors → H2𝑑(𝑋,Zℓ(𝑑))

pr*1→ H2𝑑(��,Zℓ(𝑑)) = Zℓ, (4.1.12)

(·, ·)ℓ : H2(𝑋,𝑇ℓA )Tors ×H2𝑑−1(𝑋,𝑇ℓ(A∨)(𝑑− 1))Tors → H2𝑑+1(𝑋,Zℓ(𝑑)) = Zℓ. (4.1.13)

Lemma 4.1.24. The regulator term 𝑞((𝛽𝑓𝛼)Tors) is defined iff 𝑓 is a quasi-isomorphism and then equalsdet ⟨·, ·⟩ℓdet(·, ·)ℓ

−1

ℓ

where both pairings are non-degenerate.

Proof. Using H2𝑑+1(𝑋,Zℓ(𝑑)) = Zℓ and H2𝑑(��,Zℓ(𝑑)) = Zℓ, there is a commutative diagram of pairings

H2(𝑋,𝑇ℓA )Tors× H2𝑑−1(𝑋,𝑇ℓ(A ∨)(𝑑− 1))Tors∪→

∼=��

Zℓ

(H1(��, 𝑇ℓA )Γ)Tors×

𝛽

OO

(H2𝑑−1(��, 𝑇ℓ(A ∨)(𝑑− 1))Γ)Tors∪→ Zℓ

(H1(��, 𝑇ℓA )Γ)Tors×

𝑓

OO

(H2𝑑−1(��, 𝑇ℓ(A ∨)(𝑑− 1))Γ)Tors∪→ Zℓ

H1(𝑋,𝑇ℓA )Tors×

𝛼 ∼=

OO

H2𝑑−1(𝑋,𝑇ℓ(A ∨)(𝑑− 1))Tors∪→

∼=

OO

Zℓ

By Poincare duality (using that 𝑇ℓA is a smooth sheaf since the A [ℓ𝑛] are etale), the pairing in the second lineis non-degenerate, hence the pairing in the first line is too (𝛽 is a quasi-isomorphism). (The upper right and thelower right arrows are isomorphisms since their kernel is (H2𝑑−2(��, 𝑇ℓ(A ∨)(𝑑− 1))Tors = 0 for weight reasons:(2𝑑− 2) + (−1)− 2(𝑑− 1) = −1 = 0; the lower left arrow is an isomorphism since 𝛼 is a quasi-isomorphism.)Hence if 𝑓 is a quasi-isomorphism, all pairings in the diagram are non-degenerate, and then the claimed equalityfor the regulator 𝑞((𝛽𝑓𝛼)Tors) = |coker(𝛽𝑓𝛼)Tors| follows.


Lemma 4.1.25. Let ℓ be invertible on 𝑋. Then one has a short exact sequence

0→ A (𝑋)⊗Z Zℓ𝛿→ H1(𝑋,𝑇ℓA )→ lim←−

𝑛

(H1(𝑋,A )[ℓ𝑛])→ 0.

If X(A /𝑋)[ℓ∞] = H1(𝑋,A )[ℓ∞] is finite, 𝛿 induces an isomorphism

A (𝑋)⊗Z Zℓ∼−−→ H1(𝑋,𝑇ℓA ).

Proof. Since ℓ is invertible on 𝑋, the short exact sequence of etale sheaves

0→ A [ℓ𝑛]→ Aℓ𝑛→ A → 0

induces a short exact sequence

0→ A (𝑋)/ℓ𝑛𝛿→ H1(𝑋,A [ℓ𝑛])→ H1(𝑋,A )[ℓ𝑛]→ 0

in cohomology, and passing to the limit lim←−𝑛 gives us the desired short exact sequence since the A (𝑋)/ℓ𝑛 arefinite by the Mordell-Weil theorem, so they satisfy the Mittag-Leffler condition.

The second claim follows from the short exact sequence and Lemma 2.1.7.

Lemma 4.1.26. One has 𝜌 ≥ rkZℓH1(𝑋,𝑇ℓA ). Then 1 ⇐⇒ 2 ⇐⇒ 3 and 4 ⇐⇒ 5 in the following; further

3 ⇐⇒ 4 assuming 𝜌 = rkZ A (𝑋).1. ⟨·, ·⟩ℓ and (·, ·)ℓ are non-degenerate.2. 𝑓 is a quasi-isomorphism.3. Equality holds 𝜌 = rkZℓ

H1(𝑋,𝑇ℓA ).

4. The canonical injection A (𝑋)⊗Z Zℓ𝛿→ H1(𝑋,𝑇ℓA ) is surjective.

5. The ℓ-primary part of the Tate-Shafarevich group X(A /𝑋)[ℓ∞] is finite.Furthermore, the following are equivalent:(a) 𝜌 = rkZ A (𝑋)(b) ⟨·, ·⟩ℓ is non-degenerate and the ℓ-primary part of the Tate-Shafarevich group X(A /𝑋)[ℓ∞] is finite.

Proof. 1 ⇐⇒ 2: See Lemma 4.1.24. 2 ⇐⇒ 3: This is Lemma 4.1.19. 3 ⇐⇒ 4: One has 𝜌 = rkZ A (𝑋) =rkZℓ

(A (𝑋) ⊗Z Zℓ) and by Lemma 4.1.25 rkZℓ(A (𝑋) ⊗Z Zℓ) ≤ rkZℓ

H1(𝑋,𝑇ℓA ), so this is an equality iff 𝛿in 3. is onto. 4 ⇐⇒ 5: By Lemma 2.1.7 and Lemma 2.1.8 lim←−𝑛(H1(𝑋,A )[ℓ𝑛]) = 𝑇ℓ(H

1(𝑋,A )) is trivial iff

H1(𝑋,A )[ℓ∞] = X(A /𝑋)[ℓ∞] is finite since X(A /𝑋)[ℓ∞] is a cofinitely generated Zℓ-module by Lemma 4.1.21.(a) =⇒ (b): Since 𝛿 in 4 is injective, one has rkZ A (𝑋) ≤ rkZℓ

H1(𝑋,𝑇ℓA ) ≤ 𝜌. Therefore, 𝜌 = rkZ A (𝑋)implies equality, and 3 and 4 follow, so 1–5 hold. (b) =⇒ (a): (b) is equivalent to 1–5, so from 4 one getsA (𝑋)⊗Z Zℓ

∼−−→ H1(𝑋,𝑇ℓA ), but by 3, 𝜌 = rkZℓH1(𝑋,𝑇ℓA ) = rkZ A (𝑋).

Define 𝑐 by

𝐿(A /𝑋, 𝑠) ∼ 𝑐 · (1− 𝑞1−𝑠)𝜌

∼ 𝑐 · (log 𝑞)𝜌(𝑠− 1)𝜌 for 𝑠→ 1,

see Remark 4.3.36. Note that 𝑐 ∈ Q since 𝐿(A /𝑋, 𝑠) is a rational function with Q-coefficients in 𝑞−𝑠, and 𝑐 = 0since 𝜌 is the vanishing order of the 𝐿-function at 𝑠 = 1 by definition of 𝜌 and the Riemann hypothesis.

Corollary 4.1.27. If 𝜌 = rkZℓH1(𝑋,𝑇ℓA ), then

|𝑐|−1ℓ = 𝑞((𝛽𝑓𝛼)Tors) ·

Tor H2(𝑋,𝑇ℓA )

|Tor H1(𝑋,𝑇ℓA )| ·

H2(��, 𝑇ℓA )Γ

.Proof. Using Lemma 4.1.19 for ��1(A /𝑋, 𝑡) and Corollary 4.1.18 for 𝐿0(A /𝑋, 𝑡), one gets

|𝑐|−1ℓ = 𝑞((𝛽𝑓𝛼)Tors) ·

H0(��, 𝑇ℓA )Γ

H2(��, 𝑇ℓA )Γ


|Tor H1(𝑋,𝑇ℓA )|·H0(��, 𝑇ℓA )Γ

H0(��, 𝑇ℓA )Γ

= 𝑞((𝛽𝑓𝛼)Tors) ·

1H2(��, 𝑇ℓA )Γ


|Tor H1(𝑋,𝑇ℓA )|·H0(��, 𝑇ℓA )Γ

1

.

For 0 = H0(𝑋,𝑇ℓA )∼−−→ H0(��, 𝑇ℓA )Γ use (4.1.2) with 𝑖 = 0 and (4.1.8).


Theorem 4.1.28 (The conjecture of Birch and Swinnerton-Dyer for Abelian schemes over higher-dimensionalbases, cohomological version). One has 𝜌 ≥ rkZℓ

H1(𝑋,𝑇ℓA ) and the following are equivalent:(a) 𝜌 = rkZ A (𝑋)(b) ⟨·, ·⟩ℓ and (·, ·)ℓ are non-degenerate and X(A /𝑋)[ℓ∞] is finite.If these hold, X(A /𝑋)[non-𝑝] is finite and we have

|𝑐|−1ℓ =

det ⟨·, ·⟩ℓdet(·, ·)ℓ

−1

ℓ

· |X(A /𝑋)[ℓ∞]||Tor A (𝑋)[ℓ∞]| ·

H2(��, 𝑇ℓA )Γ

.Proof. The first statement is Lemma 4.1.26. Now identify the terms in Corollary 4.1.27 using Lemma 4.1.23(cohomology groups) and Lemma 4.1.24 (regulator).

Remark 4.1.29. For example, the conjecture is true if 𝐿(A /𝑋, 1) = 0 since one then has 0 = 𝜌 ≥ rkZ A (𝑋) ≥ 0.

Remark 4.1.30. The (determinants of the) pairings ⟨·, ·⟩ℓ and (·, ·)ℓ are identified below: One has det(·, ·)ℓ = 1and det ⟨·, ·⟩ℓ is the regulator, see especially Remark 4.2.20.

Remark 4.1.31. For the vanishing of H2(��, 𝑇ℓA )Γ see Remark 4.3.40 below.

Remark 4.1.32. Assume X(A /𝑋)[ℓ∞] finite. The pairing (·, ·)ℓ has determinant 1, see the discussion below,and is thus non-degenerate. The pairing ⟨·, ·⟩ℓ equals the height pairing, see below, and is therefore by [Cona],p. 35, Theorem 9.15 non-degenerate.

Corollary 4.1.33. If, for some ℓ = 𝑝 = char 𝑘, X(A /𝑋)[ℓ∞] is finite and ⟨·, ·⟩ℓ and (·, ·)ℓ are non-degenerate,the Birch-Swinnerton-Dyer conjecture holds for every ℓ′ = 𝑝 and the prime-to-𝑝 torsion X(A /𝑋)[𝑝′] is finite aswell.

Proof. By Theorem 4.1.28 (b) for ℓ =⇒ (a) independent of ℓ =⇒ (b) for ℓ′, X(A /𝑋)[ℓ′∞] is finite for everyℓ′ = 𝑝. But since 𝑐 = 0 and the relation of |𝑐|−1

ℓ′ and |X(A /𝑋)[ℓ′∞]|, the prime-to-𝑝 torsion is finite.

4.2 Cohomological and height pairings

Theorem 4.2.1 (hard Lefschetz over algebraically closed ground fields). Let 𝑘 be an algebraically closedfield, 𝜄 : 𝑋 →˓ P𝑁

𝑘 a smooth projective scheme of dimension 𝑛, 𝜂 ∈ H2(𝑋,Zℓ(1)) the first Chern class ofO𝑋(1) = 𝜄*OP𝑁

𝑘(1) ∈ Pic(𝑋) and 𝑉 a potential pure Qℓ-sheaf. Then, for every 𝑖 ∈ N, the iterated cup product

(∪𝜂)𝑖 : H𝑛−𝑖(𝑋,𝑉 )→ H𝑛+𝑖(𝑋,𝑉 (𝑖))

is an isomorphism.

Proof. See [Del80], p. 250, Theoreme (6.2.13) and note that 𝑉 and

𝐷𝑉 [−2𝑛] = 𝑅Hom(𝑉, Qℓ(𝑛)[2𝑛])[−2𝑛] = Hom(𝑉, Qℓ(𝑛))

(for the description of 𝐷 see [Del80], p. 247, (6.2.1): The dualising complex 𝐾𝑋 for 𝑋 equidimensional ofdimension 𝑛 and smooth is Qℓ(𝑛)[2𝑛]) satisfy the condition that the 𝑖-th cohomology sheaf has dimension ofsupport ≤ 𝑛− 𝑖. This holds since the only cohomology sheaf lives in degree 𝑖 = 0 and 𝑛− 𝑖 = 𝑛 = dim𝑋.

Now assume 𝑘 = F𝑞 is finite. Using the Hochschild-Serre spectral sequence, one deduces the following

Theorem 4.2.2 (hard Lefschetz for finite ground fields). Let 𝑘 be a finite field, 𝜄 : 𝑋 →˓ P𝑁𝑘 a smooth projective

scheme of dimension 𝑑, 𝜂 ∈ H2(𝑋,Zℓ(1)) the first Chern class of O𝑋(1) = 𝜄*OP𝑁𝑘

(1) ∈ Pic(𝑋) and 𝑉 a potential

pure Qℓ-sheaf of weight 𝑤 = 0. Then the iterated cup product

(∪𝜂)𝑑−1 : H1(𝑋,𝑉 )→ H2𝑑−1(𝑋,𝑉 (𝑑− 1))

is an isomorphism.


Proof. Consider the diagram

0

��

0

��

H0(��, 𝑉 )Γ(∪𝜂)𝑑−1

//

��

H2𝑑−2(��, 𝑉 (𝑑− 1))Γ

��

H1(𝑋,𝑉 )(∪𝜂)𝑑−1

//

��

H2𝑑−1(𝑋,𝑉 (𝑑− 1))

��

H1(��, 𝑉 )Γ(∪𝜂)𝑑−1

∼=//

��

H2𝑑−1(��, 𝑉 (𝑑− 1))Γ

��

0 0

with exact columns coming from the Hochschild-Serre spectral sequence [Mil80], p. 105 f., Remark III.2.21 (b) forΓ = Gal(𝑘/𝑘), which has cohomological dimension 1. The two groups in the upper row are of weight 0 + 𝑤 = 0and (2𝑑− 2)− 2(𝑑− 1) + 𝑤 = 𝑤 = 0, respectively, hence equal to 0 by Lemma 4.1.10. As the arrow in the lowerrow is an isomorphism by the hard Lefschetz Theorem 4.2.1 for algebraically closed ground fields, the arrow inthe middle row is an isomorphism.

Theorem 4.2.3. Let 𝐴/𝑘 be an Abelian variety with principal polarisation 𝜗 ∈ Pic(𝐴)→ H2(𝐴,Zℓ(1)). Thenthe integral hard Lefschetz morphism (∪𝜗)𝑑−1 : H𝑑−𝑖(𝐴,Zℓ)→ H𝑑+𝑖(𝐴,Zℓ(𝑖)) is an isomorphism.

Proof. Using that 𝜗 is a principal polarisation, write 𝜗 =∑𝑑𝑖=1 𝑒𝑖 ∧ 𝑒′𝑖 in a symplectic basis (with respect to the

Weil pairing ∧ : 𝑇ℓ𝐴× 𝑇ℓ(𝐴∨)→ Zℓ(1); using the principal polarisation, the Weil pairing becomes a symplecticpairing 𝑇ℓ𝐴× 𝑇ℓ𝐴→ Zℓ(1)) and use that the cohomology ring H*(𝐴,Zℓ) =

⋀*H1(𝐴,Zℓ) is an exterior algebra.

By [Mil86a], p. 130, one has H*(𝐴,Zℓ) = (⋀*

𝑇ℓ𝐴)∨.

A basis of⋀1

𝑇ℓ𝐴 is 𝑒1, 𝑒′1, . . . , 𝑒𝑑, 𝑒

′𝑑, and a basis of

⋀2𝑑−1𝑇ℓ𝐴 is (a hat denotes the omission of a term)

𝑒1 ∧ 𝑒′1 ∧ . . . 𝑒𝑖 ∧ 𝑒′𝑖 ∧ . . . ∧ 𝑒𝑑 ∧ 𝑒′𝑑

and the same for 𝑒′𝑖 instead of 𝑒𝑖. Now,

𝜗𝑑−1 =

𝑑∑𝑖=1

(𝑒1 ∧ 𝑒′1 ∧ . . . 𝑒𝑖 ∧ 𝑒′𝑖 ∧ . . . ∧ 𝑒𝑑 ∧ 𝑒′𝑑).

Thus,

𝑒′𝑖 ∧ 𝜗𝑑−1 = 𝑒1 ∧ 𝑒′1 ∧ . . . 𝑒𝑖 ∧ 𝑒′𝑖 ∧ . . . ∧ 𝑒𝑑 ∧ 𝑒′𝑑

and the same for 𝑒𝑖, gives the basis of⋀2𝑑−1

𝑇ℓ𝐴.

Corollary 4.2.4. Applying this to a constant Abelian scheme A = 𝐵 ×𝑘 𝑋 and 𝑋 = 𝐴 a principallypolarised Abelian variety, gives that the hard Lefschetz morphism of a constant Abelian scheme H1(𝐴, 𝑇ℓA ) =H1(𝐴,Zℓ)× 𝑇ℓ𝐵 is an integral isomorphism.

Theorem 4.2.5 (Existence of the Poincare bundle). Let 𝑓 : A → 𝑋 be a projective Abelian scheme over 𝑋Noetherian. Then there is a Poincare bundle PA on A ×𝑋 A ∨, i. e. a line bundle such that for any 𝑋-scheme𝑇 and any line bundle L on A ×𝑋 𝑇 , there is a unique 𝑋-morphism ℎ : 𝑇 → A ∨ such that, for some linebundle N on 𝑇 ,

L ∼= (idA ×ℎ)*PA ⊗ 𝑓*𝑇 N .

Proof. See [FGI+05], p. 262 f., Exercise 9.4.3: 𝑓 has a section, the zero section. See also [FGI+05], p. 289,Remark 9.5.24.


4.2.1 The pairing ⟨·, ·⟩ℓLet ℓ = char 𝑘, 𝑋/𝑘 integral smooth projective and 𝑘 ⊆ 𝑘(𝑋) algebraically closed. Now note that from Lemma 4.1.25under the assumtion X(A /𝑋)[ℓ∞] finite, one has an isomorphism

𝛿 : A (𝑋)⊗Z Zℓ∼−−→ H1(𝑋,𝑇ℓA ).

induced by the boundary map of the long exact sequence induced by the short exact sequence

0→ A [ℓ𝑛]→ A → A → 0.

Let 𝑑 = dim𝑋. Using Theorem 4.2.2 for 𝑖 = 𝑑− 1, one gets isomorphisms

A ∨(𝑋)⊗Z Qℓ∼−−→ H1(𝑋,𝑉ℓ(A

∨))∼−−→ H2𝑑−1(𝑋,𝑉ℓ(A

∨)(𝑑− 1)).

Let us recall Conrad’s height pairing for generalised global fields [Cona]. Let 𝑘 = F𝑞 be the finiteground field and 𝐾 = 𝑘(𝑋) the function field of 𝑋. Choose a closed immersion 𝜄 : 𝑋 → P𝑁

𝑘 . For 𝑥 ∈ 𝑋(1) let

𝑐𝑥,𝜄 = 𝑞− deg𝑘,𝜄({𝑥}).

Then the discrete valuations‖ · ‖𝑥,𝜄= 𝑐ord 𝑥(·)

𝑥,𝜄 : 𝐾 → Z,

where 𝑥 ∈ 𝑋(1), satisfy the product formula by [Har83], p. 146, Exercise II.6.2 (d). Conrad calls the system ofthese valuations the structure of a generalised global field on 𝐾. This induces a height function

ℎ𝐾,𝑁,𝜄 : P𝑁𝐾(��)→ R, ℎ𝐾,𝑁,𝜄([𝑡0 : . . . : 𝑡𝑁 ]) =

1

[𝐾 ′ : 𝐾]

∑𝑥∈𝑋(1)

max𝑖

(log ‖ 𝑡𝑖 ‖𝜄,𝑥) ≥ 0

on projective spaces.Now, we construct a canonical height pairing

ℎ𝜄 : A (𝑋)×A ∨(𝑋)→ R

as follows. If L is a very ample line bundle on a projective 𝐾-variety 𝑋, the induced closed 𝐾-immersion

𝜄L : 𝑋 →˓ P(H0(𝑋,L ))

defines a height functionℎ𝐾,L ,𝜄 := ℎ𝐾,H0(𝑋,L ),𝜄 ∘ 𝜄L : 𝑋(��)→ R

modulo 𝑂(1). By linearity, since one can write any line bundle on 𝑋 as a difference of two very ample linebundles, this extends to a homomorphism

Pic(𝑋)→ R𝑋(��)/𝑂(1).

Now let A /𝑋 be an Abelian scheme. In this case, one can define a canonical height pairing, taking values inR𝑋(��), not modulo bounded functions,

ℎ𝐾,L ,𝜄 : A (𝑋) = 𝐴(𝐾)→ R

orℎ𝐾,𝜄 : A (𝑋)×A ∨(𝑋) = 𝐴(𝐾)×𝐴∨(𝐾)→ R,

respectively as in [BG06], p. 284 ff. Note that A (𝑋) = 𝐴(𝐾) by the weak Neron mapping property.

Proposition 4.2.6. Let 𝐾 be a generalised global field, 𝐴/𝐾 an Abelian variety and P ∈ Pic(𝐴 × 𝐴∨) thePoincare bundle. Then

ℎL (𝑥) = ℎP(𝑥,L )

for 𝑥 ∈ 𝐴(𝐾) and L ∈ 𝐴∨(𝐾).

Proof. See [BG06], p. 292, Corollary 9.3.7.

Now we have to show that the following diagram commutes.

4T

HE

SP

EC

IAL

𝐿-V

AL

UE

INC

OH

OM

OL

OG

ICA

LT

ER

MS

43

H1(𝑋,𝑇ℓA )Tors ×H2𝑑−1(𝑋,𝑇ℓ(A ∨)(𝑑− 1))Tors

(1)

∪→ H2𝑑(𝑋,Zℓ(𝑑))Torspr*1→ H2𝑑(𝑋,Zℓ(𝑑))

=→ Zℓ

H1(𝑋,𝑇ℓA )Tors ×H1(𝑋,𝑇ℓ(A ∨))Tors?�

(id,∪𝜂𝑑−1)

OO

(2)

∪→ H2(𝑋,Zℓ(1))Tors∪𝜂𝑑−1

→ H2𝑑(𝑋,Zℓ(𝑑))=→ Zℓ

A (𝑋)Tors ⊗Z Zℓ ×A ∨(𝑋)Tors ⊗Z Zℓ

(𝛿,𝛿) ∼=

OO

(3)

Δ*𝑋(·,·)*P→ Pic(𝑋)Tors ⊗Z Zℓ

O𝑋(1)𝑑−1

→ CH𝑑(𝑋)Tors ⊗Z Zℓdeg→ Zℓ

A (𝑋)Tors ⊗Z Zℓ ×A ∨(𝑋)Tors ⊗Z ZℓℎP(·,·)→ Z⊗Z Zℓ

=→ Zℓ


Commutativity of (1). Here, 𝜂 ∈ H2(𝑋,Qℓ(1)) is the cycle class associated to O𝑋(1) ∈ Pic(𝑋) = CH1(𝑋)(𝑋 is regular) where O𝑋(1) = 𝜄*OP𝑁

𝐾(1) for the closed immersion 𝜄 : 𝑋 →˓ P𝑁

𝐾 which defines the structure of a

generalised global field on the function field 𝐾 = 𝑘(𝑋) of 𝑋.The integral hard Lefschetz morphism (∪𝜂)𝑑−1 : H1(𝑋, 𝑇ℓ(A ∨))→ H2𝑑−1(𝑋, 𝑇ℓ(A ∨)(𝑑− 1)) is injective with

finite cokernel by the hard Lefschetz theorem Theorem 4.2.2 since it follows that the kernel and the cokerneltensored with Qℓ are trivial, hence torsion, hence finite, and recall that all groups are torsion-free, so the kernelis trivial. We call the order of the cokernel of the integral hard Lefschetz morphism integral hard Lefschetzdefect.

Commutativity of (2). Here, 𝛿 : A (𝑋) ⊗Z Zℓ∼−−→ H1(𝑋,𝑇ℓA ) is the boundary map of the long exact

sequence in Lemma 4.1.20 (this is an isomorphism since we assume X(A /𝑋)[ℓ∞] to be finite).

The morphism Pic(𝑋)→ H2(𝑋,Zℓ(1)) equals Pic(𝑋) = H1(𝑋,G𝑚)𝛿→ H2(𝑋,Zℓ(1)).

H0(𝑋,A )

𝛿

��

×H0(𝑋,A ∨)

𝛿

��

// H1(𝑋,G𝑚)

𝛿

��

H1(𝑋,A [ℓ𝑛]) ×H1(𝑋,A ∨[ℓ𝑛]) // H2(𝑋,𝜇ℓ𝑛)

Lemma 4.2.7. Let A /𝑋 be a projective Abelian scheme over a locally Noetherian scheme 𝑋. Let 𝑥 ∈ A (𝑋)and L ∈ Pic0(A /𝑋). By the universal property of the Poincare bundle Theorem 4.2.5, there is a unique𝑋-morphism ℎ : 𝑋 → A ∨ such that L ∼= (idA ×ℎ)*PA . Then 𝑥*L = (𝑥, ℎ)*PA .

Proof. Consider the composition

𝑋𝑥 // A ×𝑋 𝑋

(idA ×ℎ)// A ×𝑋 A ∨.

By pulling back, this maps PA ↦→ L ↦→ 𝑥*L and PA ↦→ 𝑥*(idA ×ℎ)*PA = (𝑥, ℎ)*PA .

Proposition 4.2.8. Let A /𝑋 be a projective Abelian scheme over a locally Noetherian scheme 𝑋. Then thefollowing diagram commutes:

A (𝑋) ×A ∨(𝑋) // Pic(𝑋)

H0(𝑋,A ) ×Ext1𝑋(A ,G𝑚)

∼=

OO

// H1(𝑋,G𝑚)

Here, the upper pairing is given by (𝑥,L ) ↦→ 𝑥*L = ∆*(𝑥,L )*PA (the equality by Lemma 4.2.7) and the lowerpairing is the Yoneda pairing.

Proof. Given 𝑥 ∈ A (𝑋), i. e. 𝑥 : 𝑋 → A , and 𝑒 : (1→ G𝑚 → 𝐺→ A → 0) ∈ Ext1𝑋(A ,G𝑚), (𝑥, 𝑒) maps to𝐺×A 𝑋 under the Yoneda pairing. This is a G𝑚-torsor on 𝑋, namely 𝑥*L if 𝐺 = L ∖ 0 (0 the zero-section).

Lemma 4.2.9.

H om𝑋(A ,G𝑚) = 0 and H om𝑋(A , 𝜇𝑛) = 0 (4.2.1)

Proof. This holds since G𝑚 and 𝜇𝑛 are affine and A is proper and connected.

So by the Ext spectral sequence H𝑝(𝑋,E xt𝑞𝑋(A ,G𝑚))⇒ Ext𝑝+𝑞𝑋 (A ,G𝑚) (see [Mil80], p. 91, Theorem III.1.22)one gets

H0(𝑋,E xt1𝑋(A ,G𝑚)) = Ext1𝑋(A ,G𝑚). (4.2.2)

By Proposition 4.2.8, the pairing A (𝑋)×A ∨(𝑋)→ Pic(𝑋) identifies with

H0(𝑋,A )× Ext1𝑋(A ,G𝑚)→ H1(𝑋,G𝑚).


Let ℓ be invertible on 𝑋. Then one has a short exact sequence of sheaves (of the shape 0→ 𝐴[ℓ𝑛]→ 𝐴ℓ𝑛→

𝐴→ 𝐴/ℓ𝑛 → 0)

0→ E xt1(A ,G𝑚)[ℓ𝑛]→ E xt1(A ,G𝑚)ℓ𝑛→ E xt1(A ,G𝑚)→ 0 (4.2.3)

since one can check E xt(A ,G𝑚)/ℓ𝑛 = A ∨/ℓ𝑛 = 0 on stalks (the points of A ∨ over a strictly Henselian ring areℓ𝑛-divisible for ℓ invertible on A : This is the exactness of the Kummer sequence 0→ A [ℓ𝑛]→ A → A → 0 atetale stalks.)

The short exact Kummer sequence (note that ℓ𝑛 is invertible on 𝑋)


yields by (4.2.1) a short exact sequence

H om(A ,G𝑚) = 0→ E xt1(A , 𝜇ℓ𝑛)→ E xt1(A ,G𝑚)ℓ𝑛→ E xt1(A ,G𝑚)→ 0 (4.2.4)

(the 0 at the right hand side by (4.2.3)).Combining (4.2.3) and (4.2.4), one gets

E xt1(A ,G𝑚)[ℓ𝑛] = E xt1(A , 𝜇ℓ𝑛) (4.2.5)

Again, the Ext spectral sequence H𝑝(𝑋,E xt𝑞𝑋(A , 𝜇ℓ𝑛)) ⇒ Ext𝑝+𝑞𝑋 (A , 𝜇ℓ𝑛) (see [Mil80], p. 91, Theo-rem III.1.22) gives us an injection

H1(𝑋,E xt1𝑋(A , 𝜇ℓ𝑛)) →˓ Ext2𝑋(A , 𝜇ℓ𝑛) (4.2.6)

since H om𝑋(A , 𝜇ℓ𝑛) = 0 by (4.2.1).The diagram

H0(𝑋,A ) ×Ext1𝑋(A ,G𝑚)

𝛿

��

// H1(𝑋,G𝑚)

𝛿

��

H0(𝑋,A ) ×Ext2𝑋(A , 𝜇ℓ𝑛) // H2(𝑋,𝜇ℓ𝑛)

commutes by [Jan13], p. 110, Satz 11.9 (iii), so we are left with proving that the lower pairing and the lowerpairing of (2) are equal. In order to show this, we prove the commutativity of

H0(𝑋,A )

𝛿

��

×Ext2𝑋(A , 𝜇ℓ𝑛) // H2(𝑋,𝜇ℓ𝑛)

H1(𝑋,A [ℓ𝑛]) ×H1(𝑋,E xt1𝑋(A , 𝜇ℓ𝑛))?�

(4.2.6)

OO

// H2(𝑋,𝜇ℓ𝑛).

Here, the lower row is induced by the Weil pairing using E xt1(A , 𝜇ℓ𝑛) = H om(A [ℓ𝑛], 𝜇ℓ𝑛) = A ∨[ℓ𝑛] by the longexact Ext sequence for the Kummer sequence 0→ A [ℓ𝑛]→ A → A → 0 (see [Mil80], p. 91, Theorem III.1.22):

0 = H om𝑋(A , 𝜇ℓ𝑛)→H om𝑋(A [ℓ𝑛], 𝜇ℓ𝑛)→ E xt1𝑋(A , 𝜇ℓ𝑛)0→ E xt1𝑋(A , 𝜇ℓ𝑛),

the first equality by (4.2.1) and the latter transition map being 0 since ℓ𝑛 kills 𝜇ℓ𝑛 .Rewrite this diagram as

Ext2𝑋(A , 𝜇ℓ𝑛) // Hom(H0(𝑋,A ),H2(𝑋,𝜇ℓ𝑛))

H1(𝑋,E xt1𝑋(A , 𝜇ℓ𝑛))?�

(4.2.6)

OO

// Hom(H1(𝑋,A [ℓ𝑛]),H2(𝑋,𝜇ℓ𝑛)).

𝛿*

OO

Now, the local-to-global Ext spectral sequence gives an embedding H1(𝑋,H om𝑋(A [ℓ𝑛], 𝜇ℓ𝑛)) →˓ Ext1𝑋(A [ℓ𝑛], 𝜇ℓ𝑛),through which the left vertical and the lower horizontal arrows factor. The two paths map

𝑒 =(1→ 𝜇ℓ𝑛 → F → A [ℓ𝑛]→ 0

)∈ Ext1𝑋(A [ℓ𝑛], 𝜇ℓ𝑛)

to (H0(𝑋,A )

𝛿→ H1(𝑋,A [ℓ𝑛])𝛿𝑒→ H2(𝑋,𝜇ℓ𝑛)

)∈ Hom(H0(𝑋,A ),H2(𝑋,𝜇ℓ𝑛)),

the first 𝛿 being the connecting homomorphism of the Kummer sequence.


Commutativity of (3). This is a generalisation of [Sch82b] and [Blo80].Let 𝑑 = dim𝑋. We want to show that the pairing

A (𝑋)× Ext1𝑋𝑓𝑝𝑝𝑓(A ,G𝑚)

∨ // H1(𝑋,G𝑚)∩O𝑋(1)𝑑−1

// CH𝑑(𝑋)deg

// Z

coincides up to a factor log 𝑞 with the generalised Bloch pairing

ℎ : 𝐴(𝐾)×𝐴∨(𝐾)→ R,

which is defined as follows: Let 𝐾 = 𝐾(𝑋) be the function field of 𝑋. Define the adele ring of 𝑋 as

A𝐾 =∏′

𝑥∈𝑋(1)𝐾𝑥,

where 𝐾𝑥 is the quotient field of the discrete valuation ring O𝑋,𝑥 with discrete valuation 𝑣𝑥 : 𝐾×𝑥 → Z and

absolute value | · |𝜄,𝑥 = 𝑞− deg𝜄 𝑥·𝑣𝑥(·). There is the natural homomorphism (the “logarithmic modulus map”),fixing a closed immersion 𝜄 : 𝑋 →˓ P𝑁

𝑘 with the very ample sheaf O𝑋(1) := 𝜄*OP𝑁 (1)

𝑙 : G𝑚(A𝐾)→ log 𝑞 · Z ⊆ R, (𝑎𝑥) ↦→∑

𝑥∈𝑋(1)

log |𝑎𝑥|𝜄,𝑥 = − log 𝑞 ·∑

𝑥∈𝑋(1)

deg𝜄 𝑥 · 𝑣𝑥(𝑎𝑥).

By the product formula (see [Har83], p. 146, Exercise II.6.2 (d)), 𝑙(G𝑚(𝐾)) = 𝑙(𝐾×) = 0. Scale the imagelog 𝑞 · Z such that 𝑙 is surjective.

Lemma 4.2.10. Let (𝑅𝑖)𝑖∈𝐼 be a familiy of rings with Pic(Spec𝑅𝑖) = 0 for every 𝑖 ∈ 𝐼. Then Pic(∏𝑖∈𝐼 Spec𝑅𝑖) =

0.

Proof. Line bundles correspond to G𝑚-torseurs, and a torseur is trivial iff it has a section. So let a line bundleon 𝑅 :=

∏𝑖∈𝐼 Spec𝑅𝑖 be represented by an affine scheme 𝑋. One has 𝑋(𝑅) =

∏𝑖∈𝐼 𝑋(𝑅𝑖), and each of the

factors has a non-trivial element by Pic(𝑅𝑖) = 0. Hence, the product is non-empty.

Corollary 4.2.11. The Picard group of the adele ring is trivial.

Proof. By the previous Lemma 4.2.10, Pic(A𝐾,𝑆) = 0, and Pic(A𝐾) = lim−→𝑆Pic(A𝐾,𝑆) by the compatiblity of

etale cohomology with limits, see [Mil80], p. 88 f., Lemma III.1.16.

Let 𝑎 ∈ 𝐴(𝐾) and 𝑎∨ = (1 → G𝑚 → X → A → 0) ∈ 𝐴∨(𝐾). By descent theory, X is a smooothcommutative 𝑋-group scheme, and by Hilbert’s theorem 90, the sequence

0→ G𝑚(𝐾)→X (𝐾)→ A (𝐾)→ H1(𝐾,G𝑚) = 0

and, by the previous corollary,

0→ G𝑚(A𝐾)→X (A𝐾)→ A (A𝐾)→ H1(A𝐾 ,G𝑚) = 0

still exact.

Proposition 4.2.12 (Adele valued points). Let 𝑋 be a 𝐾-scheme of finite type and 𝑆 ⊆ 𝑋(1) be a finite set ofplaces. Then

lim−→𝑋𝑆′(A𝐾,𝑆′) = 𝑋𝑆(A𝐾) = 𝑋(A𝐾)

and𝑋𝑆(A𝐾,𝑆) =

∏𝑣∈𝑆

𝑋𝑣(𝐾𝑣)×∏𝑣 ∈𝑆

𝑋𝑆,𝑣(O𝑋,𝑣).

Proof. See [Conc], p. 6, (3.5.1) and Theorem 3.6.

Lemma 4.2.13. The homomorphism 𝑙 : G𝑚(A𝐾)→ log 𝑞 · Z ⊆ R has a unique extension 𝑙𝑎∨ : X (A𝐾)→ R,which induces by restriction to X (𝐾) a homomorphism

𝑙𝑎∨ : 𝐴(𝐾)→ R,

since 𝑙(G𝑚(𝐾)) = 0, and one has by Bloch’s definition of ℎ : 𝐴(𝐾) × 𝐴∨(𝐾) → R, equivalently 𝐴∨(𝐾) →Hom(𝐴(𝐾),R),

ℎ(𝑎, 𝑎∨) = 𝑙𝑎∨(𝑎).


Proof. Consider the following commutative diagram (at first without the dashed arrows) with exact rows andinjective upper vertical morphisms and exact left column (define G1

𝑚 as the kernel of 𝑙 and X 1 as

X 1 = {𝑎 ∈X (A𝐾) : ∃𝑛 ∈ Z≥1, 𝑛𝑎 ∈ X 1}

the rational saturation of X 1 with

X 1 = G1𝑚 ·

∏𝑣∈𝑋(1)

X (O𝑋,𝑣)) :

0

��

0

��

1 // G1𝑚

//

��

X 1 //

��

A (A𝐾) // 0

1 // G𝑚(A𝐾) //

𝑙

��

X (A𝐾) //

𝑙𝑎∨

��

A (A𝐾) // 0

log 𝑞 · Z

��

log 𝑞 · Z

��

0 0

For this, it suffices to show:

(1) G1𝑚 = X 1 ∩G𝑚(A𝐾) ⊆X (A𝐾): One has G1

𝑚 ⊆ G𝑚(A𝐾) ∩X 1 by definition of X 1 and G1𝑚. For the

other inclusion X 1 ∩G𝑚(A𝐾) ⊆ G1𝑚, note that

𝑙( ∏𝑣∈𝑋(1)

G𝑚(O𝑋,𝑣))

= 0,

hence

G1𝑚 ·

∏𝑣∈𝑋(1)

G𝑚(O𝑋,𝑣) ⊆ G1𝑚,

so

X 1 ∩G𝑚(A𝐾) = G1𝑚 ·

∏𝑣∈𝑋(1)

G𝑚(O𝑋,𝑣) ⊆ G1𝑚,

but G𝑚(A𝐾)/G1𝑚 →˓ R is torsion-free, hence the result.

(2) X 1 � A (A𝐾): By the long exact sequence associated to the short exact sequence 1→ G𝑚 → X →A → 0 and Lemma 4.2.10, there is a surjection

X( ∏𝑥∈𝑋(1)

O𝑋,𝑥)� A

( ∏𝑥∈𝑋(1)

O𝑋,𝑥)

= A (A𝐾),

the latter equality by Proposition 4.2.12 and the valuative criterion for properness. But obviously X (∏𝑥∈𝑋(1) O𝑋,𝑥) ⊆

X 1.

By the snake lemma, the diagram completed with the dashed arrows is also exact and there exists thesought-for extension of 𝑙.

Definition 4.2.14. Define ⟨·, ·⟩ : 𝐴(𝐾)×𝐴∨(𝐾)→ log 𝑞 · Z by

𝑎 ∈ 𝐴(𝐾) = X (𝐾)/G𝑚(𝐾)→X (A𝐾)/G𝑚(𝐾)𝑙𝑎∨→ log 𝑞 · Z.

(Note that G𝑚(𝐾) ⊆ G1𝑚 by the product formula.)


Note that A (𝑋) = Hom𝑋𝑓𝑝𝑝𝑓(Z,A ) (a sheaf homomorphism 𝑓 : Z→ A induces Z = Z(𝑋)→ A (𝑋); now

𝑎 ∈ A (𝑋) is the image of 1 ∈ Z). The Yoneda pairing ∨ : Hom𝑋𝑓𝑝𝑝𝑓(Z,A )× Ext1𝑋𝑓𝑝𝑝𝑓

(A ,G𝑚)→ H1(𝑋,G𝑚)

maps (𝑎, 𝑎∨) to the extension 𝑎 ∨ 𝑎∨ defined by

𝑎 ∨ 𝑎∨ : 0 // G𝑚// Y //

��

Z //

𝑎

��

0

𝑎∨ : 0 // G𝑚// X // A // 0.

(4.2.7)

By composition, one gets an extension

𝑙𝑎∨𝑎∨ : Y (A𝐾)→X (A𝐾)𝑙𝑎∨→ R

of 𝑙 : G𝑚(A𝐾) → R to Y (A𝐾), which induces because of 𝑙(G𝑚(𝐾)) = 0 in the exact sequence 𝑎 ∨ 𝑎∨ byrestriction to Y (𝐾) a homomorphism

𝑙𝑎∨𝑎∨ : Z𝑎→ 𝐴(𝐾)

𝑙𝑎∨→ R,

so one obviously hasℎ(𝑎, 𝑎∨) = 𝑙𝑎∨(𝑎) = 𝑙𝑎∨𝑎∨(1).

The construction of 𝑙𝑎∨ in Lemma 4.2.13 shows

𝑙𝑎∨( ∏𝑥∈𝑋(1)

X (O𝑋,𝑥))

= 0, hence 𝑙𝑎∨𝑎∨( ∏𝑥∈𝑋(1)

Y (O𝑋,𝑥))

= 0,

by the diagram (4.2.7) defining 𝑎 ∨ 𝑎∨.

Lemma 4.2.15. Let (1 → G𝑚 → Y → Z → 0) = 𝑒 ∈ Ext1𝑋𝑓𝑝𝑝𝑓(Z,G𝑚) = H1(𝑋,G𝑚) = Pic𝑋 be a torseur

representing L ∈ Pic𝑋, and let 𝑙𝑒 : Y (A𝐾) → R be an extension of 𝑙 which vanishes on∏𝑥∈𝑋(1) Y (O𝑋,𝑥).

Then one has for the homomorphism 𝑙𝑒 : Z→ R (since 𝑙𝑒(G𝑚(𝐾)) = 𝑙(G𝑚(𝐾)) = 0) defined by restriction toY (𝐾): Recall 𝑑 = dim𝑋. Then one has

𝑙𝑒(1) = − log 𝑞 · deg(L ∩ O𝑋(1)𝑑−1),

where O𝑋(1)𝑑−1 denotes the (𝑑− 1)-fold self-intersection of O𝑋(1).

(Note that for every 𝑒 there is a extension 𝑙𝑒 as in the Lemma using the diagram (4.2.7) and Lemma 4.2.13.)

Proof. Considering 𝑒 as a class of a line bundle L on 𝑋, write 𝑌 (L ) := 𝑉 (L ) ∖ {0-section} for the G𝑚-torsoron 𝑋 defined by L . Then 𝑒 is isomorphic to the extension

1→ G𝑚 →∐𝑛∈Z

𝑌 (L ⊗𝑛)→ Z→ 0.

For every 𝑥 ∈ |𝑋|, choose an open neighbourhood 𝑈𝑥 ⊆ 𝑋 such that 1 ∈ Z has a preimage 𝑠𝑥 ∈ Y (𝑈𝑥) (theseexist by exactness of the short exact sequence 𝑒 of sheaves). Let further 𝑠 ∈ Y (𝐾) be a preimage of 1 ∈ Z (notethat 0→ G𝑚(𝐾)→ Y (𝐾)→ Z(𝐾)→ 0 is exact by Hilbert 90). Then one has 𝑠−1

𝑥 · 𝑠 ∈ G𝑚(𝐾) = 𝐾×. Since 𝑋is Jacobson, the 𝑈𝑥 for 𝑥 ∈ |𝑋| cover 𝑋. For every 𝑥 ∈ 𝑋(1) choose an �� ∈ |𝑋| such that 𝑥 ∈ 𝑈�� and set 𝑠𝑥 = 𝑠��and 𝑈𝑥 = 𝑈��. These define a Cartier divisor as (𝑠−1

𝑥 · 𝑠) · (𝑠−1𝑦 · 𝑠)−1 = 𝑠−1

𝑥 · 𝑠𝑦 ↦→ 1− 1 = 0 ∈ Z, so one has(𝑠−1𝑥 · 𝑠) · (𝑠−1

𝑦 · 𝑠)−1 ∈ G𝑚(𝑈𝑥 ∩ 𝑈𝑦) by the exactness of 1→ G𝑚 → Y → Z→ 0, and [(𝑈𝑥, 𝑠−1𝑥 )𝑥] = L since

Γ(𝑈𝑥,O𝑋((𝑈𝑥, 𝑠−1𝑥 )) = {𝑓 ∈ 𝐾 : 𝑓𝑠−1

𝑥 ∈ Γ(𝑈𝑥,O𝑋)} != Γ(𝑈𝑥,L ).

One has to compare the line bundle L with the G𝑚-torseur Y . Now one calculates

𝑙𝑒(1) = 𝑙𝑒(𝑠) note that 𝑙𝑒(G𝑚(𝐾)) = 𝑙(G𝑚(𝐾)) = 0 and 𝑠 ↦→ 1

= 𝑙𝑒((𝑠−1𝑥 · 𝑠)) since 𝑙𝑒(

∏𝑥∈𝑋(1)Y (O𝑋,𝑥)) = 0

= 𝑙((𝑠−1𝑥 · 𝑠)) since 𝑠−1

𝑥 · 𝑠 ∈ G𝑚(𝐾) and 𝑙𝑒 extends 𝑙, note that (𝑠−1𝑥 · 𝑠)𝑥 ∈ G𝑚(A𝐾)

= − log 𝑞 ·∑

𝑥∈𝑋(1)

deg𝜄 𝑥 · 𝑣𝑥(𝑠−1𝑥 · 𝑠) by definition of 𝑙.


On the other hand, by the above description of 𝑒, since the (𝑈𝑥, 𝑠−1𝑥 · 𝑠) define a Cartier divisor on 𝑋 with

associate line bundle isomorphic to L , one has for deg : CH𝑑(𝑋)→ Z

deg(L ∩ O𝑋(1)𝑑−1) =∑

𝑥∈𝑋(1)

deg𝜄 𝑥 · 𝑣𝑥(𝑠−1𝑥 · 𝑠)

since deg𝜄 𝑥 = deg({𝜄(𝑥)} ∩𝐻𝑑−1) for a generic hyperplane 𝐻 →˓ P𝑁𝑘 and O𝑋(1) = [𝐻] ∈ CH1(𝑋) = Pic(𝑋).

Combining the formulae gives the claim.

Applying this lemma to the above situation 𝑎 ∈ A (𝑋), 𝑎∨ ∈ 𝐴∨(𝐾) gives us

ℎ(𝑎, 𝑎∨) = 𝑙𝑎∨𝑎∨(1) = − log 𝑞 · deg([𝑎 ∨ 𝑎∨] ∩ O𝑋(1)𝑑−1) = − log 𝑞 · ⟨𝑎, 𝑎∨⟩ ,

so we get the commutative diagram

𝐴(𝐾)× 𝐴∨(𝐾)ℎ // R

𝐴(𝐾)× Ext1𝑋𝑓𝑝𝑝𝑓(A ,G𝑚)

∼=

OO

⟨·,·⟩// Z.

·(− log 𝑞)

OO

(Note tha 𝜄 and O𝑋(1) occur in 𝑙 and thus in ℎ.)

Comparison of the generalised Bloch pairing and the generalised Neron-Tate height pairing. Let𝐾𝑣 be the (completion) of 𝐾 = 𝐾(𝑋) at 𝑣 ∈ 𝑋(1), a local field.

Let ∆ be a divisor on 𝐴 defined over 𝐾𝑣 algebraically equivalent to 0 (this corresponds to 𝐴∨(𝐾𝑣) =Pic0𝐴/𝑘(𝐾𝑣)). Then there is an extension

1→ G𝑚 →XΔ → A → 0. (4.2.8)

Let LΔ be the line bundle associated to ∆. Then XΔ = 𝑉 (LΔ) ∖ {0} with LΔ = OA (∆) as a G𝑚-torseur. Theextension (4.2.8) only depends on the linear equivalence class of ∆.

Restricting to 𝐾𝑣, the extension (4.2.8) is split as a torseur over 𝐴 ∖ |∆| (since a line bundle associated to adivisor ∆ is trivial on 𝑋 ∖∆) by 𝜎Δ : 𝐴 ∖ |∆| →XΔ,𝐾𝑣

with 𝜎Δ canonical up to translation by G𝑚(𝐾𝑣) (sincethe choice of 𝜎Δ is the same as the choice of a rational section of LΔ). Let 𝑍Δ,𝐾𝑣

be the group of zero cyclesA =

∑𝑛𝑖(𝑝𝑖), 𝑝𝑖 ∈ 𝐴(𝐾𝑣), on 𝐴 defined over 𝐾𝑣 such that

∑𝑛𝑖 deg 𝑝𝑖 = 0 and suppA ⊆ 𝐴 ∖ |∆|. We get a

homomorphism 𝜎Δ : 𝑍Δ,𝐾𝑣 →XΔ(𝐾𝑣) (since XΔ is a group scheme).

We now prove a local analogue of Lemma 4.2.13.

Lemma 4.2.16. There is a commutative diagram with exact rows and columns:

0

��

0

��

0 // O×𝐾𝑣

//

��

XΔ(O𝐾𝑣) //

��

𝐴(𝐾𝑣) // 0

0 // 𝐾×𝑣

𝑙𝑣

��

// XΔ(𝐾𝑣)

𝜓Δ

��

// 𝐴(𝐾𝑣) // 0

Z

��

Z

��

0 0


Proof. The map labeled 𝑙𝑣 is the valuation map. The short exact sequence in the middle row is (4.2.8) evaluatedat 𝐾𝑣, and the short exact sequence in the upper row is (4.2.8) evaluated at O𝐾𝑣

: One is left showing thatX 1

Δ � 𝐴(𝐾𝑣) is surjective. But this follows from the long exact sequence associated to the short exact sequenceof sheaves on O𝐾𝑣

1→ G𝑚 →X → 𝐴→ 0

and Hilbert’s theorem 90: H1(Spec O𝐾𝑣,G𝑚) = 0. Further, one has A (𝐾𝑣) = A (O𝐾𝑣

) = 𝐴(𝐾𝑣) by the valuativecriterion of properness and the Neron mapping property.

Now let 𝜓Δ : XΔ(𝐾𝑣)→ Z be the map defined in the previous lemma. For A ∈ 𝑍Δ,𝐾𝑣define

⟨∆,A⟩𝑣 = 𝜓Δ𝜎Δ(A).

Theorem 4.2.17. Let 𝐾 = 𝐾(𝑋). Let 𝑎 ∈ 𝐴(𝐾) and 𝑎∨ ∈ 𝐴∨(𝐾). Let ∆ resp. A be a divisor algebraicallyequivalent to 0 defined over 𝐾 resp. a zero cycle of degree 0 over 𝐾 on 𝐴 such that [∆] = 𝑎∨ resp. A maps to 𝑎.Assume supp ∆ and suppA disjoint. Then

⟨𝑎, 𝑎∨⟩ = log 𝑞 ·∑

𝑣∈𝑋(1)

⟨∆,A⟩𝑣

with ⟨𝑎, 𝑎∨⟩ defined as in Definition 4.2.14.

Proof. Let1→ G𝑚 →XΔ → A → 0

be the G𝑚-torseur represented by 𝑎∨, and 𝜎Δ : 𝑍Δ,𝐾 →XΔ(𝐾) be as in the local case. One has to show thatthe map

𝑙𝑎∨ : XΔ(A𝐾)→ log 𝑞 · Z

(of the above definition in Lemma 4.2.13; note that XΔ(A𝐾) →˓X (A𝐾)) coincides with the sum of the localmaps

𝜓Δ,𝑣 : XΔ(𝐾𝑣)→ Z

multiplied by log 𝑞 · deg𝜄 𝑣 for 𝑣 ∈ 𝑋(1) defined above.Consider the commutative diagram

G1𝑚∑

𝑣 deg𝜄 𝑣·𝑣(·)��

// XΔ(A𝐾)/∏𝑣 XΔ(O𝐾𝑣 )

∑𝑣 deg𝜄 𝑣·𝜓Δ,𝑣

��

// XΔ(A𝐾)/X 1Δ

𝑙𝑎∨

��

// 0

0 // ker(∑

) //⨁

𝑣∈𝑋(1) Zlog 𝑞·

∑// log 𝑞 · Z // 0.

One has∏𝑣 XΔ(O𝐾𝑣

) ⊆ X 1Δ and G1

𝑚 ⊆ XΔ(A𝐾) by the commutative diagram in Lemma 4.2.13, so theexactness of the upper row follows. The exactness of the lower row is clear.

The left square commutes obviously. The right square commutes since by Lemma 4.2.13 the extensionof 𝑙 to X (A𝐾) is unique and

∑𝑣 deg𝜄 𝑣 · 𝜓Δ,𝑣 is well-defined (since 𝜓Δ,𝑣 vanishes on XΔ(O𝐾𝑣) and in the

adele ring, almost all components lie in XΔ(O𝐾𝑣 )) and restricts to 𝑙 : G𝑚(A𝐾)→ R since 𝜓Δ,𝑣 restricts to 𝑙𝑣by Lemma 4.2.16.

Now let 𝑥 ∈XΔ(A𝐾)/X 1Δ with 𝑙𝑎∨(𝑥) = ℎ. Lift it to �� ∈XΔ(A𝐾)/

∏𝑣 XΔ(O𝐾𝑣

). Then log 𝑞 ·∑𝑣 deg𝜄 𝑣 ·

𝜓Δ,𝑣(��) = ℎ by commutativity of the right square. If one chooses another lift, their difference comes from𝑑 ∈ G1

𝑚, which has height 0, so log 𝑞 ·∑𝑣 deg𝜄 𝑣 · 𝜓Δ,𝑣(��) only depends on 𝑥.

Proposition 4.2.18. The local pairings ⟨∆,A⟩𝑣 coincide with the local Neron height pairings ⟨∆,A⟩Neron,𝑣.

Proof. This follows from Neron’s axiomatic characterisation in [BG06], p. 304 f., Theorem 9.5.11, which holds for⟨∆,A⟩𝑣 by the same argument as in [Blo80], p. 73 ff., (2.11)–(2.15).

Corollary 4.2.19. The height pairing coincides with the canonical Neron-Tate height pairing.

Proof. This is clear since the local Neron-Tate height pairings sum up to the canonical Neron-Tate height pairing,see [BG06], p. 307, Corollary 9.5.14.


Remark 4.2.20. Note that the cohomological pairing ⟨·, ·⟩ℓ does not depend on an embedding 𝜄 : 𝑋 →˓ P𝑁𝑘 ,

but all other pairings depend on a line bundle 𝜗 or cohomology class 𝜂 ∈ H2(𝑋,Zℓ(1)), which manifests in theintegral hard Lefschetz defect in the commutative square (1). The two choices, in the integral hard Lefschetzdefect in the commutative square (1) and in the other pairings, cancel. Note that in the case of Corollary 4.2.4,one can choose 𝜂 such that the integral hard Lefschetz defect is trivial.

4.2.2 The pairing (·, ·)ℓ

One has a commutative diagram

0

��

0

��

0

��

0 // (A ∨(𝑋)⊗ Zℓ)Tors

��

∼= // H1(𝑋,𝑇ℓA ∨)Tors

��

� � (∪𝜂)𝑑−1

// H2𝑑−1(𝑋,𝑇ℓ(A ∨)(𝑑− 1))Tors

��

A ∨(𝑋)⊗Qℓ

��

∼= // H1(𝑋,𝑉ℓA ∨)

��

∼=(∪𝜂)𝑑−1

// H2𝑑−1(𝑋,𝑉ℓ(A ∨)(𝑑− 1))

��

A ∨(𝑋)⊗Qℓ/Zℓ

��

∼= // Div H1(𝑋,A ∨[ℓ∞])

��

(∪𝜂)𝑑−1

// // Div H2𝑑−1(𝑋,A ∨[ℓ∞](𝑑− 1))

��

// 0

0 0 0

using Lemma 4.1.26 for the upper left arrow. For the lower left arrow: By Lemma 4.1.20, one has a short exactsequence

0→ A (𝑋)⊗Qℓ/Zℓ → H1(𝑋,A [ℓ∞])→ H1(𝑋,A )[ℓ∞]→ 0.

Since A (𝑋)⊗Qℓ/Zℓ is divisible, one gets an inclusion A (𝑋)⊗Qℓ/Zℓ →˓ Div H1(𝑋,A [ℓ∞]). Since X(A /𝑋)[ℓ∞] =H1(𝑋,A )[ℓ∞] is finite, if an element from Div H1(𝑋,A [ℓ∞]) is mapped to H1(𝑋,A )[ℓ∞], it has finite order andis divisible, so it is 0, hence it comes from A (𝑋)⊗Qℓ/Zℓ.

The upper and middle right arrows are induced by the hard Lefschetz theorem Theorem 4.2.2, and the lowerone by functoriality of the coker-functor. So the middle one is an isomorphism, hence the upper one is injectiveand the lower one surjective by the snake lemma.

For the exactness of the columns: Left column: This column arises from tensoring

0→ Zℓ → Qℓ → Qℓ/Zℓ → 0

with A (𝑋)Tors ∼= ZrkA (𝑋), and TorZ1 (Z,−) = 0. (By the theorem of Mordell-Weil, A (𝑋) is a finitely generatedAbelian group). Middle and right column: This follows from Lemma 4.1.22.

Applying the snake lemma gives us that

Hom(H2𝑑−1(𝑋,𝑇ℓ(A∨)(𝑑− 1))Tors,Zℓ)→ Hom((A ∨(𝑋)⊗ Zℓ)Tors,Zℓ) and

Hom(Div H2𝑑−1(𝑋,A ∨[ℓ∞](𝑑− 1)),Qℓ/Zℓ)→ Hom(A ∨(𝑋)⊗Qℓ/Zℓ,Qℓ/Zℓ)

are injective with with finite (since it is finitely generated and killed by tensoring with Qℓ) cokernels of the sameorder (even isomorphic): Write

0 // 𝐴′ //

𝑓

��

𝐴 //

∼=��

𝐴′′ //

𝑔

��

0

0 // 𝐵′ // 𝐵 // 𝐵′′ // 0

in short for the big diagram.Applying Hom(−,Zℓ) to the short exact sequence 0→ 𝐴′ → 𝐵′ → coker(𝑓)→ 0 gives

0→ Hom(coker(𝑓),Zℓ)→ Hom(𝐵′,Zℓ)→ Hom(𝐴′,Zℓ)→ Ext1(coker(𝑓),Zℓ)→ Ext1(𝐵′,Zℓ).


Since coker(𝑓) is finite, the first term vanishes and Ext1(coker(𝑓),Zℓ) ∼= coker(𝑓), and since 𝐵′ is torsion-freeand finitely generated, hence projective, the last term vanishes.

Applying the exact functor Hom(−,Qℓ/Zℓ) to the short exact sequence 0→ ker(𝑔)→ 𝐴′′ → 𝐵′′ → 0 gives

0→ Hom(𝐵′′,Qℓ/Zℓ)→ Hom(𝐴′′,Qℓ/Zℓ)→ Hom(ker(𝑔),Qℓ/Zℓ)→ 0

and Hom(ker(𝑔),Qℓ/Zℓ) ∼= ker(𝑔).

Poincare duality for the absolute situation [Mil80], p. 183, Corollary V.2.3 (easily generalised to higherdimensions) gives perfect pairings

H2(𝑋,A [ℓ𝑛])×H2𝑑−1(𝑋,A ∨[ℓ𝑛](𝑑− 1))→ Qℓ/Zℓ.

This is the same as isomorphisms

H2(𝑋,A [ℓ𝑛])∼−−→ Hom(H2𝑑−1(𝑋,A ∨[ℓ𝑛](𝑑− 1)),Qℓ/Zℓ),

and passing to the projective limit gives us an isomorphism

H2(𝑋,𝑇ℓA )∼−−→ Hom(H2𝑑−1(𝑋,A ∨[ℓ∞](𝑑− 1)),Qℓ/Zℓ).

Let 𝐴 = H2(𝑋, 𝑇ℓA ) and 𝐵 = H2𝑑−1(𝑋,A ∨[ℓ∞](𝑑− 1)). These are finitely and cofinitely generated, respectively.One has

𝐴Tors = 𝐵∨/ lim−→𝐵∨[ℓ𝑛] = 𝐵∨/(lim←−𝐵/ℓ𝑛)∨ = 𝐵∨/��∨

since 0→ Div𝐵 → 𝐵𝑓→ �� is exact with �� the ℓ-adic completion of 𝐵. As 𝐵 ∼= (Qℓ/Zℓ)

𝑟 ⊕ 𝑇 with 𝑇 finite, onehas �� ∼= 𝑇 and 𝑓 surjective. Dualising gives 0→ ��∨ → 𝐵∨ → (Div𝐵)∨ → 0, so 𝐵∨/��∨ = (Div𝐵)∨. Thereforeone has an isomorphism

H2(𝑋,𝑇ℓA )Tors∼−−→ Hom(Div H2𝑑−1(𝑋,A ∨[ℓ∞](𝑑− 1)),Qℓ/Zℓ) (4.2.9)

induced by the cup product.

Now consider the commutative diagram

Hom(H2𝑑−1(𝑋,𝑇ℓ(A ∨)(𝑑− 1))Tors,Zℓ) // Hom((A ∨(𝑋)⊗ Zℓ)Tors,Zℓ)

∼=

��

H2(𝑋,𝑇ℓA )Tors

OO

∼=(4.2.9)

��

Hom(Div H2𝑑−1(𝑋,A ∨[ℓ∞](𝑑− 1)),Qℓ/Zℓ) // Hom(A ∨(𝑋)⊗Qℓ/Zℓ,Qℓ/Zℓ).

The right vertical map is an isomorphism: A homomorphism 𝑓 : Zℓ → Zℓ induces a map Qℓ → Qℓ by tensoringwith Q and hence a map between the cokernels Qℓ/Zℓ → Qℓ/Zℓ. This is an isomorphism by Lemma 2.1.15since, by the Mordell-Weil theorem, A (𝑋)Tors ∼= ZrkA (𝑋), and since Hom(Qℓ/Zℓ,Qℓ/Zℓ) = (Qℓ/Zℓ)

∨ =(lim−→

1ℓ𝑛Z/Z)∨ = lim←−Z/ℓ𝑛Z = Zℓ.

It follows from Poincare duality for the absolute situation [Mil80], p. 183, Corollary V.2.3 that one has aperfect pairing

(·, ·)ℓ : H2(𝑋,𝑇ℓA )Tors ×H2𝑑−1(𝑋,𝑇ℓ(A∨)(𝑑− 1))Tors

∪→ H2𝑑+1(𝑋,Zℓ(𝑑)) = Zℓ,

so

H2(𝑋,𝑇ℓA )Tors → Hom(H2𝑑−1(𝑋,𝑇ℓ(A∨)(𝑑− 1))Tors,Zℓ)

is injective with cokernel of order det(·, ·)ℓ. By comparison of the terms, it follows that det(·, ·)ℓ = 1.


4.3 The case of a constant Abelian scheme

In this section, we specialise to the case where A = 𝐴×𝑘 𝑋 is a constant Abelian variety. Let Γ = 𝐺𝑘 be theabsolute Galois group of the finite ground field 𝑘 = F𝑞, 𝑞 = 𝑝𝑛.

Lemma 4.3.1. 1. There is an isomorphism A [𝑚]∼−−→ 𝐴[𝑚] ×𝑘 𝑋 of finite flat group schemes resp. of

constructible sheaves (for char 𝑘 - 𝑚) on 𝑋.

2. There is an isomorphism 𝑇ℓA = (𝑇ℓ𝐴)×𝑘 𝑋 of ℓ-adic sheaves on 𝑋 for ℓ = char 𝑘.

3. There is an isomorphism of Abelian groups

A (𝑋) = Mor𝑋(𝑋,A )∼−−→ Mor𝑘(𝑋,𝐴), (𝑓 : 𝑋 → A ) ↦→ pr1 ∘𝑓,

and under this isomorphism Tor A (𝑋) corresponds to the subset of constant morphisms

Tor A (𝑋)∼−−→ {𝑓 : 𝑋 → 𝐴 | 𝑓(𝑋) = {𝑎}} = Hom𝑘(𝑘,𝐴) = 𝐴(𝑘).

Proof. 1. Consider the fibre product diagram

𝐴[𝑚] //

��

𝑘

0

��

𝐴[𝑚]

// 𝐴

and apply −×𝑘 𝑋.

2. This follows from 1.

3. The inverse is given by (𝑓 : 𝑋 → 𝐴) ↦→ ((𝑓, id𝑋) : 𝑋 → 𝐴×𝑘 𝑋 = A ).

For the second statement: If 𝑓 : 𝑋 → 𝐴 is constant 𝑎, (𝑓, id𝑋) has finite order ord 𝑎 in 𝐴(𝑘) since 𝑘 andthus 𝐴(𝑘) is finite. Conversely, if 𝑓 : 𝑋 → A has finite order 𝑛, the image of pr1 ∘𝑓 lies in the discrete setof 𝑛-torsion points (since pr1 : 𝐴×𝑘 𝑋 → 𝐴 is a morphism of group schemes), so is constant because 𝑋 isconnected.

Example 4.3.2. The rank of the Mordell-Weil group of a constant Abelian variety over a projective space hasrank 0, since there are no non-constant 𝑘-morphisms P𝑛

𝑘 → 𝐴, see [Mil86a], p. 107, Corollary 3.9.

Corollary 4.3.3. Assume 𝑋 has a 𝑘-rational point 𝑥0. Then there is a commutative diagram with exact rows

0 // 𝐴(𝑘) //

∼=��

A (𝑋) // Hom𝑘(Alb𝑋/𝑘, 𝐴) //

∼=��

0

0 // Tor A (𝑋) // A (𝑋) // A (𝑋)Tors // 0,

and, if 𝑘 is a finite field,

rk A (𝑋) = 𝑟(𝑓𝐴, 𝑓Alb𝑋/𝑘),

where 𝑟(𝑓𝐴, 𝑓𝐵) for 𝐴 and 𝐵 Abelian varieties over a finite field is defined in [Tat66a], p. 138.

Proof. The lower row is trivially exact. By the universal property of the Albanese (use that 𝑋 has a 𝑘-rationalpoint 𝑥0), one has {𝑓 ∈ Mor𝑘(𝑋,𝐴) | 𝑓(𝑥0) = 0} = Hom𝑘(Alb𝑋/𝑘, 𝐴). Thus the upper row is exact. The lefthand vertical arrow is an isomorphism because of Lemma 4.3.1 3. Now the five lemma implies that the righthand vertical arrow is an isomorphism since it is a well-defined homomorphism: Precompose 𝑓 : Alb𝑋/𝑘 → 𝐴with the Abel-Jacobi map 𝜙 : 𝑋 → Alb𝑋/𝑘 associated to 𝑥0.

The equality for the rank follows from [Tat66a], p. 139, Theorem 1 (a).


4.3.1 Heights

Lemma 4.3.4. Let 𝑀 and 𝑁 be torsion-free finitely generated Zℓ-modules, resp. continuous Zℓ[Γ]-modules.Then one has

𝑀 ⊗Zℓ𝑁 = HomZℓ−Mod(𝑀∨, 𝑁), (4.3.1)

(𝑀 ⊗Zℓ𝑁)Γ = HomZℓ[Γ]−Mod(𝑀∨, 𝑁), (4.3.2)

where (−)∨ denotes the Zℓ-dual.

Proof. This is clear.

Theorem 4.3.5. Let 𝑓 : A → A ′ be an 𝑋-isogeny of Abelian schemes. The Weil pairing

⟨·, ·⟩𝑓 : ker(𝑓)×𝑋 ker(𝑓∨)→ G𝑚

is a non-degenerate and biadditive pairing of finite flat 𝑋-group schemes, i. e. it defines a canonical 𝑋-isomorphism

ker(𝑓∨)∼−−→ (ker(𝑓))∨.

Moreover, it is functorial in 𝑓 .If 𝑋 = Spec 𝑘, it induces a perfect pairing of torsion-free finitely generated Zℓ[Γ]-modules.

𝑇ℓA × 𝑇ℓ(A ∨)→ Zℓ(1) (4.3.3)

Hom(𝑇ℓA ,Zℓ) = 𝑇ℓ(A∨)(−1) (4.3.4)

Proof. See [Mum70], p. 186 (for Abelian varieties) and [Oda69], p. 66 f., Theorem 1.1 (for Abelian schemes).

Remark 4.3.6. By [Mil80], p. 53, Theorem II.1.9, a Zℓ[Γ]-module is a Zℓ-sheaf on (the etale site of) 𝑘 = F𝑞.

Lemma 4.3.7. Let A = 𝐴×𝑘𝑋 be a constant Abelian scheme. Then one has H𝑖(��, 𝑇ℓA ) = H𝑖(��,Zℓ)⊗Zℓ𝑇ℓ𝐴

as ℓ-adic sheaves on the etale site of 𝑘.

Proof. By the projection formula for 𝜋 : �� → 𝑘, one has

H𝑖(��, 𝑇ℓA ) = R𝑖𝜋*(𝑇ℓA )

= R𝑖𝜋*(Zℓ ⊗Zℓ𝜋*𝑇ℓ𝐴)

= R𝑖𝜋*(Zℓ)⊗Zℓ𝑇ℓ𝐴

= H𝑖(��,Zℓ)⊗Zℓ𝑇ℓ𝐴

as ℓ-adic sheaves on the etale site of 𝑘.

Theorem 4.3.8 (The height pairing). Let 𝑋/𝑘 be a smooth projective geometrically connected variety withAlbanese 𝐴 such that Pic𝑋/𝑘 is reduced. Denote the constant Abelian scheme 𝐵 ×𝑘 𝑋/𝑋 by A /𝑋. Then thetrace pairing


Tr→ Z

tensored with Zℓ equals the cohomological pairing

⟨·, ·⟩ℓ : H1(𝑋,𝑇ℓA )Tors ×H2𝑑−1(𝑋,𝑇ℓ(A∨)(𝑑− 1))Tors → H2𝑑(𝑋,Zℓ(𝑑))

pr*1→ H2𝑑(��,Zℓ(𝑑)) = Zℓ.

Proof. We have

H1(𝑋,𝑇ℓA )Tors = H1(��, 𝑇ℓA )Γ by (4.1.6)

=(H1(��,Zℓ(1))⊗Zℓ

(𝑇ℓ𝐴)(−1))Γ

by Lemma 4.3.7

= (𝑇ℓ Pic(𝑋)⊗Zℓ(𝑇ℓ𝐴)(−1))

Γby the Kummer sequence

= HomZℓ[Γ]−Mod

(((𝑇ℓ𝐴)(−1))

∨, 𝑇ℓ Pic(𝑋)

)by (4.3.2)

= HomZℓ[Γ]−Mod (Hom(𝑇ℓ(𝐴∨),Zℓ)

∨, 𝑇ℓ Pic(𝑋)) by (4.3.4)

= HomZℓ[Γ]−Mod (𝑇ℓ(𝐴∨), 𝑇ℓ Pic(𝑋))

= Hom𝑘(𝐴∨,Pic(𝑋))⊗Z Zℓ by the Tate conjecture [Tat66a].


Note that H1(��, 𝑇ℓA )Γ is torsion-free since H1(��, 𝑇ℓA ) is so, and this holds because of the Kunneth formulaand since H1(��,Zℓ(1)) = 𝑇ℓ Pic(𝑋) is torsion-free because of Lemma 2.1.9. Therefore, in (4.1.6) above,ker𝛼 = H0(��, 𝑇ℓA )Γ is the whole torsion subgroup of H1(𝑋,𝑇ℓA ).

Remark 4.3.9. Note that 𝑇ℓ𝐴 = 𝑇ℓ𝐴 as Zℓ[Γ]-modules.

𝑇ℓA has weight −1 and 𝑇ℓ(A ∨)(𝑑− 1) has weight −1− 2(𝑑− 1) = −2𝑑+ 1 and we have

0 // H2𝑑−1(��, 𝑇ℓ(A∨)(𝑑− 1))Γ

𝛽// H2𝑑(𝑋,𝑇ℓ(A

∨)(𝑑− 1)) // H2𝑑(��, 𝑇ℓ(A∨)(𝑑− 1))Γ // 0

0 // H2𝑑−2(��, 𝑇ℓ(A∨)(𝑑− 1))Γ // H2𝑑−1(𝑋,𝑇ℓ(A

∨)(𝑑− 1))𝛼 // H2𝑑−1(��, 𝑇ℓ(A

∨)(𝑑− 1))Γ

𝑓

mm

// 0,

(4.3.5)

where only the four groups connected by 𝑓 , 𝛼 and 𝛽 can be infinite.The perfect Poincare duality pairing

H1(��,Zℓ(1))×H2𝑑−1(��,Zℓ(𝑑− 1))→ H2𝑑(��,Zℓ(𝑑))∼−−→ Zℓ (4.3.6)

identifies H2𝑑−1(��,Zℓ(𝑑− 1)) with 𝑇ℓ Pic(𝑋)∨.

Theorem 4.3.10. Let 𝑋/𝑘 be a smooth projective geometrically connected variety with a 𝑘-rational point.Then the reduced Picard variety (Pic0𝑋/𝑘)red is dual to Alb(𝑋) and Pic0𝑋/𝑘 is reduced iff dimPic0𝑋/𝑘 =

dim𝑘 H1Zar(𝑋,O𝑋).

Proof. By [Moc12], Proposition A.6 (i) or [FGI+05], p. 289 f., Remark 9.5.25, (Pic0𝑋/𝑘)red is dual to Alb(𝑋).

By [FGI+05], p. 283, Corollary 9.5.13, the Picard variety is reduced (and then smooth and an Abelian scheme)iff equality holds in dimPic0𝑋/𝑘 ≤ dim𝑘 H1

Zar(𝑋,O𝑋).

Remark 4.3.11. The integer 𝛼(𝑋) := dim𝑘 H1Zar(𝑋,O𝑋)− dimPic0𝑋/𝑘 is called the defect of smoothness.

Example 4.3.12. One has e. g. 𝛼(𝑋) = 0 if (b) from Theorem 4.3.23 below is satisfied. This holds true for 𝑋an Abelian variety or a curve.

H2𝑑−1(𝑋,𝑇ℓ(A∨)(𝑑− 1))Tors = H2𝑑−1(��, 𝑇ℓ(A

∨)(𝑑− 1))Γ by (4.1.6)

=(H2𝑑−1(��,Zℓ(𝑑− 1))⊗Zℓ

𝑇ℓ(𝐴∨))Γ

by Lemma 4.3.7

= (𝑇ℓ Pic(𝑋)∨ ⊗Zℓ𝑇ℓ(𝐴

∨))Γ

by (4.3.6)

= HomZℓ[Γ]−Mod (𝑇ℓ Pic(𝑋), 𝑇ℓ(𝐴∨)) by (4.3.2)

= Hom𝑘(Pic(𝑋), 𝐴∨)⊗Z Zℓ by the Tate conjecture [Tat66a].

Example 4.3.13. In particular, if the characteristic polynomials of the Frobenius on Pic(𝑋) and 𝐴∨ are coprime,Hom𝑘(𝐴∨,Pic(𝑋)) = 0 = Hom𝑘(Pic(𝑋), 𝐴∨) and the discriminants of the parings ⟨·, ·⟩ℓ and (·, ·)ℓ are equal to 1.

In the general case, we have to investigate if the following diagram commutes:

Lemma 4.3.14. Let 𝑓 : 𝐴→ 𝐵 an homomorphism of an Abelian variety 𝐴 and 𝑒𝐴 : 𝑇ℓ𝐴× 𝑇ℓ𝐴∨ → Zℓ(1) and𝑒𝐵 : 𝑇ℓ𝐵 × 𝑇ℓ𝐵∨ → Zℓ(1) be the perfect Weil pairings. Then

𝑒𝐵(𝑓(𝑎), 𝑏) = 𝑒𝐴(𝑎, 𝑓∨(𝑏))

for all 𝑎 ∈ 𝑇ℓ𝐴 and 𝑏 ∈ 𝑇ℓ𝐵∨.

Proof. See [Mum70], p. 186, (I).

56

4T

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SP

EC

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𝐿-V

AL

UE

INC

OH

OM

OL

OG

ICA

LT

ER

MS

H1(𝑋, 𝑇ℓA )Tors × H2𝑑−1(𝑋, 𝑇ℓ(A∨)(𝑑 − 1))Tors

pr*1

��

(1)

∪→ H2𝑑(𝑋,Zℓ(𝑑)) → H2𝑑(𝑋,Zℓ(𝑑)) → Zℓ

H1(𝑋, 𝑇ℓA )Γ × H2𝑑−1(𝑋, 𝑇ℓ(A∨)(𝑑 − 1))Γ

(2)

∪→ H2𝑑(𝑋,Zℓ(𝑑)) → Zℓ

(H1(𝑋,Zℓ(1)) ⊗Zℓ

(𝑇ℓ𝐴)(−1))Γ ×

(H2𝑑−1(𝑋,Zℓ(𝑑 − 1)) ⊗Zℓ

𝑇ℓ(𝐴∨)

)Γ∼=

OO

∼=

��

(3)

→ H2𝑑(𝑋,Zℓ(𝑑)) ⊗ZℓZℓ → Zℓ

(𝑇ℓ Pic(𝑋) ⊗Zℓ

(𝑇ℓ𝐴)(−1))Γ ×

(𝑇ℓ Pic(𝑋)∨ ⊗Zℓ

𝑇ℓ(𝐴∨)

)Γ

∼=

��

(4)

→ End(𝑇ℓ Pic(𝑋)) ⊗ZℓZℓ

Tr→ Zℓ

HomZℓ[Γ]−Mod( (

(𝑇ℓ𝐴)(−1))∨⏟ ⏞

𝑇ℓ(𝐴∨)

, 𝑇ℓ Pic(𝑋))× HomZℓ[Γ]−Mod

(𝑇ℓ Pic(𝑋), 𝑇ℓ(𝐴

∨))

∼=

��

(5)

→ HomZℓ[Γ]−Mod( (

(𝑇ℓ𝐴)(−1))∨⏟ ⏞

𝑇ℓ(𝐴∨)

, 𝑇ℓ(𝐴∨)

)→ Zℓ

HomZℓ[Γ]−Mod

(𝑇ℓ(𝐴

∨), 𝑇ℓ Pic(𝑋))

× HomZℓ[Γ]−Mod

(𝑇ℓ Pic(𝑋), 𝑇ℓ(𝐴

∨))

(6)

∘→ EndZℓ[Γ]−Mod(𝑇ℓ𝐴∨)

Tr→ Zℓ

Hom𝑘(𝐴∨,Pic(𝑋)) ⊗Z Zℓ × Hom𝑘(Pic(𝑋), 𝐴∨) ⊗Z Zℓ

∼=

OO

(7)

∘→ End𝑘(𝐴∨) ⊗Z ZℓTr→ Zℓ

Hom𝑘(Pic(𝑋), 𝐴∨) ⊗Z Zℓ × Hom𝑘(𝐴∨,Pic(𝑋)) ⊗Z Zℓ

∼= 𝜎

OO

(8)

∘→ End𝑘(Pic(𝑋)∨) ⊗Z ZℓTr→ Zℓ

Hom𝑘(Alb(𝑋), 𝐴) ⊗Z Zℓ × Hom𝑘(𝐴,Alb(𝑋)) ⊗Z Zℓ

∼= (−)∨

OO

∘→ End𝑘(Alb(𝑋)) ⊗Z ZℓTr→ Zℓ


Corollary 4.3.15. Let 𝑓 : 𝐴→ 𝐴 an endomorphism of an Abelian variety 𝐴. Then

Tr𝑇ℓ(𝐴)(𝑓) = Tr𝑇ℓ(𝐴∨)(𝑓∨).

Proof. The relation 𝑒𝐴(𝑓(𝑥), 𝑦) = 𝑒𝐴(𝑥, 𝑓∨(𝑦)) means that 𝑓∨ is the transpose of 𝑓 with respect to the bilinearform 𝑉ℓ(𝐴)× 𝑉ℓ(𝐴∨)→ Qℓ(1) (the Weil pairing). Choosing orthonormal bases, the matrix of 𝑓∨ is the transposeof the matrix of 𝑓 , so they have the same trace.

Remark 4.3.16. One has 𝑇ℓ(𝐴∨) = 𝑇ℓ(𝐴)∨ as ℓ-adic sheaves on the etale site of 𝑘 since their weights are

different; however, the Weil pairing gives us under a non-canonical isomorphism Zℓ(1) ∼= Zℓ a non-canonicalisomorphism 𝑇ℓ(𝐴

∨) = Hom(𝑇ℓ(𝐴),Zℓ(1)) ∼= 𝑇ℓ(𝐴)∨, which does not respect the weights. In fact, one has𝑇ℓ(𝐴

∨) = 𝑇ℓ(𝐴)∨(1).

(1) commutes since ∪-product commutes with restrictions.(2) commutes because of the associativity of the ∪-product.(3) commutes since, in general, one has a commutative diagram of finitely generated free modules over a ring

𝑅

𝐴×𝐵∼=��

⟨·,·⟩// 𝑅

𝐶 × 𝐶∨ // 𝑅

identifying 𝐵 with the dual of 𝐶 ∼= 𝐴 with a perfect pairing ⟨·, ·⟩ and the canonical pairing 𝐶 ×𝐶∨ → 𝐾: Choosea basis (𝑎𝑖) of 𝐴 and the dual basis (𝑏𝑖) of 𝐵; these are mapped to the bases (𝑐𝑖) and (𝑐′𝑖) of 𝐶 and 𝐶∨. Then,under the top horizontal map, ⟨𝑎𝑖, 𝑏𝑗⟩ = 𝛿𝑖𝑗 with the Kronecker symbol 𝛿𝑖𝑗 , and under the bottom horizontalmap (𝑐𝑖, 𝑐

′𝑗) = 𝛿𝑖𝑗 .

(4) commutes since, in general, one has a commutative diagram of finitely generated free modules over a ring𝑅

(𝑀 ⊗𝑅 𝑁∨)× (𝑀∨ ⊗𝑅 𝑁)

∼=��

// End𝑅−Mod(𝑀)⊗𝑅 End𝑅−Mod(𝑁)Tr𝑀 ⊗𝑅 Tr𝑁 // 𝑅

Hom𝑅−Mod(𝑁,𝑀)×Hom𝑅−Mod(𝑀,𝑁)∘ // End𝑅−Mod(𝑁)

Tr𝑁 // 𝑅

For proving this, choose bases (𝑎𝑖) of 𝑀 and (𝑏𝑖) of 𝑁 and their dual bases (𝑎′𝑖) of 𝑀∨ and (𝑏′𝑖) of 𝑁∨. Theelement (𝑎𝑖⊗ 𝑏′𝑗 , 𝑎′𝑘⊗ 𝑏𝑙) of (𝑀 ⊗𝐾 𝑁∨)× (𝑀∨⊗𝐾 𝑁) is sent by the upper horizontal arrows to 𝛿𝑖𝑘𝛿𝑗𝑙, and by theleft vertical arrow to (𝑏𝑚 ↦→ 𝑏′𝑗(𝑏𝑚)𝑎𝑖, 𝑎𝑛 ↦→ 𝑎′𝑘(𝑎𝑛)𝑏𝑙). The latter element is mapped by the lower left horizontalarrow to 𝑏𝑚 ↦→ 𝑎′𝑘(𝑏′𝑗(𝑏𝑚)𝑎𝑖)𝑏𝑙 and the trace of this endomorphism is 𝛿𝑗𝑙𝛿𝑘𝑖. Therefore, the diagram commutes.

(5) commutes because of precomposing with the isomorphism (𝑇ℓA (−1))∨∼−−→ 𝑇ℓ(A ∨) coming from the

perfect Weil pairing.(6) commutes because of [Lan58], p. 186 f., Theorem 3.(7) commutes because of Tr(𝛼𝛽) = Tr(𝛽𝛼), see [Lan58], p. 187, Corollary 1.(8) commutes because of Corollary 4.3.15 and since Pic(𝑋) is dual to Alb(𝑋) by Theorem 4.3.10 since Pic𝑋/𝑘

is reduced.

The case of a curve as a basis. Let 𝑋/𝑘 be a smooth projective geometrically connected curve with functionfield 𝐾 = 𝐾(𝑋), base point 𝑥0 ∈ 𝑋(𝑘), Albanese 𝐴 and Abel-Jacobi map 𝜙 : 𝑋 → 𝐴 and canonical principalpolarisation 𝑐 : 𝐴

∼−−→ 𝐴∨, and 𝐵/𝑘 be an Abelian variety. Let 𝛼 ∈ Hom𝑘(𝐴,𝐵) and 𝛽 ∈ Hom𝑘(𝐵,𝐴) behomomorphisms.

Let

⟨𝛼, 𝛽⟩ = Tr(𝛽 ∘ 𝛼 : 𝐴→ 𝐴) ∈ Z

be the trace pairing, the trace being taken as an endomorphism of 𝐴 as in [Lan58]. By [Lan58], p. 186 f.,Theorem 3, this equals the trace taken as an endomorphism of the Tate module 𝑇ℓ𝐴 or H1(𝐴,Zℓ) (they are dualto each other, so for the trace, it does not matter which one we are taking, see Theorem 4.1.13).

We now show that our trace pairing is equivalent to the usual Neron-Tate height pairing on curves and isthus a sensible generalisation in the case of a higher dimensional base.


Proposition 4.3.17. Let 𝑋,𝑌 be Abelian varieties over 𝑘 and 𝑓 ∈ Hom𝑘(𝑋,𝑌 ). Then

(𝑓 × id𝑌 ∨)*P𝑌∼= (id𝑋 ×𝑓∨)*P𝑋

in Pic(𝑋 ×𝑘 𝑌 ∨).

Proof. By the universal property of the Poincare bundle P𝑋 applied to (𝑓 × idP𝑦)*P𝑌 , there exists a unique

map 𝑓 : 𝑋∨ → 𝑌 ∨ such that

(𝑓 × id𝑌 ∨)*P𝑌∼= (id𝑋 ×𝑓)*P𝑋 . (4.3.7)

It remains to show that 𝑓 = 𝑓∨.Let 𝑇/𝑘 be a variety and L ∈ Pic00(𝑌 ×𝑘 𝑇 ) arbitrary. By the universal property of the Poincare bundle P𝑌 ,

there exists 𝑔 : 𝑇 → 𝑌 ∨ such that L = (id𝑌 ×𝑔)*P𝑌 . We want to show 𝑓* : 𝑌 ∨(𝑇 )→ 𝑋∨(𝑇 ), 𝑔 ↦→ 𝑓𝑔 equals𝑓∨ : Pic00(𝑌 ×𝑘 𝑇 )→ Pic00(𝑋 ×𝑘 𝑇 ),L ↦→ 𝑓*L . Now we have

𝑓∨(L ) = (𝑓 × id𝑇 )*L

= (𝑓 × id𝑇 )*(id𝑌 ×𝑔)*P𝑌

= (𝑓 × 𝑔)*P𝑌

= (id𝑋 ×𝑔)*(𝑓 × id𝑌 ∨)*P𝑌

= (id𝑋 ×𝑔)*(id𝑋 ×𝑓)*P𝑋 by (4.3.7)

= (id𝑋 ×𝑓𝑔)*P𝑋

= 𝑓*(L )

Proposition 4.3.18. Let 𝑐𝐴 = 𝑚*𝜗− − pr*1 𝜗− − pr*2 𝜗

− ∈ Pic(𝐴×𝑘 𝐴). Then

(𝜙, id𝐴)*𝑐𝐴 = −𝛿1 (4.3.8)

and(id𝐴, 𝜙𝜗−)*P𝐴 = 𝑐𝐴. (4.3.9)

Proof. See [BG06], p. 279, Propositions 8.10.19–20.

Theorem 4.3.19 (The trace and the height pairing for curves). Let 𝑋/𝑘 be a smooth projective geometricallyconnected curve with Albanese 𝐴. Then the trace pairing


Tr→ Z

equals the following height pairing

𝛾(𝛼) : 𝑋𝜙→ 𝐴

𝛼→ 𝐵,

𝛾′(𝛽) : 𝑋𝜙→ 𝐴

𝑐→ 𝐴∨ 𝛽∨

→ 𝐵∨,

(𝛾(𝛼), 𝛾′(𝛽))ℎ𝑡 = deg𝑋(−(𝛼𝜙, 𝛽∨𝑐𝜙)*P𝐵),

where 𝜙 : 𝑋 → 𝐴 is the Abel-Jacobi map associated to a rational point of 𝑋 and 𝑐 : 𝐴∼−−→ 𝐴 the canonical

principal polarisation associated to the theta divisor, and this is equivalent to the usual Neron-Tate canonicalheight pairing.

Proof. By [Mil68], p. 100, we have⟨𝛼, 𝛽⟩ = deg𝑋((id𝑋 , 𝛽𝛼𝜙)*𝛿1),

where 𝛿1 ∈ Pic(𝑋 ×𝑘 𝐴) is a divisorial correspondence such that

(id𝑋 , 𝜙)*𝛿1 = ∆𝑋 − {𝑥0} ×𝑋 −𝑋 × {𝑥0},

which we define to be ∆* (∆*𝑋 is the idempotent cutting out ℎ(𝑋)− ℎ0(𝑋)− ℎ2(𝑋) = ℎ1(𝑋) for 𝑋 a smooth

projective geometrically connected curve).


There is the property Proposition 4.3.18 of the Theta divisor Θ of the Jacobian 𝐴 of 𝐶 on 𝐴 (which is definedin [BG06], p. 272, Remark 8.10.8) and let Θ− = [−1]*Θ with 𝜗 and 𝜗− denoting the respecting divisor class).The Theta divisor induces the canonical principal polarisation 𝜙𝜗 = 𝑐 : 𝐴

∼−−→ 𝐴∨. Therefore

(𝛾(𝛼), 𝛾′(𝛽)) = (𝛼𝜙, 𝛽∨𝑐𝜙)*P𝐵 by definition

= (𝛼𝜙× 𝑐𝜙)*(id𝑋 , 𝛽∨)*P𝐵

= (𝛼𝜙× 𝑐𝜙)*(𝛽, id𝐴∨)*P𝐴 by Proposition 4.3.17

= (𝛽𝛼𝜙, 𝑐𝜙)*P𝐴

= (𝛽𝛼𝜙, 𝜙𝜗𝜙)*P𝐴

= (𝛽𝛼𝜙, 𝜙)*(id𝐴×𝜙𝜗)*P𝐴

= (𝛽𝛼𝜙, 𝜙)*𝑐𝐴 by (4.3.9)

= (𝜙, 𝛽𝛼𝜙)*𝑐𝐴 by symmetry

= −(id𝑋 , 𝛽𝛼𝜙)*𝛿1 by (4.3.8)

Summing up, one has

(𝛾(𝛼), 𝛾′(𝛽))ℎ𝑡 = deg𝑋(−(id𝑋 , 𝛽𝛼𝜙)*𝛿1)

= −⟨𝛼, 𝛽⟩ .

By [MB85], p. 72, Theoreme 5.4, this pairing equals the Neron-Tate canonical height pairing.

Remark 4.3.20. Note that our height coming from the trace pairing is normalised since it is bilinear.

4.3.2 The special 𝐿-value

Lemma 4.3.21. Let 𝑓 : 𝐴→ 𝐵 be an isogeny of Abelian varieties over a field 𝑘 and ℓ = char 𝑘. Then 𝑓 inducesan Galois equivariant isomorphism 𝑉ℓ𝐴

∼−−→ 𝑉ℓ𝐵 of rationalised Tate modules.

Proof. There is the exact sequence of etale sheaves over 𝑘

0→ ker(𝑓)→ 𝐴→ 𝐵 → 0.

This induces an exact sequence of 𝐺𝑘-modules

0→ ker(𝑓)(𝑘)→ 𝐴(𝑘)→ 𝐵(𝑘)→ 0.

Since for an abelian group 𝑀 , one has 𝑇ℓ𝑀 = Hom(Qℓ/Zℓ,𝑀), applying Hom(Qℓ/Zℓ,−) to the above exactsequence yields (writing, by abuse of notation, 𝑇ℓ𝐴 for 𝑇ℓ𝐴(𝑘))

0→ 𝑇ℓ ker(𝑓)(𝑘)→ 𝑇ℓ𝐴→ 𝑇ℓ𝐵 → Ext1(Qℓ/Zℓ, ker(𝑓)(𝑘)).

Since ker(𝑓) is a finite group scheme, we have 𝑇ℓ ker(𝑓)(𝑘) = 0 by Lemma 2.1.7

0→ 𝑇ℓ𝐴→ 𝑇ℓ𝐵 → Ext1(Qℓ/Zℓ, ker(𝑓)(𝑘)).

Since 𝑇ℓ𝐴 and 𝑇ℓ𝐵 have the same rank as 𝑓 is an isogeny (or since Ext1(Qℓ/Zℓ, ker(𝑓)(𝑘)) is finite), tensoringwith Qℓ yields the desired isomorphism.

Lemma 4.3.22. Let 𝑘 = F𝑞 be a finite field and 𝐴/𝑘 be an Abelian variety of dimension 𝑔. Denote the

eigenvalues of the Frobenius Frob𝑞 on 𝑉ℓ𝐴 by (𝛼𝑖)2𝑔𝑖=1. Then 𝛼𝑖 ↦→ 𝑞/𝛼𝑖 is a bijection.

Proof. The Weil pairing induces a perfect Galois equivariant pairing

𝑉ℓ𝐴× 𝑉ℓ𝐴∨ → Qℓ(1),

and, choosing a polarisation 𝑓 : 𝐴 → 𝐴𝑉 , by Lemma 4.3.21, we also have by precomposing a perfect Galoisequivariant pairing

⟨·, ·⟩ : 𝑉ℓ𝐴× 𝑉ℓ𝐴→ Qℓ(1).


Now let 𝑣𝑖 be an eigenvector of Frob𝑞 on 𝑉ℓ𝐴 with eigenvalue 𝛼𝑖. Then, since the pairing ⟨·, ·⟩ is perfect, thereis an eigenvector 𝑣𝑗 of Frob𝑞 on 𝑉ℓ𝐴 such that ⟨𝑣𝑖, 𝑣𝑗⟩ = 𝑥 = 0 (otherwise, we would have ⟨𝑣𝑖, 𝑣𝑗⟩ = 0 for alleigenvectors 𝑣𝑗 , but there is a basis of eigenvectors on the Tate module since the Frobenius acts semi-simply). Now,since the pairing is Galois equivariant, 𝑞𝑥 = Frob𝑞(𝑥) = Frob𝑞 ⟨𝑣𝑖, 𝑣𝑗⟩ = ⟨Frob𝑞 𝑣𝑖,Frob𝑞 𝑣𝑗⟩ = ⟨𝛼𝑖𝑣𝑖, 𝛼𝑗𝑣𝑗⟩ =𝛼𝑖𝛼𝑗 ⟨𝑣𝑖, 𝑣𝑗⟩ = 𝛼𝑖𝛼𝑗𝑥. Since 𝑥 = 0, the statement follows.

Theorem 4.3.23. Let 𝑘 = F𝑞, 𝑞 = 𝑝𝑛 be a finite field and 𝑋/𝑘 a smooth projective and geometrically connectedvariety and assume �� = 𝑋 ×𝑘 𝑘 satisfies

(a) the Neron-Severi group of �� is torsion-free and(b) the dimension of H1

Zar(��,O��) as a vector space over 𝑘 equals the dimension of the Albanese of ��/𝑘.If 𝐵/𝑘 is an Abelian variety, then H1(𝑋,𝐵) is finite and its order satisfies the relation

𝑞𝑔𝑑∏𝑎𝑖 =𝑏𝑗

(1− 𝑎𝑖

𝑏𝑗

)=H1(𝑋,𝐵)

|det ⟨𝛼𝑖, 𝛽𝑗⟩| ,

where 𝐴/𝑘 is the Albanese of 𝑋/𝑘, 𝑔 and 𝑑 are the dimensions of 𝐴 and 𝐵, respectively, (𝑎𝑖)2𝑔𝑖=1 and (𝑏𝑗)

2𝑑𝑗=1 are

the roots of the characteristic polynomials of the Frobenius of 𝐴/𝑘 and 𝐵/𝑘, (𝛼𝑖)𝑟𝑖=1 and (𝛽𝑖)

𝑟𝑖=1 are bases for

Hom𝑘(𝐴,𝐵) and Hom𝑘(𝐵,𝐴), and ⟨𝛼𝑖, 𝛽𝑗⟩ is the trace of the endomorphism 𝛽𝑗𝛼𝑖 of 𝐴.

Remark 4.3.24. Note that the Hom𝑘(𝐴,𝐵) and Hom𝑘(𝐵,𝐴) are free Z-modules of the same rank 𝑟 =𝑟(𝑓𝐴, 𝑓𝐵) ≤ 4𝑔𝑑 by [Tat66a], p. 139, Theorem 1 (a), with 𝑓𝐴 and 𝑓𝐵 the characteristic polynomials of theFrobenius of 𝐴/𝑘 and 𝐵/𝑘. (Another argument for them having the same rank is that the category of Abelianvarieties up to isogeny is semi-simple.) Furthermore, H1(𝑋,𝐵) = H1(𝑋,𝐵 ×𝑘 𝑋) = X(𝐵 ×𝑘 𝑋/𝑋) since for𝑈 → 𝑋, one has 𝐵(𝑈) = (𝐵 ×𝑘 𝑋)(𝑈) by the universal property of the fibre product.

Proof. See [Mil68], p. 98, Theorem 2.

Example 4.3.25. (a) and (b) are satisfied for 𝑋 = 𝐴 an Abelian variety or a curve: (a) because of [Mum70],p. 178, Corollary 2, and (b) since 𝐴∨ = Pic0𝐴/𝑘 is an Abelian variety, in particular smooth and reduced. Seealso Theorem 4.3.10, Remark 4.3.11 and Example 4.3.12.

Combining the above, we get

Corollary 4.3.26. In the situation of Theorem 4.3.23, one has

𝑞𝑔𝑑∏𝑎𝑖 =𝑏𝑗

(1− 𝑎𝑖

𝑏𝑗

)= |X(𝐵 ×𝑘 𝑋/𝑋)|𝑅(𝐵).

Definition 4.3.27. Define the 𝐿-function of 𝐵 ×𝑘 𝑋/𝑋 as the 𝐿-function of the motive

ℎ1(𝐵)⊗ (ℎ0(𝑋)⊕ ℎ1(𝑋)) = ℎ1(𝐵)⊕ (ℎ1(𝐵)⊗ ℎ1(𝑋)),

namely

𝐿(𝐵 ×𝑘 𝑋/𝑋, 𝑠) =𝐿(ℎ1(𝐵)⊗ ℎ1(𝑋), 𝑠)

𝐿(ℎ1(𝐵), 𝑠).

Here, the Kunneth projectors are algebraic by [Jan92], p. 451, Remarks 2).

Theorem 4.3.28. The two 𝐿-functions Definition 4.1.3 and Definition 4.3.27 agree for constant Abelian schemes.

Proof. One has 𝑉ℓ𝐵 = H1(��,Qℓ)∨ by Theorem 4.1.13, (𝑉ℓ𝐵)∨ = (𝑉ℓ𝐵

∨)(−1) by Remark 4.3.16, 𝑉ℓ(𝐵) ∼= 𝑉ℓ(𝐵∨)

by Lemma 4.3.21 and the existence of a polarisation, H𝑖(��, 𝑉ℓA ) = H𝑖(��,Qℓ) ⊗ 𝑉ℓ𝐵 by Lemma 4.3.7 and𝑉ℓA = (𝑉ℓ𝐵)×𝑘 𝑋 by Lemma 4.3.1 since A /𝑋 is constant. Using this, one gets

𝐿(ℎ𝑖(𝑋)⊗ ℎ1(𝐵), 𝑡) = det(1− Frob−1𝑞 𝑡 | H𝑖(��,Qℓ)⊗H1(��,Qℓ))

= det(1− Frob−1𝑞 𝑡 | H𝑖(��,Qℓ)⊗ 𝑉ℓ(𝐵∨)(−1))

= det(1− Frob−1𝑞 𝑡 | H𝑖(��,Qℓ)⊗ 𝑉ℓ(𝐵)(−1))

= det(1− Frob−1𝑞 𝑡 | H𝑖(��, (𝑉ℓ𝐵)×𝑘 𝑋)(−1))

= det(1− Frob−1𝑞 𝑞−1𝑡 | H𝑖(��, 𝑉ℓA ))

= 𝐿𝑖(A /𝑋, 𝑞−1𝑡),

for 𝑖 = 0, 1. Now conclude using ℎ1(𝐵) = ℎ0(𝑋)⊗ ℎ1(𝐵) since 𝑋 is connected.


Remark 4.3.29. Note that

ord𝑡=1

𝐿(A /𝑋, 𝑡) = ord𝑠=1

𝐿(A /𝑋, 𝑞−1𝑞𝑠).

Remark 4.3.30. Note that this 𝐿-function does not satisfy a functional equation coming from Poincare duality.

Remark 4.3.31. Now let us explain how we came up with this definition of the 𝐿-function.

I omit the characteristic Polynomials 𝐿𝑖(A /𝑋, 𝑡) in higher dimensions 𝑖 > 1 since otherwise cardinalitiesof cohomology groups would turn up in the special 𝐿-value for which I have no interpretation as in the case𝑖 = 0 and the cardinality of the ℓ-torsion of the Mordell-Weil group or in the case 𝑖 = 1 and the cardinalityof the ℓ-torsion of the Tate-Shafarevich group. In the case of a curve 𝐶 as a basis, my definition is the sameas the classical definition of the 𝐿-function up to an 𝐿2(𝑡)-factor. This factor contributes basically only afactor |Tor A ∨(𝑋)[ℓ∞]| in the denominator. In the classical curve case dim𝑋 = 1, the 𝐿-function can also berepresented as a product over all closed points 𝑥 ∈ |𝑋| of Euler factors.

I came up with my definition of the 𝐿-function for a higher dimensional basis via the constant Abelian schemecase: The main contribution of the motive of an Abelian variety is ℎ1 since the motive of an Abelian scheme isan exterior algebra over ℎ1. Thus it suggests itself to take ℎ1(𝐵) as a piece for the motive of the 𝐿-function. Inthe classical case is the motive of 𝐵 ×𝑘 𝐶/𝐶 for a curve 𝐶/𝑘 just ℎ1(𝐵)⊗ ℎ*(𝐶). Here, ℎ*(𝐶) decomposes asℎ0(𝐶)⊕ ℎ1(𝐶)⊕ ℎ2(𝐶). The summand ℎ1(𝐵)⊗ ℎ2(𝐶) contributes only the cardinality of Tor A ∨(𝑋), whichis well-understood. For a higher dimensional basis 𝑋 one has to omit the higher terms ℎ𝑖(𝑋) for 𝑖 > 1 sinceotherwise factors in the special 𝐿-value would turn up which I do not have an interpretation for.

We expand

𝐿(𝐵 ×𝑘 𝑋/𝑋, 𝑠) =𝐿(ℎ1(𝐵)⊗ ℎ1(𝑋), 𝑠)

𝐿(ℎ1(𝐵), 𝑠)

=

∏2𝑑𝑗=1

∏2𝑔𝑖=1(1− 𝑎𝑖𝑏𝑗𝑞−𝑠)∏2𝑑

𝑗=1(1− 𝑏𝑗𝑞−𝑠).

By Lemma 4.3.22, one has for the numerator

2𝑑∏𝑗=1

2𝑔∏𝑖=1

(1− 𝑎𝑖𝑏𝑗𝑞−𝑠) =

2𝑑∏𝑗=1

2𝑔∏𝑖=1

(1− 𝑎𝑖

𝑏𝑗𝑞1−𝑠

), (4.3.10)

and the denominator has no zeros at 𝑠 = 1 by the Riemann hypothesis. Therefore

ord𝑠=1

𝐿(𝐵 ×𝑘 𝑋/𝑋, 𝑠) = 𝑟(𝑓𝐴, 𝑓𝑏)

is equal to the number 𝑟(𝑓𝐴, 𝑓𝐵) of pairs (𝑖, 𝑗) such that 𝑎𝑖 = 𝑏𝑗 , which equals by [Tat66a], p. 139, Theorem 1 (a)the rank 𝑟 of (𝐵 ×𝑘 𝑋)(𝑋):

𝑟(𝑓𝐴, 𝑓𝐵) = rk Hom𝑘(𝐴,𝐵)

= rk Hom𝑘(𝑋,𝐵) by the universal property of the Albanese

= rk Hom𝑋(𝑋,𝐵 ×𝑘 𝑋),

see Corollary 4.3.3.

Lemma 4.3.32. The denominator evaluated at 𝑠 = 1 equals

2𝑑∏𝑗=1

(1− 𝑏𝑗𝑞−1) =|Tor (𝐵 ×𝑘 𝑋)(𝑋)|

𝑞𝑑.


Proof.

2𝑑∏𝑗=1

(1− 𝑏𝑗𝑞−1) =

2𝑑∏𝑗=1

(1− 1

𝑏𝑗

)by Lemma 4.3.22

=

2𝑑∏𝑗=1

𝑏𝑗 − 1

𝑏𝑗

=

2𝑑∏𝑗=1

1− 𝑏𝑗𝑏𝑗

since 2𝑑 is even

=deg(id−Frob𝑞)

𝑞𝑑by Lemma 4.3.22 and [Lan58], p. 186 f., Theorem 3

=|𝐵(F𝑞)|𝑞𝑑

since id−Frob𝑞 is separable

=|Tor (𝐵 ×𝑘 𝑋)(𝑋)|

𝑞𝑑by Lemma 4.3.1 3.

Remark 4.3.33. Note that, if 𝑋/𝑘 is a smooth curve, (𝐵 ×𝑘 𝑋)(𝑋) = 𝐵(𝐾) with 𝐾 = 𝑘(𝑋) the functionfield of 𝑋 by the valuative criterion since 𝑋/𝑘 is a smooth curve and 𝐵/𝑘 is proper. For general 𝑋, settingA = 𝐵 ×𝑘 𝑋 and 𝐾 = 𝑘(𝑋) the function field, (𝐵 ×𝑘 𝑋)(𝑋) = A (𝑋) = 𝐴(𝐾) also holds true because of theweak Neron mapping property.

Remark 4.3.34. One has |Tor (𝐵 ×𝑘 𝑋)(𝑋)| = |𝐵(𝑘)| = |𝐵∨(𝑘)| = |Tor (𝐵 ×𝑘 𝑋)∨(𝑋)| by Lemma 4.3.1 3and Lemma 4.1.15.

Definition 4.3.35. Define the regulator 𝑅(𝐵) by det(⟨·, ·⟩).

Remark 4.3.36. We have

1− 𝑞1−𝑠 = 1− exp(−(𝑠− 1) log 𝑞)

= (log 𝑞)(𝑠− 1) +𝑂((𝑠− 1)2

)for 𝑠→ 1

using the Taylor expansion of exp.

Putting everything together,

lim𝑠→1

𝐿(A /𝑋, 𝑠)

(𝑠− 1)𝑟=

𝑞𝑑(log 𝑞)𝑟

|Tor A (𝑋)|∏𝑎𝑖 =𝑏𝑗

(1− 𝑎𝑖

𝑏𝑗

)by Lemma 4.3.32 and (4.3.10)

= 𝑞(𝑔−1)𝑑(log 𝑞)𝑟|X(A /𝑋)| ·𝑅(𝐵)

|Tor A (𝑋)|by Corollary 4.3.26.

We conclude with our main theorem.

Theorem 4.3.37 (The conjecture of Birch and Swinnerton-Dyer for constant Abelian schemes over higher-di-mensional bases, version with the height pairing). In the situation of Theorem 4.3.23, one has:

1. The Tate-Shafarevich group X(A /𝑋) is finite.

2. The vanishing order equals the Mordell-Weil rank 𝑟: ord𝑠=1

𝐿(A /𝑋, 𝑠) = rk𝐴(𝐾).

3. There is the equality for the leading Taylor coefficient

𝐿*(A /𝑋, 1) = 𝑞(𝑔−1)𝑑(log 𝑞)𝑟|X(A /𝑋)| ·𝑅(𝐵)

|Tor A (𝑋)|.

Remark 4.3.38. Note that the factor 𝑞(𝑔−1)𝑑 also appears in [Bau92], p. 286, Theorem 4.6 (ii).

REFERENCES 63

Combining Theorem 4.1.28 and Theorem 4.3.37 and using Theorem 4.3.28, one can identify the remainingtwo expressions in Theorem 4.1.28:

Corollary 4.3.39. Assuming (a) and (b) of Theorem 4.3.23, in Theorem 4.1.28 resp. Lemma 4.1.26, allequalities hold and one has

|det(·, ·)ℓ|−1ℓ ·

H2(��, 𝑇ℓA )Γ

= 1.

Since both factors are positive integers, it follows that

|det(·, ·)ℓ|−1ℓ = 1,

H2(��, 𝑇ℓA )Γ

= 1.

Remark 4.3.40. In fact,H2(��, 𝑇ℓA )Γ

= 1 for constant Abelian schemes (under the assumption (a) above

that NS(��) is torsion-free) can also be seen directly: The long exact sequence associated to the short exactsequence of etale sheaves on ��


yields the exactness of

0→ H1(��,G𝑚)/ℓ𝑛 → H2(��, 𝜇ℓ𝑛)→ H2(��,G𝑚)[ℓ𝑛]→ 0.

Combining with the exactness of

0→ Pic0(��)→ Pic(��)→ NS(��)→ 0

and the divisibility of Pic0(��) (hence H1(��,G𝑚)/ℓ𝑛 = NS(��)/ℓ𝑛) and passage to the inverse limit lim←−𝑛 gives us

0→ NS(��)⊗Z Zℓ → H2(��,Zℓ(1))→ 𝑇ℓH2(��,G𝑚)→ 0

since the NS(��)/ℓ𝑛 are finite by [Mil80], p. 215, Theorem V.3.25, so they satisfy the Mittag-Leffler condition.As NS(��) is torsion-free (by assumption (a) above) and 𝑇ℓH

2(��,G𝑚) too (by Lemma 2.1.9), it follows thatH2(��,Zℓ(1)) is torsion-free, so also (here we are using that A /𝑋 is constant)

H2(��, 𝑇ℓA )Γ = (H2(��,Zℓ(1))⊗Zℓ𝑇ℓ𝐴(−1))Γ,

so H2(��, 𝑇ℓA )Γ

= 1

since this group is finite (having weight 2− 1 = 1 = 0) and torsion-free (as a subgroup of a tensor product oftorsion-free groups).

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Timo Keller, Fakultat fur Mathematik, Universitat Regensburg, 93040 Regensburg, Germany

on the conjecture of birch and swinnerton-dyer for abelian ... · birch-swinnerton-dyer and the...

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