holomorphic projection of the linear combination of products of some
modular forms. We call Z the geometric kernel and call Φ the analytic kernel.
Then the geometric kernel Z gives the Néron-Tate height, and the analytic
kernel Φ gives the L-function. The identity (4.60) in the proof of the original
Gross-Zagier formula is a particular case of (4.61).
Proof of the fundamental identity. Let us give a brief explanation on how
to prove the equality (4.61). The definition of Z involves the Néron-Tatepairing
and Hecke correspondence. Hence the terms of Z can be regrouped into
letting the ®-
nite subgroup H 1f ;S K;T on T be the exact annihilator of H 1f ;S K ;T under the Tatepairing. This respects minimally rami®ed structures for p3 l; by , Proposition 3.8 it also
respects crystalline structures if T nZl Ql is deRham.
1.1.5. Archimedean structures. We brie¯y consider the archimedean case. Let K
denote either R or C and let T be an l-adic GK -module. The cohomology group H
1 K ;T is
trivial, so that there is only one choice for the ®nite/singular structure, unless K R and
l 2. We refer to , Remark 1.3.7 for the natural
that if the abelian variety AF has semistable ordinary reduction at all the primes of F above
p then the p-adic height pairing on AF defined using the unit-root splitting and the one of
Let us now describe the idea of the proof. p-adic height pairings are the Qp-valued
counterparts of the real-valued Néron-Tate height pairings on abelian varieties. As the
Néron-Tatepairings they can be decomposed into local contributions, one for each finite
place of the ground field F . At the places not dividing p, these local contributions are
the Cassels–Tatepairing equals that of the Artin–Verdier pairing ½a2; b 0,
where b 0 A H 1
U ;TZ=nZðM Þ
is a preimage of a 0. A diagram chasing now shows that a2
comes from ðcvÞ A
. It follows that ½a2; b 0 equals the sum of the local
pairings hcv; b 0viv for v A S, where b
v is the image of b
0 in H 1
Our assumption that ha; a 0i ¼ 0 for all a 0 A D1ðU ;M Þ½n thus implies that ðcvÞ
satisfies the assumptions of the lemma, and hence up to modifying it by an element
H0ðkv;MÞ (which does not change a), we may
field in one variable over a finite field, provided
that one ignores the p-primary torsion part of the groups under consideration, where
p ¼ char k. We leave the verification of this to the readers.
6. Comparison with the Cassels-Tatepairing
In this section, we give a definition of the pairing of Theorem 0.2 purely in terms
of Galois cohomology and show that in the case M ¼ ½0! A it reduces to the classical
Cassels-Tatepairing for abelian varieties.
The idea is to use the diminished cup-product construction discovered by Poonen and
Stoll (see , pp. 1117
H 1ðKw;TÞ=H 1FðKw;TÞ
where the sum is over all places w of K . A Selmer structure F is self-dual if the submodule
H 1FðKw;TÞ is maximal isotropic under the (symmetric) local Tatepairing
H 1ðKw;TÞ H 1ðKw;TÞ !
for every finite place w A SF.
Remark 2.1.2. Note that p3 2 implies H 1ðKw;TÞ ¼ 0 for w archimedean. By Tate
local duality, H 1FðKw;TÞ ¼ H 1unrðKw;TÞ is maximal isotropic for all w B SF.
Remark 2.1.3. If S is a submodule (resp. quotient) of T and ðF;SFÞ is a Selmer
structure on T , then there is an induced Selmer structure, still denoted ðF
duality gives a perfect pairing
H1ðFq;TÞ H1ðFq;W Þ ! F=O
satisfying ðlx; yÞ ¼
for any l A LðFÞ. In order to make the Tatepairing LðFÞ-
equivariant, we agree to view X ðFÞ as a LðFÞ-module via ðl f ÞðxÞ ¼ f
for X ðFÞ, X SðFÞ, etc.
Denote by Cy, Dy, and Ky the cyclotomic Zp-extension, the anticyclotomic Zp-
extension, and the unique Z2p-extension of K, respectively. Set G ¼ GalðDy=KÞ, so
GGZp. Thus we have a field diagram:
We will often work with the Iwasawa algebras LðKyÞ and LðDyÞ, so it is convenient to
define the ideal
pairing of finite groups
CA;F ;S C1B;F ;S ! Q=Z;
where C1B;F ;S is the group (3.7) associated to B.
89González-Avil és, On Néron class groups of abelian varieties
Proof. By the previous corollary and the existence of the perfect ‘‘Cassels–Tatepair-
ing’’ [1ðAÞ [1ðBÞ ! Q=Z (, Corollary 4.9, and , Corollary 6.7), the dual of the
second exact sequence appearing in Proposition 3.4 for B is an exact sequence
0! ðC1B;F ;SÞ
D ! D1ðU ;AÞ ![1ðAÞ ! 0:
The compatibility of the Cassels–Tatepairing with the pairings (4.3) and (4.4) for G ¼ 0
and G 0 ¼ F 0 (see
the local conditions F ¼Fcl or F ¼FL to be
H 1Fðk;TÞ :¼ ker
H 1ðk;TÞ !
H 1ðkl;TÞ=H 1Fðkl;TÞ
where the product runs over all rational primes l.
1.2. Local duality and Selmer groups for T*. We now define the dual Selmer struc-
ture to ðT ;FLÞ and the dual Selmer group. Later in this section, we will compute the
classical and the modified Selmer groups explicitly and compare their sizes to each other.
T ¼ HomZpðT ;Qp=ZpÞð1ÞGQp=Zp n w
be the Cartier dual of T . For any prime l of k, let h ; il denote the local Tatepairing
h ; il : H
1ðkl;TÞ H 1ðkl;T Þ ! Qp