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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-Tate pairing 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 Tate pairing. This respects minimally rami®ed structures for p3 l; by [1], 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 ˆ R and l ˆ 2. We refer to [37], Remark 1.3.7 for the natural

.6, states 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 Mazur-Tate coincide. 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-Tate pairings 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–Tate pairing 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 L v AS H 1 kv;TZ=nZðMÞ . It follows that ½a2; b 0 equals the sum of the local pairings hcv; b 0viv for v A S, where b 0 v is the image of b 0 in H 1 kv;TZ=nZðM Þ . 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 of L v AS 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-Tate pairing 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-Tate pairing for abelian varieties. The idea is to use the diminished cup-product construction discovered by Poonen and Stoll (see [20], 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 Tate pairing H 1ðKw;TÞ H 1ðKw;TÞ ! W H 2 Kw;Rð1Þ GR 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 x; iðlÞy for any l A LðFÞ. In order to make the Tate pairing LðFÞ- equivariant, we agree to view X ðFÞ as a LðFÞ-module via ðl f ÞðxÞ ¼ f iðlÞx and similarly 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: Ky Cy Dy G K : We will often work with the Iwasawa algebras LðKyÞ and LðDyÞ, so it is convenient to define the ideal

perfect 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–Tate pair- ing’’ [1ðAÞ [1ðBÞ ! Q=Z ([11], Corollary 4.9, and [8], 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–Tate pairing 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 ! Q l 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. Let 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 Tate pairing h ; il : H 1ðkl;TÞ H 1ðkl;T Þ ! Qp