On Néron class groups of abelian varieties

Abstract

Abstract. Let F be a global field, let S∞ be the set of archimedean primes of F and let S be any nonempty finite set of primes of F containing S∞. In this paper we study the Néron S-class group CA,F,S of an abelian variety A defined over F . In the well-known analogy that exists between the Birch and Swinnerton-Dyer conjecture for A over F and the analytic class number formula for the field F (in the number field case), the finite group CA,F,S ∞ (not the Tate-Shafarevich group of A) is a natural analog of the ideal class group of F .

Torsion and Tamagawa numbers

“…-Let K be a number field and let G/K be a smooth group scheme with an integral model of finite type G/ Spec(O K ). The class group C(G) is defined for instance in [26].…”

Brauer groups and Neron class groups

“…If v / ∈ S, we will write K v for the completion of K at v, O v for the ring of integers of K v , k(v) for the corresponding residue field and K nr v for the maximal unramified extension of K v inside a fixed separable algebraic closure of K v . Let A be an abelian variety over K. The Néron S-class group of A, introduced in [9], is the finite abelian group (0.1)…”

“…where, for every prime v / ∈ S, Φ v (A) is the étale k(v)-group scheme of connected components of the Néron model of A Kv over O v and the v-component of the map ρ is the canonical reduction map A(K) → Φ v (A)(k(v)). The first objective of this paper is to extend the duality theorem established in [9] for the group (0.1) by removing from [op.cit.] the hypothesis that the Tate-Shafarevich group X 1 (A) of A is finite.…”

“…(=Corollary 2.5 ) There exists a canonical perfect pairing of finite abelian groups C A, S /ϕ A,S ( T X 1 (A) div ) × C 1 A t , S → Q/Z, where C 1 A t , S is the group (1.6) attached to the abelian variety A t dual to A. The proof of Theorem 0.1 is completely different from and substantially simpler than the proof of [9,Theorem 4.9]. Its main ingredient is the generalized Cassels-Tate dual exact sequence established in [6].…”

“…

Let K be a global field and let S be a finite set of primes of K containing the archimedean primes. We generalize the duality theorem established in [9] for the Néron S-class group of an abelian variety A over K by removing the hypothesis that the Tate-Shafarevich group of A is finite. We also derive an exact sequence that relates the indicated group associated to the Jacobian variety of a proper, smooth and geometrically connected curve X over K to a certain finite subquotient of the Brauer group of X.

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