## 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 *C*_{A, 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 *C*_{A, F, S∞} (not the Tate–Shafarevich group of *A*) is a natural analog of the ideal class group of *F*.

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