Selmer groups as flat cohomology groups

Abstract

Given a prime number p, Bloch and Kato showed how the p Selmer group of an abelian variety A over a number field K is determined by the p-adic Tate module. In general, the pm1-Selmer group Selpmn A need not be determined by the mod pm Galois representation A[pm]; we show, however, that this is the case if p is large enough. More precisely, we exhibit a finite explicit set of rational primes E depending on K and A, such that Selpm A is determined by A[pm] for all ... In the course of the argument we describe the flat cohomology group ... of the ring of integers of K with coefficients in the pm- torsion A[pm] of the Neron model of A by local conditions for p V E, compare them with the local conditions defining Selm 2A, and prove that A[p't ] itself is determined by A[pm] for such p. Our method sharpens the relationship between Selpm A and ... which was observed by Mazur and continues to work for other isogenies 0 between abelian varieties over global fields provided that deg o is constrained appropriately. To illustrate it, we exhibit resulting explicit rank predictions for the elliptic curve 11A1 over certain families of number fields. Standard glueing techniques developed in the course of the proofs have applications to finite flat group schemes over global bases, permitting us to transfer many of the known local results to the global setting.