# Journal of the Ramanujan Mathematical Society

Volume 31, Issue 1, March 2016  pp. 31–61.

Selmer groups as flat cohomology groups

Authors:  Kestutis Cesnavicius
Author institution:Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA

Summary:  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 p m-Selmer group Sel p m A need not be determined by the mod p m Galois representation A[p m]; we show, however, that this is the case if p is large enough. More precisely, we exhibit a finite explicit set of rational primes Σ depending on K and A, such that Sel p m A is determined by A[p m] for all p ∉ Σ. In the course of the argument we describe the flat cohomology group H fppf 1(O K, A[pm]) of the ring of integers of K with coefficients in the p m-torsion A[p m] of the Néron model of A by local conditions for p ∉ Σ, compare them with the local conditions defining Sel p m A, and prove that A[p m] itself is determined by A[p m] for such p. Our method sharpens the known relationship between Sel p m A and H fppf 1 (O K, A [p m]) and continues to work for other isogenies φ between abelian varieties over global fields provided that deg φ is constrained appropriately. To illustrate it, we exhibit resulting explicit rank predictions for the elliptic curve 11A1 over certain families of number fields.