Selmer Groups As Flat Cohomology Groups

Abstract

Given a prime number p, Bloch and Kato showed how the pa��-Selmer group of an abelian variety A over a number field K is determined by the p-adic Tate module. In general, the pm-Selmer group Selpm 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 Σ depending on K and A, such that Selpm A is determined by A[pm] for all p a��Σ. In the course of the argument we describe the flat cohomology group H1 fppf(OK ,A[pm]) of the ring of integers of K with coefficients in the pm-torsion A[pm] of the Néron model of A by local conditions for p a�� Σ, compare them with the local conditions defining Selpm A, and prove that A[pm] itself is determined by A[pm] for such p. Our method sharpens the known relationship between Selpm A and H1 fppf(OK ,A[pm]) 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.