Selmer Groups as Flat Cohomology Groups

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 Selp^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 Selp^m A is determined by A[p^m] for all p ∉Σ. In the course of the argument we describe the flat cohomology group H1 fppf(O_K ,A[p^m]) 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 Selp^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 H1 fppf(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.