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Authors
Affiliations
1 Department of Mathematics, University of California, Berkeley, CA 94720-3840, US
Source
Journal of the Ramanujan Mathematical Society, Vol 31, No 1 (2016), Pagination: 31-61
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 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 ∉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 ∉ 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.