# 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.

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