Selmer groups as flat cohomology groups

Abstract

Given a prime number $p$, Bloch and Kato showed how the $p^\infty$-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 $\mathrm{Sel}_{p^m} A$ 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 $\Sigma$ depending on $K$ and $A$, such that $\mathrm{Sel}_{p^m} A$ is determined by $A[p^m]$ for all $p \not \in \Sigma$. In the course of the argument we describe the flat cohomology group $H^1_{\mathrm{fppf}}(O_K, \mathcal{A}[p^m])$ of the ring of integers of $K$ with coefficients in the $p^m$-torsion $\mathcal{A}[p^m]$ of the N\'{e}ron model of $A$ by local conditions for $p\not\in \Sigma$, compare them with the local conditions defining $\mathrm{Sel}_{p^m} A$, and prove that $\mathcal{A}[p^m]$ itself is determined by $A[p^m]$ for such $p$. Our method sharpens the known relationship between $\mathrm{Sel}_{p^m} A$ and $H^1_{\mathrm{fppf}}(O_K, \mathcal{A}[p^m])$ and continues to work for other isogenies $\phi$ between abelian varieties over global fields provided that $\mathrm{deg} \phi$ is constrained appropriately. To illustrate it, we exhibit resulting explicit rank predictions for the elliptic curve $11A1$ over certain families of number fields.Comment: 22 pages; final version, to appear in Journal of the Ramanujan Mathematical Societ