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 $p\notin \Sigma$. In the course of the argument we describe the flat cohomology group $H_{\mathrm{fppf}}^1(O_K,\mathcal{A}[p^m])$ of the ring of integers of $K$ with coefficients in the $p^m$-torsion $\mathcal{A}[p^m]$ of the Néron model of $A$ by local conditions for $p\notin \Sigma$, 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 $H_{\mathrm{fppf}}^1(O_K,\mathcal{A}[p^m])$ and continues to work for other isogenies $\varphi$ between abelian varieties over global fields provided that $\mathrm{deg}\varphi$ is constrained appropriately. To illustrate it, we exhibit resulting explicit rank predictions for the elliptic curve $11A1$ over certain families of number fields.

22 pages; final version, to appear in Journal of the Ramanujan Mathematical Society

References in corpus (1)

• Selmer groups and class groups

Cited by in corpus (6)

• Selmer groups and class groups
• The geometric average size of Selmer groups over function fields
• $p$-Selmer growth in extensions of degree $p$
• On Tate-Shafarevich groups of 1-motives over Galois extensions
• Arithmetic rank bounds for abelian varieties over function fields
• The $\ell$-parity conjecture over the constant quadratic extension