Abelian surfaces good away from 2

TL;DR

This paper investigates the relationship between good reduction properties of abelian varieties and their torsion extensions, showing that for abelian surfaces over , good reduction away from 2 is sufficient, but not in higher dimensions.

Contribution

It establishes that for abelian surfaces over , good reduction away from 2 implies certain torsion extension properties, and provides explicit examples where this fails in higher dimensions.

Findings

Good reduction away from 2 suffices for abelian surfaces over .

Explicit counterexamples exist for higher-dimensional abelian varieties.

Extension of results to elliptic curves and certain number fields.

Abstract

Fix a number field $k$ and a rational prime $\ell$. We consider abelian varieties whose $\ell$-power torsion generates a pro-$\ell$ extension of $k(\mu_{\ell^\infty})$ which is unramified away from $\ell$. It is a necessary, but not generally sufficient, condition that such varieties have good reduction away from $\ell$. In the special case of $\ell = 2$, we demonstrate that for abelian surfaces $A/\mathbb{Q}$, good reduction away from $\ell$ does suffice. The result is extended to elliptic curves and abelian surfaces over certain number fields unramified away from $\{2,\infty\}$. An explicit example is constructed to demonstrate that good reduction is not sufficient, at $\ell = 2$, for abelian varieties of sufficiently high dimension.