Abelian surfaces good away from 2

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.

9 pages, 1 table

References in corpus (1)

• Arithmetic of abelian varieties with constrained torsion