Abelian Surfaces Good Away From 2

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