  A precise and testable modularity conjecture for rational abelian surfaces A
with trivial endomorphisms, End_Q A = Z, is presented. It is consistent with
our examples, our non-existence results and recent work of C. Poor and D. S.
Yuen on weight 2 Siegel paramodular forms. We obtain fairly precise information
on ell-division fields of semistable abelian varieties A, mainly when A[ell] is
reducible, by considering extension problems for groups schemes of small rank.
Our general results imply, for instance, that the least prime conductor of an
abelian surface is 277.
