Arithmetic of abelian varieties with constrained torsion

Abstract

Let $K$ be a number field. We present several new finiteness results for isomorphism classes of abelian varieties over $K$ whose $\ell$-power torsion fields are arithmetically constrained for some rational prime $\ell$. Such arithmetic constraints are related to an unresolved question of Ihara regarding the kernel of the canonical outer Galois representation on the pro-$\ell$ fundamental group of $P^1 - \{0,1,\infty\}$. Under GRH, we demonstrate the set of classes is finite for any fixed $K$ and any fixed dimension. Without GRH, we prove a semistable version of the result. In addition, several unconditional results are obtained when the degree of $K/\Q$ and the dimension of abelian varieties are not too large, through a careful analysis of the special fiber of such abelian varieties. In some cases, the results (viewed as a bound on the possible values of $\ell$) are uniform in the degree of the extension $K/\Q$.