A finiteness conjecture on abelian varieties with constrained prime power torsion

Abstract

The pro-Galois representation attached to the arithmetic fundamental group of a curve influences heavily the arithmetic of its branched ' -covers.' In many cases, the -power torsion on the Jacobian of such a cover is fixed by the kernel of this representation, giving explicit information about this kernel. Motivated by the relative scarcity of interesting examples for -covers of the projective line minus three points, the authors formulate a conjecture to quantify this scarcity. A proof for certain genus one cases is given, and an exact set of curves satisfying the required arithmetic conditions in the base case is determined.