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The pro-$\ell$ Galois representation attached to the arithmetic fundamental group of a curve influences heavily the arithmetic of its branched `$\ell$-covers.' In many cases, the $\ell$-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 $\ell$-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.

A finiteness conjecture on abelian varieties with constrained prime power torsion