Mathematical Research Letters

Volume 15 (2008)

Number 6

A finiteness conjecture on abelian varieties with constrained prime power torsion

Pages: 1223 – 1231

DOI: http://dx.doi.org/10.4310/MRL.2008.v15.n6.a12

Authors

Christopher Rasmussen (Wesleyan University)

Akio Tamagawa (Research Institute for Mathematical Sciences)

Abstract

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.

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