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arXiv · 1309.6568

On the Frey-Mazur conjecture over low genus curves

Benjamin Bakker·Jacob Tsimerman
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Abstract

The Frey--Mazur conjecture states that an elliptic curve over $\mathbb{Q}$ is determined up to isogeny by its $p$-torsion Galois representation for $p\geq 17$. We study a geometric analog of this conjecture, and show that the map from isogeny classes of "fake elliptic curves"---abelian surfaces with quaternionic multiplication---to their $p$-torsion Galois representations is one-to-one over function fields of small genus complex curves for sufficiently large $p$ relative to the genus.

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Benjamin Bakker, Jacob Tsimerman. 2013-09-25. On the Frey-Mazur conjecture over low genus curves. https://arxiv.org/abs/1309.6568

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