On the Frey-Mazur conjecture over low genus curves

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 $p\ge 17$. 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.

17 pages. v2: result improved. v3: minor corrections

References in corpus (1)

• Shimura curves of genus at most two

Cited by in corpus (2)

• Rank 2 Local Systems, Barsotti-Tate Groups, and Shimura Curves
• P-torsion monodromy representations of elliptic curves over geometric function fields