On the Frey-Mazur conjecture over low genus curves

2013/09/25 by Benjamin Bakker, Jacob Tsimerman, Bakker, Benjamin +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1309.6568

openalex publication_date 2013/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Frey--Mazur conjecture states that an elliptic curve over ℚ is determined up to isogeny by its p-torsion Galois representation for p≥ 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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