P-torsion monodromy representations of elliptic curves over geometric\n function fields

2014/03/27 by Jacob Tsimerman, Tsimerman, Jacob, Benjamin Bakker +1 · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.1403.7168

Abstract

Given a complex quasiprojective curve B and a non-isotrivial family\n\E of elliptic curves over B, the p-torsion \E[p]\nyields a monodromy representation \ρ_\E[p]:\π₁(B)\→\n\GL₂( mathbbF_p). We prove that if \ρ _{mathcal E}[p]\≅\n\ρ _{mathcal E'}[p] then \E and mathcal E' are isogenous,\nprovided p is larger than a constant depending only on the gonality of B.\nThis can be viewed as a function field analog of the Frey--Mazur conjecture,\nwhich states that an elliptic curve over \ℚ is determined up to\nisogeny by its p-torsion Galois representation for p> 17. The proof relies\non hyperbolic geometry and is therefore only applicable in characteristic 0.\n

Cited by

• Effective surjectivity of Galois representations of products of elliptic curves over function fields

Related

• On the Frey-Mazur conjecture over low genus curves
• Polynomial bounds on torsion from a fixed geometric isogeny class of elliptic curves
• Uniform polynomial bounds on torsion from rational geometric isogeny classes
• Pursuing polynomial bounds on torsion
• Elliptic curves with maximally disjoint division fields