p-torsion monodromy representations of elliptic curves over geometric function fields

• 成果类型: Article
• 署名作者: Bakker, Benjamin; Tsimerman, Jacob
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2016.184.3.2
• 发表日期: 2016
• 页码: 709-744
• 关键词: riemann surface

摘要

Given a complex quasiprojective curve B and a nonisotrivial family epsilon of elliptic curves over B, the p-torsion epsilon[p] yields a monodromy representation p epsilon[p] : pi(1)(B) -> GL(2)(F-p). We prove that if p epsilon[p] congruent to p epsilon'[p], then epsilon and epsilon' are isogenous, provided p is larger than a constant depending only on the gonality of B. This can be viewed as a function field analog of the Frey-Mazur conjecture, which states that an elliptic curve over Q is determined up to isogeny by its p-torsion Galois representation for p > 17. The proof relies on hyperbolic geometry and is therefore only applicable in characteristic 0.