Effective open image theorem for products of principally polarized abelian varieties

Abstract: Let $E/\mathbb{Q}$ be an elliptic curve without complex multiplication. By Serre's open image theorem, the mod $\ell$ Galois representation $\overline{\rho}_{E, \ell}$ of $E$ is surjective for each prime number $\ell$ that is sufficiently large. Partially motivated by Serre's uniformity question, there has been research into an effective version of this result, which aims to find an upper bound on the largest prime $\ell$ such that $\overline{\rho}_{E, \ell}$ is nonsurjective. In this talk, we consider an analogue of the problem for a product of principally polarized abelian varieties $A_1, \ldots, A_n$ over $K$, where the varieties are pairwise non-isogenous over $\overline{K}$. We will present an effective version of the open image theorem for $A_1\times \ldots \times A_n$ due to Hindry and Ratazzi. This is joint work with Jacob Mayle.

algebraic geometrynumber theory

Audience: researchers in the discipline

( paper )

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SFU NT-AG seminar

Series comments: The Number Theory and Algebraic Geometry (NT-AG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry hosted by the NT-AG group (Nils Bruin, Imin Chen, Stephen Choi, Katrina Honigs, Nathan Ilten, Marni Mishna).

We acknowledge the support of PIMS, NSERC, and SFU.

For Fall 2025, the organizers are Katrina Honigs and Peter McDonald.

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