SFU Number Theory and Algebraic Geometry Seminar: Tian Wang

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.