Shiva Chidambaram : Arithmetic of genus 3 Jacobians with imaginary multiplication

Posted by Frederick Saia , part of the Number Theory Seminar.

At

Nov. 21, 2025, noon

In

636 SEO

Abstract

Let $C$ be a genus 3 curve whose Jacobian is geometrically simple and has imaginary multiplication. I will discuss an algorithm, developed jointly with Pip Goodman, to compute the set of primes $\ell$ for which the image of the associated mod-$\ell$ Galois representation is not maximal. There are two natural families of genus 3 Jacobians with imaginary multiplication by $\mathbb{Z}[i]$ and $\mathbb{Z}[\zeta_3]$, coming from curves with a $\mu_4$ or $\mu_6$ action. Can we construct any new families? How do we rigorously, and not just numerically, certify these extra endomorphisms when they don't come from automorphisms on the curve? I will also discuss these questions. These are ongoing joint works with Pip Goodman and Francesc Fite.