Computing Galois images of Picard curves

Abstract: Let $C$ be a genus $3$ curve whose Jacobian is geometrically simple and has geometric endomorphism algebra equal to an imaginary quadratic field. In particular, consider Picard curves $y^3 = f_4(x)$ where the geometric endomorphism algebra is $\mathbb{Q}(\zeta_3)$. We study the associated mod-$\ell$ Galois representations and their images. I will discuss an algorithm, developed in ongoing joint work with Pip Goodman, to compute the set of primes $\ell$ for which the images are not maximal. By running it on several datasets of Picard curves, the largest non-maximal prime we obtain is $13$. This may be compared with genus 1, where Serre's uniformity question asks if the mod-$\ell$ Galois image of non-CM elliptic curves over $\Q$ is maximal for all primes $\ell > 37$.

number theory

Audience: researchers in the topic

Harvard number theory seminar