Freddy Saia : Shimura curve Atkin--Lehner quotients of genus at most two

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

At

Sept. 19, 2025, noon

In

636 SEO

Abstract

This talk, based on joint work with Oana Padurariu, will concern quotients of the Shimura curves $X_0^D(N)$ over $\mathbb{Q}$ by Atkin—Lehner involutions, which parameterize abelian surfaces with potential quaternionic multiplication. We prove that there are exactly $3711$ quotients $X_0^D(N)/W$, with $D>1$ and $W$ a non-trivial Atkin—Lehner subgroup, having genus $g \leq 2$. We also investigate the arithmetic of these curves; we determine isomorphism classes over $\mathbb{Q}$ of the Jacobians of genus $1$ quotients, we prove a theorem on infinitude of rational points on Atkin—Lehner quotients, and we produce equations for over $500$ non-elliptic genus $1$ and bielliptic genus $2$ quotients. One of our main tools is an algorithm we implement to compute, when $N$ is squarefree, the dual graph of the minimal regular model of $X_0^D(N)/W$ over $\mathbb{Z}_p$ for each prime divisor $p$ of $D$ using the theory of Cerednik--Drinfeld reductions. We will aim to give an approachable overview of this theory in this context, with concrete examples along the way.