Number Theory Seminar
Freddy Saia
UIC
Shimura curve Atkin--Lehner quotients of genus at most two
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.
Friday September 19, 2025 at 12:00 PM in 636 SEO