Number Theory Seminar
Freddy Saia
UIC
Bielliptic Shimura curves $X_0^D(N)$
Abstract: Since Mazur's work on rational isogenies and rational torsion of elliptic curves over $\mathbb{Q}$, there has been concerted effort towards understanding low degree points on modular curves such as $X_0(N)$ and $X_1(N)$. Considerably less is known for Shimura curves, which parameterize abelian surfaces with quaternionic multiplication and analogous torsion structures. By a result of Shimura, these curves have no real points, hence no odd degree points, so we focus first on degree 2 points. We will discuss the determination of the Shimura curves $X_0^D(N)$ with infinitely many quadratic points, resulting from a study of the bielliptic curves in this family. This is based on joint work with Oana Padurariu.
Friday October 11, 2024 at 9:00 AM in 636 SEO