Bielliptic Shimura curves \(X_0^D(N)\) with nontrivial level

We work towards completely classifying all bielliptic Shimura curves \(X_0^D(N)\) with nontrivial level \(N\) coprime to \(D\), extending a result of Rotger that provided such a classification for level one. Combined with prior work, this allows us to determine the list of all relatively prime pairs...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2024-09
Main Authors: Padurariu, Oana, Saia, Frederick
Format: Article
Language:English
Subjects:
Classification
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We work towards completely classifying all bielliptic Shimura curves \(X_0^D(N)\) with nontrivial level \(N\) coprime to \(D\), extending a result of Rotger that provided such a classification for level one. Combined with prior work, this allows us to determine the list of all relatively prime pairs \((D,N)\) for which \(X_0^D(N)\) has infinitely many degree \(2\) points. As an application, we use these results to make progress on determining which curves \(X_0^D(N)\) have sporadic points. Using tools similar to those that appear in this study, we also determine all of the geometrically trigonal Shimura curves \(X_0^D(N)\) with \(\gcd(D,N)=1\) (none of which are trigonal over \(\mathbb{Q}\)).
ISSN:2331-8422