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

Oana Padurariu; Frederick Saia

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

Number Theory 2025-10-09 v3

Authors: Oana Padurariu , Frederick Saia

Abstract

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 $\mathbb{Q}$ ).

Keywords

Shimura curves bielliptic level degree 2 points sporadic points trigonal

Cite

@article{arxiv.2401.08829,
  title  = {Bielliptic Shimura curves $X_0^D(N)$ with nontrivial level},
  author = {Oana Padurariu and Frederick Saia},
  journal= {arXiv preprint arXiv:2401.08829},
  year   = {2025}
}

Comments

23 pages. This version adds a reference to a paper of Hindry for a result which we previously only attributed to Harris--Silverman. It also removes a theorem we previously used but found to be incorrect, and replaces all prior instances of its use with other methods which already appeared elsewhere in the paper so that none of our main results are affected

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

Abstract

Keywords

Cite

Comments

Related papers