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

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 $\mathbb{Q}$).

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