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

We work towards completely classifying all bielliptic Shimura curves \(X_0^D(N)\) $X_0^D(N)$ with nontrivial level N coprime to D and their bielliptic involutions, 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 \(X_0^D(N)\) $X_0^D(N)$ has infinitely many degree 2 points. As an application, we use these results to make progress on determining which curves \(X_0^D(N)\) $X_0^D(N)$ have sporadic points. Using tools similar to those that appear in this study, we also determine all of the geometrically trigonal Shimura curves \(X_0^D(N)\) $X_0^D(N)$ with \(\gcd (D,N)=1\) $gcd(D,N)=1$ (none of which are trigonal over \(\mathbb {Q}\) $\mathbb{Q}$ ).