Finiteness properties for Shimura curves and modified diagonal cycles

We prove that only finitely many Shimura curves can have gonality bounded by a given number, and we study the computability of this finite set. Motivated by the relation between hyperellipticity (that is, gonality 2) and the vanishing of the modified diagonal cycle, we conjecture that such vanishing occurs for only finitely many Shimura curves. We establish several finiteness and classification results toward this conjecture and, as a by-product, obtain explicit examples of curves with vanishing modified diagonal cycles. Our computations are based on modular form data from the database $\mathsf{\text{LMFDB}}$, and some of them are carried out using the computer algebra system $\mathsf{\text{Sage}}$.