Finiteness properties for Shimura curves and modified diagonal cycles

Abstract: We conjecture that only finitely many Shimura curves can have vanishing modified diagonal cycles in the Chow groups of their triple products. We prove some finiteness results toward this conjecture in terms of an automorphic criterion for such vanishing. In particular, we establish an analog of the classification of hyperelliptic Shimura curves by Ogg. As a by-product, we also obtain some new examples of curves with vanishing modified diagonal cycles.