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We conjecture that only finitely many Shimura curves can have vanishingmodified diagonal cycles in the Chow groups of their triple products. We provesome finiteness results toward this conjecture in terms of an automorphiccriterion for such vanishing. In particular, we establish an analog of theclassification of hyperelliptic Shimura curves by Ogg. As a by-product, we alsoobtain some new examples of curves with vanishing modified diagonal cycles.

Finiteness properties for Shimura curves and modified diagonal cycles