邱聪灵博士, Yale University

For a smooth projective variety over a number field, the Beilinson—Bloch conjecture predicts that Chow groups of algebraic cycles modulo rational equivalence can be determined using cohomological data. In particular, it predicts the vanishing of some cycles under suitable cohomological conditions. We verify such vanishing on powers of Shimura curves, for CM 0-cycles and modified diagonal 1-cycles. Automorphy and modularity play essential roles in the proofs.