We discuss generalizations of Kolyvagin's work on bounding Selmer ranks of elliptic curves using Heegner points. The speaker's thesis proves the analytic rank zero case of the Bloch–Kato conjecture for many conjugate self-dual motives via the theta correspondence within the Gan–Gross–Prasad framework.

Extending this strategy to the orthogonal setting, we show that the nonvanishing of certain Gross–Prasad periods or diagonal cycle classes determines Selmer ranks for the corresponding Rankin–Selberg motives over suitable coefficient fields. These diagonal cycles are higher-dimensional analogues of Heegner points, Gross–Schoen cycles, and Hirzebruch–Zagier cycles. This yields analytic rank zero cases of the Bloch–Kato conjecture for many self-dual motives, including all Cartesian powers of non-CM elliptic curves over Q. Key ingredients are the geometry of the basic loci in the special fibers of GSpin Shimura varieties at primes of bad reduction and the theory of automorphic forms.