Hao Peng: On the Beilinson-Bloch-Kato conjecture for polarized motives

Number Theory Seminar: Hao Peng (MIT)

The (Beilinson-)Bloch–Kato conjecture vastly generalizes the rank part of the Birch–Swinnerton-Dyer conjecture for modular elliptic curves. Building on results of Liu-Tian-Xiao-Zhang-Zhu, we prove the Bloch-Kato conjecture at analytic rank zero for a large class of conjugate self-dual (unitary) motives, including those arising from odd symmetric powers of non-CM modular elliptic curves. As an application, we obtain families of nontrivial high-dimensional varieties for which the Hodge, Tate, and Bloch–Kato conjectures all hold for large \ell, namely, powers of certain non-CM modular elliptic curves. If time permits, I will also discuss work in progress toward the Bloch-Kato conjecture at analytic rank at most one for certain standard and Rankin-Selberg orthogonal motives over Q, where an almost minimal R = T theorem for orthogonal Shimura varieties plays a crucial role.