Beilinson-Bloch conjecture for unitary Shimura varieties
For certain automorphic representations $\pi$ on unitary groups, we show that if $L(s, \pi)$ vanishes to order one at the center $s=1/2$, then the associated $\pi$-localized Chow group of a unitary Shimura variety is nontrivial. This proves part of the Beilinson-Bloch conjecture for unitary Shimura varieties, which generalizes the BSD conjecture. Assuming the modularity of Kudla's generating series of special cycles, we further prove a precise height formula for $L'(1/2, \pi)$. This proves the conjectural arithmetic inner product formula, which generalizes the Gross-Zagier formula to Shimura varieties of higher dimension. We will motivate these conjectures and discuss some aspects of the proof. This is joint work with Yifeng Liu.