Joint IAS/Princeton University Number Theory Seminar

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 Kudla's modularity conjecture, we further prove the arithmetic inner product formula for $L'(1/2, \pi)$, which generalizes the Gross-Zagier formula. We will motivate these conjectures and discuss some aspects of the proof. We will also mention recent extensions applicable to symmetric power L-functions of elliptic curves. This is joint work with Yifeng Liu.