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Passcode: The three digit integer that is the cube of the sum of its digits

For certain automorphic representations $\pi$ onÌýunitaryÌýgroups, 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 aÌýunitaryÌýShimuraÌývariety is nontrivial. This proves part of theÌýµþ±ð¾±±ô¾±²Ô²õ´Ç²Ô-µþ±ô´Ç³¦³óÌý³¦´Ç²ÔÂá±ð³¦³Ù³Ü°ù±ðÌý´Ú´Ç°ùÌý³Ü²Ô¾±³Ù²¹°ù²âÌý³§³ó¾±³¾³Ü°ù²¹Ìý±¹²¹°ù¾±±ð³Ù¾±±ð²õ, which generalizes the BSDÌýconjecture. Assuming Kudla's modularity conjecture, we further prove the arithmetic inner product formula for $L'(1/2, \pi)$, which generalizes the Gross-Zagier formula.ÌýWe 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.