Faltings' height of CM cycles and Derivative of $L$-functions

Faltings' height of CM cycles and Derivative of $L$-functions

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Tonghai Yang, University of Wisconsin, Madison
Fine Hall 214

In this talk, we first describe a systematic way to construct `automorphic Green functions' for Kudla's special divisors on a Shimura variety of orthogonal type $(n, 2)$. We then give an explicit formula for their values at a CM cycle. This formula suggests a direct relation between the Faltings' height of these CM cycles with the central derivative of some Rankin-Selberg $L$-function. As an application, we also give an `analytic proof' of the Gross-Zagier formula without computing the local intersection numbers at finite primes. This is a joint work with Jan Bruinier.