Central derivatives of some automorphic L-functions

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The PI's proposal consists of two problems on the central derivative of L-functions. The first problem is to generalize the derivative of the triple product formula from the split case to the semi-split case. He will use the idea of Yuan--Zhang--Zhang from the split case. The extra difficulties mainly come from the generating function and the height pairing. The second problem is to prove the derivative formula in the Gan--Gross--Prasad conjecture for the case (U(2),U(3)). He will use the relative trace formula proposed by Wei Zhang. Both problems are higher-dimensional analogues of the original Gross--Zagier formula. They will have applications to the Beilinson--Bloch conjecture, which is the higher-dimensional analogue of the Birch and Swinnerton-Dyer conjecture.A central topic of number theory is to solve Diophantine equations. Namely, given a system of polynomial equations of several variables with rational coefficients, one seeks for rational solutions. The Gross--Zagier formula characterizes some important solution of a cubic equation of two variables by some quantity defined by complex analytic functions. The current proposal attempts to generalize the Gross--Zagier formula to high dimensions, i.e., systems of more variables and more equations. The proofs and the understanding of the proposed problems will help us understand the theories of automorphic forms, group representations, Shimura varieties, and Arakelov geometry.