Central derivatives of some automorphic L-functions

PI的建议包含两个关于L函数中心导数的问题。第一个问题是将三重积公式的导数从分裂情形推广到半分裂情形。他将使用拆分案件中的元-张-张的想法。额外的困难主要来自母函数和高度配对。第二个问题是证明Gan-Gross-Prasad猜想(U(2)，U(3))中的导数公式。他将使用张伟提出的相对迹公式。这两个问题都是原始Gross-Zagier公式的高维类似。它们将应用于Beilinson-Bloch猜想，Beilinson-Bloch猜想是Birch和Swinnerton-Dyer猜想的高维类似。数论的一个中心主题是求解丢番图方程。也就是说，给定一个具有有理系数的多变量多项式方程组，就可以寻求有理解。Gross-Zagier公式用复解析函数定义的量刻画了二元三次方程的一些重要解。目前的建议试图将Gross-Zagier公式推广到高维，即由更多变量和更多方程组成的系统。对所提出问题的证明和理解将有助于我们理解自同构形式、群表示、Shimura簇和Arakelov几何的理论。

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.