Central derivatives of some automorphic L-functions

PI的建议包括两个关于L-函数的中心导数的问题。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。 他将使用元-张-张的想法从分裂的情况下。额外的困难主要来自于生成函数和高度配对。第二个问题是证明Gan-Gross-Prasad猜想中关于（U（2），U（3））的导数公式。他将使用张炜提出的相对迹公式。这两个问题都是原来的格罗斯-扎吉尔公式的高维类似物。它们将应用于Beilinson-Bloch猜想，这是Birch和Swinnerton-Dyer猜想的高维模拟。数论的一个中心主题是求解丢番图方程。 也就是说，给定一个系统的多项式方程的几个变量与合理的系数，一个寻求合理的解决方案。格罗斯-扎吉尔公式用复解析函数定义的某些量来刻画二元三次方程的某些重要解。目前的建议试图将Gros-Zagier公式推广到高维，即，多变量多方程的系统对这些问题的证明和理解将有助于我们理解自守形式、群表示、志村簇和阿拉克洛夫几何等理论。

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.