A lower bound for average values of dynamical Green's functions

Average CM-values of Higher Green's Function and Factorization

In this paper, we prove an averaged version of an algebraicity conjecture in \cite{GKZ87} concerning the values of higher Green's function at CM points. Furthermore, we give the factorization of the ideal generated by such algebraic value in the spirit of the famous work of Gross and Zagier on singular moduli.

双曲3-空间上格林函数特殊值的研究

Special values of Green's functions on hyperbolic $3$-space

本文将 Gross-Kohnen-Zagier 关于模曲线高阶格林函数特殊值代数性的平均结果推广到双曲3-空间，证明其平均值可用素数对数和实二次域单位对数表示，并进一步研究扭曲平均得到代数数。

射影簇上动力系统的 Arakelov-Green 函数

Arakelov-Green's functions for dynamical systems on projective varieties

本文引入极化动力系统的 Arakelov-Green 函数并证明其 Elkies 型下界，进而得到乘积公式域上 Abel 簇非挠点典范高度的 Lehmer 型下界。

Nilsystems and ergodic averages along primes

A celebrated result by Bourgain and Wierdl states that ergodic averages along primes converge almost everywhere for $L^p$-functions, $p>1$, with a polynomial version by Wierdl and Nair. Using an anti-correlation result for the von Mangoldt function due to Green and Tao we observe everywhere convergence of such averages for nilsystems and continuous functions.

CM values of higher automorphic Green functions for orthogonal groups

Gross and Zagier conjectured that the CM values (of certain Hecke translates) of the automorphic Green function $G_s(z_1,z_2)$ for the elliptic modular group at positive integral spectral parameter $s$ are given by logarithms of algebraic numbers in suitable class fields. We prove a partial average version of this conjecture, where we sum in the first variable $z_1$ over all CM points of a fixed discriminant $d_1$ (twisted by a genus character), and allow in the second variable the evaluation at individual CM points of discriminant $d_2$. This result is deduced from more general statements for…