A lower bound limiting solutions in the hyperbolic case of the generalized Fermat equation

Sato-Tate distributions of twists of the Fermat and the Klein quartics

We determine the limiting distribution of the normalized Euler factors of an abelian threefold A defined over a number field k when A is geometrically isogenous to the cube of a CM elliptic curve defined over k. As an application, we classify the Sato-Tate distributions of the Jacobians of twists of the Fermat and Klein quartics, obtaining 54 and 23, respectively, and 60 in total. We encounter a new phenomenon not visible in dimensions 1 or 2: the limiting distribution of the normalized Euler factors is not determined by the limiting distributions of their coefficients.

Almost periodic functions and hyperbolic counting

In this paper we prove the existence of asymptotic moments, and an estimate on the tails of the limiting distribution, for a specific class of almost periodic functions. Then we introduce the hyperbolic circle problem, proving an estimate on the asymptotic variance of the remainder that improves a result of Chamizo. Applying the results of the first part we prove the existence of limiting distribution and asymptotic moments for three functions that are integrated versions of the remainder, and were considered originally (with due adaptations to our settings) by Wolfe, Phillips and Rudnick, and…

几何有限双曲子群的轨道方向

Directions in Orbits of Geometrically Finite Hyperbolic Subgroups

本文证明了几何有限双曲子群轨道在边界上的极限精细统计定理，刻画高阶空间统计，并应用于球堆积，推导出矩收敛、两点相关函数及二维间隙分布公式。