Locally Divergent Orbits on Hilbert Modular Spaces

Entropy and Escape of Mass for Hilbert Modular Spaces

We study the relation between metric entropy and escape of mass for the Hilbert modular spaces with the action of a diagonal element.

Primitive rational points on expanding horospheres in Hilbert modular surfaces

In recent work by Einsiedler, Mozes, Shah and Shapira the limiting distributions of primitive rational points on expanding horospheres was examined in arbitrary dimension, and a suspended version of this result was announced. Motivated by this, we prove an analog to the announcement for primitive rational points in Hilbert modular surfaces via effective mixing.

Periodic Points in Genus Two: Holomorphic Sections over Hilbert Modular Varieties, Teichmuller Dynamics, and Billiards

We show that all GL(2, R)-equivariant point markings over orbit closures of primitive genus two translation surfaces arise from marking pairs of points exchanged by the hyperelliptic involution, Weierstrass points, or the golden points in the golden eigenform locus. As corollaries, we classify the holomorphically varying families of points over orbifold covers of genus two Hilbert modular surfaces, solve the finite blocking problem on genus two translation surfaces, and show that there is at most one nonarithmetic rank two orbit closure in the minimal stratum in genus four.

发散测地线、歧义闭测地线与二元加性除数问题

Divergent geodesics, ambiguous closed geodesics and the binary additive divisor problem

本文研究负曲率局部对称空间中发散测地线间的公共垂线计数，给出非纯指数增长的渐近公式，并应用于模轨道流形中歧义测地线的计数，恢复Sarnak结果并证实推广Motohashi猜想。