测度叶状结构与希尔伯特第12问题

对“测度叶状结构与Hilbert第12问题”的补充

Addendum to "Measured foliations and Hilbert 12th problem"

本文基于前期研究结果，研究了实二次数域阿贝尔扩张的数值例子，是对“测度叶状结构与Hilbert第12问题”的补充研究。

Cusp shapes of Hilbert-Blumenthal surfaces

We introduce a new fundamental domain for the cusp stabilizer of a Hilbert modular group over a real quadratic field K=Q(sqrt n). This is constructed as the union of Dirichlet domains for the maximal unipotent group, over the leaves in a foliation of the biplane. The region is the Cartesian product of the positive reals with a 3-dimensional tower formed by deformations of lattices in the ring of integers of K, and makes explicit the cusp cross section's Sol 3-manifold structure and Anosov diffeomorphism. We include computer generated images and data illustrating various examples.

Hilbert spaces and low-lying zeros of L-functions

Generalizing previous work of Iwaniec, Luo, and Sarnak (2000), we use information from one-level density theorems to estimate the proportion of non-vanishing of $L$-functions in a family at a low-lying height on the critical line (measured by the analytic conductor). To solve the Fourier optimization problems that arise, we provide a unified framework based on the theory of reproducing kernel Hilbert spaces of entire functions (there is one such space associated to each symmetry type). Explicit expressions for the reproducing kernels are given. We also revisit the problem of estimating the hei…