Hyperbolic distribution problems on Siegel 3-folds and Hilbert modular varieties

TL;DR

This paper extends the study of equidistribution and Weyl sum limits from classical modular forms to higher-dimensional Hilbert modular varieties and Siegel 3-folds, revealing new distribution properties and vanishing results.

Contribution

It generalizes equidistribution results of Heegner points and geodesics to higher dimensions and proves vanishing of certain Weyl sums for Siegel 3-folds.

Findings

Equidistribution of special points on Hilbert modular varieties.

Vanishing of cuspidal Weyl sums in Siegel 3-folds.

Extension of classical results to higher-dimensional modular varieties.

Abstract

We generalize to Hilbert modular varieties of arbitrary dimension the work of W. Duke (Inventiones 1988) on the equidistribution of Heegner points and of primitive positively oriented closed geodesics in the Poincare upper half plane, subject to certain subconvexity results. We also prove vanishing results for limits of cuspidal Weyl sums associated with analogous problems for the Siegel upper half space of degree 2. In particular, these Weyl sums are associated with families of Humbert surfaces in Siegel 3-folds and of modular curves in these Humbert surfaces.