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

Abstract

We generalize to Hilbert modular varieties of arbitrary dimension the work of W. Duke [16] on the equidistribution of Heegner points and of primitive positively oriented closed geodesics in the Poincaré 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.