arXiv subjects
Publications and source records attributed to Patricio Pérez-Piña.
We study a distribution problem over global function fields. More precisely, we describe the asymptotic distribution of rank $2$ CM Drinfeld modules among the irreducible components of the analytic reduction of the Drinfeld modular curve. Our approach relies on the properties of the building map and the spectral decomposition of the adjacency operator on a quotient of the Bruhat-Tits tree.
We prove an equidistribution result about Hecke orbits on the Picard group of Shimura curves coming from definite quaternion algebras over function fields. In particular, we show the equidistribution of Hecke orbits of supersingular Drinfeld modules of rank 2. Our approach is via the automorphic method, using bounds for coefficients of cuspidal automorphic forms of Drinfeld type as the main tool.
We prove the equidistribution of some cycles of S-arithmetic nature that are related to RM points and Stark-Heegner points. We also prove the equidistribution of Picard orbits of ATR cycles as defined by Darmon, Rotger and Zhao.
We propose a p-adic version of Duke's Theorem on the equidistribution of closed geodesics on modular curves. Our approach concerns quadratic fields split at p as well as a p-adic covering of the modular curve. We also prove an equidistribution result of Heegner points in the p-adic space attached to Shimura curves.