Equidistribution of Algebraic Points and Arithmetic Invariants of Modular Curves

Princeton University, Princeton NJ

Investigators

Abstract

The investigator main topic is number theory. He is using Arakelov theory with a special eye on modular forms and automorphic forms. He is studying equidistribution phenomenas for sequences of algebraic points on varieties over number fields. His main result is the discreteness of the set of algebraic points of a curve of genus greater than one embedded in his Jacobian for the Neron-Tate topology. He tries to have results in the direction of the Andre-Oort conjecture concerning the possible Zariski closures of a set of points with complex multiplication in the moduli space of abelian varieties. Equidistribution of Hecke (obtained by the author and Clozel) is a step towards equidistribution of Galois orbits of points with complex multiplication. In an other direction, the investigator studies the arithmetic invariants of modular curve from the Arakelov view-point. Diophantine geometry is one of the most aesthetic subject of mathematics. Even if the questions are phrased in a very elementary way: Find all the rational or algebraic solutions of a system of polynomial equations; the solution to this type of problems requires in general the most advanced techniques of mathematics. This work explains that the set of algebraic points of a curve (which is infinite) is not very big: We can think about it as a discrete set in a natural arithmetic set.