Technical description: This is a project in the arithmetic algebraic geometry of diophantine problems. The PI is continuing his work which was successfully used by Faltings in the solution of the Mordell Conjecture and by E. Ullmo, S. Zhang and the PI in the solution of the Bogomolov Conjecture. The use of the modern theory of heights has been very effective. The PI views the Equidistribution Theorem as the solution to the problem of finding lower bounds for solutions of algebraic equations. The question of finding effective upper bounds for solutions of algebraic equations leads to many conjectures. Some are very well known and may be unattainable (the abc conjecture, or the discriminant conjecture for elliptic curves) but the PI has always believed that attacking difficult problems is the secret of success in doing high-level mathematics. The project will concentrate on:
a) The study of the degree of Belyi maps (these are coverings of the Riemann sphere ramified in only 3 points and they characterize curves defined over the field of algebraic numbers). b) The study of the Zariski closure of the non-zero p-division points (for p big enough) in an abelian variety as a finite scheme. c) The consequences for the Tate-Shafarevich group of recent results of the PI and J. Pesenti on the discriminant inequality for potential good reduction. d) Dynamical Systems (first on the Sphere then on towers of Shimura varieties): The canonical height associated to these objects should lead to equidistribution statements, for example for CM points (cf the work of Duke).
Non technical description: The PI and his collaborators are studying a subject first investigated by Diophantus in ancient Greece: find the solutions in integers of algebraic equations. The modern attack uses algebraic geometry, analysis, and geometry. Many problems in the natural world (asking: How many times? How to decipher?) require a solution in integers. This no doubt explains why number theory, like physics, has been a constant motivation for the development of mathematics.
