Investigations in Arithmetic Geometry and Arithmetic Dynamics

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The PI will investigate two problems in arithmetic geometry and several problems in arithmetic dynamics. In the first arithmetic geometry project, the PI will prove new cases of a fundamental conjecture of Paul Vojta describing the size of rational points on certain blowup varieties. The second project in arithmetic geometry is related to Heegner's construction of special points on modular elliptic curves associated to imaginary quadratic fields and to a classical result of Deuring that explains how to lift "mod p" points to Heegner points. Recently Darmon showed how one might construct Heegner-type points associated instead to real quadratic fields. The PI will investigate the possibility of a Deuring-type lifting result for these Darmon-Heegner points. The PI's other projects are in the area of arithmetic dynamics, which is a new field in which one studies algebraic, number theoretic, and p-adic properties of discrete dynamical systems associated to iteration of polynomial or rational functions. The PI plans to investigate four problems in this area: (1) Transformation properties of height functions under regular affine automorphisms. (2) Arithmetic properties of Misiurewicz points in the Mandelbot set. (3) p-adic dynamics and nonarchimedean Green functions on projective space and other projective varieties. (4) Classification of Latths maps in finite characteristic.The solution of polynomial equations using integers or rational numbers has been studied since antiquity. A fundamental conjecture of Paul Vojta from the 1980's describes the size of such solutions in terms of geometry. The PI plans to prove Vojta's conjecture for new classes of equations. Elliptic curves, which are defined by a particular type of polynomial equation, have been extensively studied during the past 80 years. In the 1930's, Deuring described how to lift certain "mod p" solutions to actual solutions. These lifted solutions are called Heegner points. Recently Darmon constructed a new type of Heegner point. The PI will study ways to lift "mod p"solutions to these new Darmon-Heegner points. The PI's other area of research is in the field of arithmetic dynamics. Dynamical systems is the study of what happens to different starting points when a function is iterated, i.e., take a function f(x) and a starting point b and look at the sequence f(b), f(f(b)), f(f(f(b))),... . The new field of arithmetic dynamics asks for number theoretic properties of these iterated values. The PI will investigate problems in arithmetic dynamics, including studying the complexity of the iterated values, number theoretic properties of certain special points in the famous Mandelbrot set, and problems involving iteration of certain functions defined using elliptic curves.