Mathematical Sciences: Height Functions and Arithmetic Geometry

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This research concerns questions in the general area of arithmetic geometry which involve or are applications of the theory of height functions. The research will involve proofs of formulas for the number of points having height less than a given bound on certain varieties, a precise formula for the variation of the Neron-Tate height of points in an algebraic family of elliptic curves, and a lower bound for the Neron-Tate height of points on certain abelian varieties. This research is in the general area of algebraic arithmetic geometry. This is a subject which concerns the integral points or integral coordinates which lie on algebraic surfaces. In particular, this research involves counting the number of integral points on curves of genus two.