Mathematical Sciences: Investigations in the Arithmetic of Curves and Surfaces

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This project supports research on a number of problems concerning the arithmetic geometry of curves and surfaces, as well as possible higher-dimensional generalizations. The research involves a detailed study of K3 surfaces with infinite automorphism group, centered around the canonical heights defined on such surfaces. The project will study the variation of the p-adic height on elliptic surfaces, extending recent results for the classical height, with possible applications to L-series. The research will also involve the study of various general position theorems in algebraic geometry, quantitative bounds for the number of rational points on curves of genus at least two, and the development of a new algorithm for computing canonical heights on elliptic curves which does not require a complete factorization of the discriminant. This project is in the general area of number theory and algebraic geometry. This is an area of research that focuses on the solutions sets, which are curves or surfaces in lower dimensions, of families of algebraic polynomials.