Topics in Arithmetic Geometry

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The project will explore various topics within algebraic number theory and algebraic geometry. The study of integer and rational number solutions to polynomial equations has been an ongoing area of research since the time of the ancient Greeks, and the project will develop a theory of "number field fragments" that may yield a new approach to answering some such questions, in particular those related to Fermat's last theorem and its generalizations. In algebraic geometry, the project aims, among other things, to show that one of the simplest parameter spaces already has bad singularities, and in fact contains every singularity imaginable. Many of the specific research topics contained in the project are at a level accessible to beginning graduate students, and one of the goals is to involve students in the research.More specifically, given a Galois extension K of Q with Galois group G, the project will extend geometry-of-numbers methods to study the "fragments" of K cut out by an idempotent of the group ring QG, in the hope of developing a new way to prove the nonexistence of certain Galois representations. It will also study correspondences from a curve to itself that are unramified on one side, from the point of view of arithmetic dynamics, since such setups yield new examples of almost-everywhere-unramified arboreal representations. In algebraic geometry proper, the project will study infinitesimal neighborhoods of the vertex of a cone to try to prove that the Hilbert scheme of points satisfies Murphy's law. Finally, the project will explore whether there is an idele class group variant of the Cohen-Lenstra heuristics.