Measuring Galois Actions and Moduli Spaces

Colorado State University, Fort Collins CO

Investigators

Abstract

The field of arithmetic geometry focuses on the study of shapes (called varieties) that are defined by polynomial equations and the number of solutions to those equations. The PI will investigate properties of curves and related varieties that are of particular interest to number theorists. The project includes studying Fermat curves (connected to the celebrated proof of Fermat's Last Theorem by Andrew Wiles), curves with a large set of symmetries (for which the interplay between the arithmetic and the geometry provides a rich structure), and arithmetic invariants of more general curves. The PI also proposes two projects to train students in number theory. The first is to train undergraduates at Colorado State University in mathematics research during summer 2020. The second is to host a video-conference training seminar in arithmetic geometry, with world-wide open access. This will train graduate students in this field and build connections among researchers in this area. The PI will continue to mentor and train students and to provide service to the math community. The PI plans to study 3 topics: (1) Galois cohomology of Fermat curves: The proof of Fermat's Last Theorem determined all of the points on the Fermat curves that are defined over the rational field Q. The PI proposes to study a map in Galois cohomology that measures an obstruction for rational points and to show that this obstruction does not vanish for the Fermat curves defined over cyclotomic fields. (2) Special Shimura varieties for Jacobians and Pryms: For curves of genus g < 3, the image of the Torelli morphism is open and dense in the moduli space of abelian varieties of dimension g. The PI proposes to study families of curves with extra automorphisms for which the image of the Torelli morphism is open and dense in an associated Shimura variety; these families are called special. The PI proposes to find and analyze more special families of curves and Prym varieties. The PI also proposes a method to deduce results about non-special families, working inductively starting with the input of special families. (3) Variation of p-torsion invariants for Galois covers: An elliptic curve defined over a finite field can be ordinary or supersingular, as studied by Hasse, Deuring, and Igusa. For g > 1, there are invariants that generalize the supersingular property for a curve of genus g or an abelian variety of dimension g. For example, the Newton polygon characterizes information about the Frobenius operator on the cohomology. Measuring these invariants is important because they determine arithmetic and geometric information about the curve. It is a tantalizing open question to determine which of these invariants occur for curves. The PI proposes numerous projects about the Newton polygon and p-torsion group scheme of Jacobians of curves. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.