Curves, covers, and cohomology

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An underlying theme of arithmetic geometry is the existence of solutions to polynomial equations whose coefficients lie in fields other than the complex numbers. Historically, equations having extra symmetry have been a focus of inquiry since the automorphisms of the corresponding geometric object provide extra structure which helps illuminate the set of solutions. The PI's research is about functions on curves defined by polynomial equations with coefficients in a finite field. The goal is to analyze invariants of algebraic structures attached to these curves, to determine how these invariants vary across parameter spaces for the curves, and to investigate which exceptional structures occur. As a broader impact of the proposal, the PI will collaborate with the Fort Collins Museum of Discovery to build activity kits about cryptography for children. The PI is also involved with other initiatives that have broader impacts: the PI is a leader of the WIN network, whose goal is to vitalize the research careers of women in number theory via new research collaborations; and the PI was a co-organizer of the Arizona Winter School from 2010-2015. Curves and abelian varieties exhibit new phenomena in positive characteristic p. These phenomena typically arise due to the underlying presence of morphisms of degree p. They lead to important cohomological invariants, such as the p-rank, Newton polygon, and Ekedahl-Oort type. The goal of the PI's proposal is to prove: (non)-existence results about abelian varieties and curves with given invariants; and structural results about the stratifications induced by the invariants on various moduli spaces. In particular, the PI will investigate supersingular abelian varieties and Jacobians; the interplay between automorphisms and p-torsion invariants; formulae, akin to the Deuring-Shafarevich formula, for the variation in more refined invariants of the Ekedahl-Oort type in terms of ramification data; and rational points of curves over non-algebraically closed fields. Key techniques include: intersection of divisors on moduli spaces; degeneration to singular curves of compact type; results about monodromy of families of curves; cohomology calculations (de Rham or of resolutions); and actions of Frobenius and Vershiebung.