The PI proposes to study stratifications and filtrations associated with curves defined over an algebraically closed field k of characteristic p. The p-rank and a-number are invariants of the p-torsion group scheme of the Jacobian of a k-curve; these invariants induce stratifications of moduli spaces of curves. The first part of the proposal is to analyze the geometry of such strata, including questions about irreducibility, dimension, and boundaries. The second part of the proposal is about Galois covers of the affine line over k and the ramification filtrations of the wild inertia groups. These filtrations play a key role in answering questions about lifting and deformation of wildly ramified covers of curves. These projects all require geometric techniques (degeneration to boundaries of moduli spaces, deformation/lifing/formal patching). Yet all parts of the proposal are about the arithmetic of p-group covers of k-curves and involve arithmetic objects (group schemes, action of Frobenius, Galois groups, norm groups, formal groups). In addition, the projects yield concrete applications about the existence of curves over a finite field of characteristic p with given automorphism group or p-torsion invariants.
Galois theory arose classically as a means of understanding symmetries of equations and of classifying subfields of the complex numbers. Number theory arose classically as a way of finding integer solutions to polynomial equations. The PI's research is about functions on curves defined by polynomial equations with coefficients in a finite field. This topics has some applications to cryptosystems and data-transfer codes. The PI will lead several summer research workshops for graduate students about curves and Galois covers. The goal is to give the students experience with collaborative research and to increase knowledge about problems relevant to this proposal. The PI is also involved with other initiatives that have broader impacts including: co-organization of WIN (women in numbers) initiatives to increase the research training of women in number theory; co-organization of the Arizona Winter School; and a new liaison between the math departments of CSU and the Universidad de Costa Rica.
