Oregon Number Theory Days

Oregon Number Theory Days is a triannual number theory seminar rotating between Oregon State University, Portland State University, and the University of Oregon. Our May speakers include Rachel Pries and Özlem Ejder.

Rachel Pries,

Lecture I: Newton polygons of cyclic covers of the projective line.

An elliptic curve in characteristic p can be ordinary or supersingular. For a curve of higher genus, there are finer invariants on the Jacobian called the Newton polygon and the Dieudonne module. They give information about the Frobenius morphism. Studying cyclic covers of the projective line, we verify many new examples of Newton polygons and Dieudonne modules which occur for Jacobians of smooth curves. For the proof, we study the Newton polygon and Ekedahl-Oort stratification of PEL-type Shimura varieties and compute slopes of Frobenius on the crystalline cohomology. As an application, we give new examples of supersingular curves of genus 5-11. This is joint work with Li, Mantovan, and Tang.

Lecture II: Generalizing a Galois action on the homology of the Fermat curve.

The Fermat curves play an important role in arithmetic geometry, not only because of Fermat's Last Theorem, but also because they characterize abelian covers of the projective line branched at 3 points. Anderson studied the action of the absolute Galois group of Q on the homology of the Fermat curve. In earlier work, we gave explicit formulas for this action. In this project, we determine the Galois action on the second quotient of the fundamental group of the Fermat curve. The proof involves some fun combinatorics, commutator identities, a cup product in cohomology. This is joint work with Davis and Wickelgren.

Özlem Ejder:

Sporadic points on X_1(n).

The points on the modular curve X_1(n) (roughly) classifies the pairs (E,P) up to isomorphism where E is an elliptic curve and P is a point of order n on E. We call a closed point x on X_1(n) sporadic if there are only finitely many closed points of degree at most deg(x); hence classifying sporadic points on X_1(n) is closely related to determining the torsion subgroups of elliptic curves over a degree d field. When d=1 or 2, Mazur and Kamienny's work show that there are no sporadic points of degree d on X_1(n). In this talk, I will discuss that the sporadic j-invariants of bounded degree is finite. This is joint with A. Bourdon, Y. Liu, F. Odumudu and B. Viray.

Program:

10:30-11:30am: Rachel Pries (Colorado State University), Lecture I

11:50-12:20pm: Özlem Ejder (Colorado State University)

12:20-2:30pm: Lunch

2:30-3:30pm: Rachel Pries (Colorado State University), Lecture II