Southern California Number Theory Day 2014
Schedule:
MATT BAKER (Georgia Institute of Technology)
Riemann-Roch for Graphs and Applications (10AM)
We will begin by formulating the Riemann-Roch theorem for graphs due to the speaker and Norine. We will then describe some refinements and connections with Berkovich analytic spaces. Applications include a new proof by Jensen and Payne of the Gieseker-Petri theorem in algebraic geometry, a generalization by Amini and the speaker of the Eisenbud-Harris theory of limit linear series, and a new Chabauty-Coleman style bound for the number of rational points on an algebraic curve over the rationals, proved recently by Katz and Zureick-Brown.
ELENA FUCHS (University of California, Berkeley)
Thin Monodromy Groups (11:30 AM)
In recent years, it has become interesting from a number-theoretic point of view to be able to determine whether a finitely generated subgroup of GL_n(Z) is a so-called thin group. In general, little is known as to how to approach this question. In this talk we discuss this question in the case of hypergeometric monodromy groups, which were studied in detail by Beukers and Heckman in 1989. We will convey what is known, explain some of the difficulties in answering the thinness question, and show how one can successfully answer it in many cases where the group in question acts on hyperbolic space. This work is joint with Meiri and Sarnak.
CHRISTOPHER SKINNER (Princeton University)
Elliptic curves of rank at most one and the p-part of the BSD conjecture (2:30 PM)
This talk will describe some p-adic and mod p criteria for an elliptic curve over the rationals to have algebraic and analytic rank either zero or one as well as connections with Iwasawa theory and the p-part of the BSD conjecture.
MANJUL BHARGAVA (Princeton University)
The average number of elements in the 5-Selmer groups of elliptic curves, and applications (4:00 PM)
We determine the average number of elements in the 5-Selmer groups of elliptic curves. Together with an analysis of root numbers, this yields new upper bounds on the average rank of elliptic curves. (Joint work with Arul Shankar.) Using the above together with the results of the previous talk, we also describe new *lower* bounds on the average rank of elliptic curves.
Dinner (5:30pm) - Dinner after the seminar will be held at Saladang Restaurant. The charge will be $25 per person ($15 per person for students) and does not include beverages. Please pay by cash or check (made out to Caltech) the day of the dinner.