Number Theory Seminar

January 19

Tommy Occhipinti (UC Irvine)
Some Mordell-Weil Groups of Large Rank

The existence of elliptic curves of large rank over number fields is an open question, but it has been known for decades that there exist elliptic curves of arbitrarily large rank over global function fields. In this talk we will discuss some results of Ulmer that showcase the ubiquity of large ranks over function fields, as well as some newer work in the area.

January 26

Ruochuan Liu (University of Michigan)
Triangularities of refined families

TBA

February 2

Dino Lorenzini (University of Georgia)
The index of an algebraic variety

Let K be a field. Suppose that the algebraic variety is given be the set of common solutions to a system of polynomials in n variables with coefficients in K. Given a solution P=(a₁,…,a_n) of this system with coordinates in the algebraic closure of K, we associate to it an integer called the degree of P, and defined to be the degree of the extension K(a₁,…,a_n) over K. When all coordinates a_i belong to K, P is called a K-rational point, and its degree is 1. The index of the variety is the greatest common divisor of all possible degrees of points on P. It is clear that if there exists a K-rational point on the variety, then the index equals 1. The converse is not true in general. We shall discuss in this talk various properties of the index, including how to compute it when K is a complete local field using data pertaining only to a reduction of the variety. This is joint work with O. Gabber and Q. Liu.

February 9

Ron Evans (UCSD)
Some character sum

TBA

Tuedsay
February 14
AP&M 5402
2pm

Everett W. Howe (Center for Communications Research)
Producing genus-4 curves with many points

I will talk about a computational problem inspired by the desire to improve the tables of curves over finite fields with many points (http://www.manypoints.org). Namely, if q is a large prime power, how does one go about producing a genus-4 curve over F_q with many points? I will discuss the background to this problem and give a number of algorithms, one of which one expects (heuristically!) to produce a genus-4 curve whose number of points is quite close to the Weil upper bound in time O(q^{3/4 + ε} ).

Tuesday
February 21
AP&M 6402
2pm

John Voight (University of Vermont)
Arithmetic aspects of triangle groups

Triangle groups, the symmetry groups of tessellations of the hyperbolic plane by triangles, have been studied since early work of Hecke and of Klein--the most famous triangle group being SL(2,Z). We present a construction of congruence subgroups of triangle groups (joint with Pete L. Clark) that gives rise to curves analogous to the modular curves, and provide some applications to arithmetic. We conclude with some computations that highlight the interesting features of these curves.

Wednesday
February 22
AP&M 6402
4:30pm

Mirela Çiperiani (The University of Texas at Austin)
The divisibility of the Tate-Shafarevich group of an elliptic curve in the Weil-Chatelet group

In this talk I will report on progress on the following two questions, the first posed by Cassels in 1961 and the second considered by Bashmakov in 1974. The first question is whether the elements of the Tate-Shafarevich group are infinitely divisible when considered as elements of the Weil-Chatelet group. The second question concerns the intersection of the Tate-Shafarevich group with the maximal divisible subgroup of the Weil-Chatelet group. This is joint work with Jakob Stix.

March 1

Francesc Fité (UPC Barcelona)
Sato-Tate groups and Galois endomorphism modules in genus 2

The (general) Sato-Tate Conjecture for an abelian variety A of dimension g defined over a number field k predicts the existence of a compact subgroup ST(A) of the unitary symplectic group USp(2g) that is supposed to govern the limiting distribution of the normalized Euler factors of A at the primes where it has good reduction. For the case g=1, there are 3 possibilities for ST(A) (only 2 of which occur for k=Q). In this talk, I will give a precise statement of the Sato-Tate Conjecture for the case of abelian surfaces, by showing that if g=2, then ST(A) is limited to a list of 52 possibilites, exactly 34 of which can occur if k=Q. Moreover, I will provide a characterization of ST(A) in terms of the Galois-module structure of the R-algebra of endomorphisms of A defined over a Galois closure of k. This is a joint work with K. S. Kedlaya, V. Rotger, and A. V. Sutherland

March 8

NO SEMINAR

March 15

Harold Stark (UCSD)
Poincaré series and modular forms

TBA