January 19 
Tommy Occhipinti (UC Irvine) 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) TBA 
February 2 
Dino Lorenzini (University of Georgia) 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=(a_{1},…,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(a_{1},…,a_{n}) over K. When all coordinates a_{i} belong to K,
P is called a Krational 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 Krational 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) TBA 
Tuedsay 
Everett W. Howe (Center for Communications Research) 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 genus4 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 genus4 curve whose number of points is quite close to the Weil upper bound in time O(q^{3/4 + ε} ). 
Tuesday 
John Voight (University of Vermont) Triangle groups, the symmetry groups of tessellations of the
hyperbolic plane by triangles, have been studied since early work of
Hecke and of Kleinthe 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 
Mirela Çiperiani (The University of Texas at Austin) 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 TateShafarevich group are
infinitely divisible when considered
as elements of the WeilChatelet group. The second question
concerns the intersection of
the TateShafarevich group with the maximal divisible subgroup
of the WeilChatelet group.
This is joint work with Jakob Stix.

March 1 
Francesc Fité (UPC Barcelona) The (general) SatoTate 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 SatoTate
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 Galoismodule structure of the Ralgebra 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) TBA 