November 12, 2010
1360 East Hall, University of Michigan
9:15–9:30 Opening ceremonies
9:30–10:30 C. Douglas Haessig (University of Rochester)
On the evolution of the Chevalley–Warning theorem
We will begin by recalling a result of Chevalley and Warning which was conjectured by Emil Artin in 1935. This conjecture asks whether we may be guaranteed that a multivariable polynomial has a zero over a finite field under a simple condition related to the polynomial's degree and the number of variables. A reinterpretation of this result may be given in terms of estimating the p-adic absolute value of an exponential sum, a viewpoint that is at the heart of a lot of current research. We will discuss a few current research questions which evolved from this topic.
10:45–11:45 Aaron Levin (Michigan State University)
Integral points on affine varieties
I will introduce and discuss several topics and open problems all
revolving around the theme of integral points on affine varieties
which have many components "at infinity". In particular, I will discuss
recent work related to the Corvaja-Zannier Subspace theorem approach to
integral points, the effective method of Runge, and applications of these
techniques and results to specific problems and varieties of interest.
1:45–2:45 Michael Larsen (Indiana University)
Some applications of group theory to number theory
I will describe some results, old and new, in number theory which bear
on the problem of characterizing images of Galois representations.
3:00–4:00 Alina Carmen Cojocaru (University of Illinois at Chicago)
One parameter families of elliptic curves with maximal Galois
representations
Let E be an elliptic curve over Q and let Q(E[n]) be its n-th division
field. In 1972, Serre showed that if E is without complex multiplication,
then the Galois group of Q(E[n])/Q is as large as possible, that is,
GL_2(Z/nZ), for all integers n coprime to a constant integer c(E,Q)
depending (at most) on E/Q. Serre also showed that the best one can hope
for is to have |GL_2(Z/nZ):Gal(Q(E[n])/Q)| at most 2 for all nonzero
integers n. I will discuss the frequency of this optimal situation in a
one-parameter family of elliptic curves over Q. This is joint work with
David Grant and Nathan Jones.
4:15–5:15 Roman Holowinsky (The Ohio State University)
Rankin-Selberg L-functions and subconvexity
We'll discuss the roll and analysis of Rankin-Selberg L-functions in the
recent proof of mass equidistribution of Hecke eigenforms. This will
motivate the consideration of various subconvexity problems. In
particular, we will focus on the level aspect subconvexity problem for the
Rankin-Selberg convolution of two varying Hecke eigenforms. We shall
present a method of obtaining subconvexity bounds for Rankin-Selberg
convolutions when both of the involved forms contribute distinctly to the
size of the conductor and are sufficiently distinguishable. Much of the
talk will be based on joint work with Ritabrata Munshi.
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