Midwest Number Theory Day 2009

November 6, 2009

Image Courtesy of Joy Wang

The math department at the University of Wisconsin-Madison will host Midwest Number Theory Day on Friday, November 6, 2009. This conference is an
opportunity for number theorists from the Midwest and elsewhere to hear about recent research in number theory and to meet local colleagues. We especially encourage people new to the region to come!
This year's speakers are:

- 9:45 AM: Michael Zieve (Michigan)
*Intersections of polynomial orbits, and a dynamical Mordell-Lang conjecture*I will prove this theorem:

Let*f*and*g*be nonlinear polynomials (in one variable) with complex coefficients. If*a*and*b*are complex numbers for which the orbits {*a, f(a), f(f(a)), f(f(f(a))), ...*} and {*b, g(b), g(g(b)), g(g(g(b))), ...*} have infinite intersection, then*f*and*g*have a common iterate, i.e., some*f(f(f(f(...(f(x))...)))) = g(g(g(...(g(x))...)))*.The proof involves Siegel's theorem on integral points on curves, a result of Bilu and Tichy on Diophantine equations with infinitely many solutions, specialization arguments, and other ingredients.

I will also discuss variants and generalizations of this result, including connections to Lang's generalization of the Mordell conjecture, according to which the intersection of a subvariety and a finitely-generated subgroup of a "nice" algebraic group (semiabelian variety) consists of finitely many cosets of subgroups.

- 11:00 AM: Joseph Rabinoff (Harvard)
*Canonical subgroups for p-divisible groups*Katz and Lubin proved in the 1970's that if E is an elliptic curve over a p-adic field K and the reduction type of E is "not too supersingular", then there is a canonical line inside of its p-torsion subgroup E[p]. This canonical subgroup has seen some important applications in the study of p-adic modular forms. I will present an analogous existence result in higher dimensions which is valid for all p-divisible groups, independent of its height and dimension. I'll also try to indicate the roles played by Zink's displays and tropical geometry in the construction.

- 1:30 PM: Sug-Woo Shin (Chicago)
*Multiplicity and density of automorphic representations*I will explain how to use the trace formula to obtain information on the multiplicity and (Zariski) density of automorphic representations satisfying prescribed local conditions.

- 2:45 PM: Kirsten Wickelgren (AIM and Harvard)
*Etale pi_1 obstructions to rational points*Grothendieck's section conjecture says that for a hyperbolic curve over a number field, rational points are in bijection with sections of etale pi_1 of the structure map. We use cohomological obstructions of Jordan Ellenberg to study such sections. We will relate Ellenberg's obstructions to Massey products, and explicitly compute versions of the first and second for P^1- {0,1, infty}. Over R, we show the first obstruction alone determines the connected components of real points of the curve from those of the Jacobian.

On Saturday and Sunday, November 7-8, 2009, we will host the Sixth Midwest Number Theory Conference for Graduate Students. We encourage graduate students who are attending the latter to come a day early for Midwest Number Theory Day.

If you plan to attend MNTD, please register here. Registration is requested only for determining the quantity of coffee and snacks. We will not turn away unregistered participants, nor will we deny them coffee and snacks.

To come. We'll start around 9:30 and be done by 4 (so as not to conflict with the colloquium by Diaconis.) .

Madison is in south-central Wisconsin, and is situated along highways I-39, I-90 and I-94. It is about 79 miles west of Milwaukee, 148 miles northwest of Chicago, and 270 miles southeast of Minneapolis. Northwest, United, American and Continental Airlines service Madison's Dane County Regional Airport. In addition, Madison is served by Greyhound. Also, Badger Coaches offers service from the Milwaukee airport to Madison and Van Galder Bus Company offers service from Chicago's O'Hare airport to Madison.

For information on dining, see FoodSpot.com, an extensive Wisconsin restaurant guide.

The Midwest Number Theory Day is being organized by Jordan Ellenberg, Ken Ono, Nigel Boston, and Tonghai Yang.

Last Modified Wednesday, September 12, 2009.