University of Michigan Student Arithmetic Seminar

Wednesday 3:10 - 4:00

3866 East Hall

If you would like to be added to the email list, give a talk, or hear a talk on a particular topic, email Alexander Mueller.

Date: Speaker: Title: WebVideo:
1/12/2011 Ari Shnidman The Stark Conjectures and Organizational Meeting None
1/19/2011 Wushi Goldring Unifying Themes Suggested by Belyi's Theorem None
1/26/2011 Julian Rosen p-adic Gamma functions and p-adic L-functions None
2/2/2011 Hunter Brooks p-adic L-functions, II None
2/9/2011 Ari Shnidman Heegner Points None
2/16/2011 Ben Weiss Abhyankar's Lemma and Applications None
2/23/2011 Alex Mueller Fun With The Weil Conjectures None
3/2/2011 Spring Break Cancun None
3/9/2011 Zach Scherr Introduction to Carlitz Modules None
3/16/2011 ... Arizona Winter School None
3/23/2011 Zach Scherr Introduction to Drinfeld Modules None
3/30/2011 Julian Rosen K-theory None
4/6/2011 Hunter Brooks ... None
4/13/2011 Ari Shnidman The Gross-Zagier Formula None
4/20/2011 Alex Mueller ... None


1/12/2011 The Stark Conjectures and Organizational Meeting
Ari will give a short talk on the Stark conjectures. We will also have an organizational meeting to plan the remainder of our seminars. Come prepared with ideas for talks you would like to give and ideas for talks you would like to hear.


1/19/2011 Unifying Themes Suggested by Belyi's Theorem
Belyi's Theorem, first proved in 1979, states that every curve defined over the algebraic numbers admits a map to the projective line with at most three branch points. We will begin by presenting a second, much less well-known, yet more direct proof, also due to Belyi. The rest of the talk will be concerned with generalizations of Belyi's Theorem and connections with other topics, such as (1) Moduli spaces of curves with marked points, (2) abc type problems and (time permitting) (3) The arithmetic and modularity of elliptic curves.


1/26/2011 p-adic Gamma functions and p-adic L-functions
Many classical complex analytic functions, including the Gamma function and Dirichlet L-functions, have analogs over the p-adic numbers. Like over C, values of "special" functions hold arithmetic significance. There are many different method of constructing p-adic analogs of functions, and there are obstructions that make the theory of analytic functions more complicated than over C. We will discuss several methods for constructing p-adic Gamma and L-functions functions.


1/26/2011 p-adic L-functions, II
Last week, the speaker introduced us to the family of Kubota-Leopoldt p-adic L-functions, which interpolate special values of classical Dirichlet L-functions at negative integers. Iwasawa realized that some classical results relating the p-divisibility of Bernoulli numbers with the class numbers of cyclotomic fields can be strengthened when thought of as results that p-adic L-functions govern the arithmetic of cyclotomic fields. He proved several theorems along these lines and proposed a famous "Main Conjecture," which is now a theorem due to Wiles.


2/9/2011 Heegner Points
I will explain the basic construction of Heegner points on elliptic curves defined over Q. Such curves are known to be modular, i.e. admit a morphism from a modular curve X_0(N), and the Heegner points are images of certain CM-points under this map. After defining these Heegner points, I will prove some of their (Euler system) properties. Assuming there is time, I will briefly explain how Kolyvagin used systems of Heegner points to prove special cases of the Birch and Swinnerton-Dyer conjecture.


2/16/2011 Abhyankar's Lemma and Applications
I'll discuss some of the main lemmas used in classifying equations F(X) - G(Y) = 0 which are irreducible and describe a genus 0 curve. These techniques should be applicable to many problems classifying ramification of polynomials.


2/23/2011 Fun With The Weil Conjectures
I will use the Weil conjectures to translate from Diophantine to geometric information and back. My focus will be on giving several concrete examples of varieties for which the Betti numbers (zeta functions) are known or easily computable and applying the Weil conjectures to glean information about their zeta functions (Betti numbers). I will not discuss the proof of the Weil conjectures, and I will be assuming only a very basic familiarity with algebraic topology and rational points on varieties over finite fields.


3/9/2011 Introduction to Drinfel'd Modules
This talk will give a brief introduction to Drinfel'd Modules. We will start out by recalling facts about Carlitz Modules, and then discuss how the theory of Drinfel'd Modules gives a generalization of these ideas. If time permits, I will discuss the connection of Drinfel'd Modules to the Langland Conjectures for GL_2 of a function field.


3/30/2011 K-theory
This talk will be a friendly introduction to K-theory. The first half of the talk will deal with the topological theory, which concerns classification of vector bundles on a topological space. The second half will be about the algebraic theory. The third half will be some applications to number theory.


4/13/2011 The Gross-Zagier Formula
The Gross-Zagier formula relates the height of a Heegner point on an elliptic curve E/Q to the derivative of the zeta function of E. I will give some background which will help explain what the formula is saying. I will also discuss the consequences for the Birch and Swinnerton-Dyer conjecture and for class numbers of quadratic fields.

Fall 2010 Seminar

2009-2010 Seminar

2008-2009 Seminar

2008-2009 Seminar

2007-2008 Seminar

2006-2007 Seminar

2005-2006 Seminar