University of Michigan Student Arithmetic Seminar

Date:	Speaker:	Title:
9/13/06		Organizational Meeting
9/20/06	Craig Spencer	Counting Rational and Integral Points with the Circle Method
9/27/06	Eugene Eisenstein	Applications of the Chebotarev Density Theorem
10/4/06	Johnson Jia	A Glance at Iwasawa Theory
10/11/06	Johnson Jia	Converse to Herbrand's Theorem
10/18/06	Craig Spencer	A Brief Introduction to Additive Combinatorics
10/25/06	Leo Goldmakher	Pretentious Functions
11/1/06	Nick Ramsey	The Eigencurve
11/8/06	Wansu Kim	Faltings's Proof of the Mordell Conjecture: Part 1
11/15/06	Wansu Kim	Faltings's Proof of the Mordell Conjecture: Part 2
11/29/06	Mahesh Agarwal	Automorphic forms on GL(2)
12/6/06	Matthew Smith	Roth's Theorem
12/13/06	Rizwanur Khan	The Large Gaps Between Primes Problem
1/24/07	Jonathan Bober	Primality Testing I
1/31/07	Hester Graves	Euclidean Ideal Classes
2/7/07	Craig Spencer	Forms in Many Variables
2/14/07	Greg McNulty	Low Lying Zeros of L-Functions
3/7/07	Craig Spencer	Zeros of p-adic Forms
3/21/07	Johnson Jia	How to construct Galois representations from modular forms
3/28/07	Johnson Jia	CM Theory of Elliptic Curves
4/11/07	Hester Graves	S-Ray Class Fields

10/4/06

Johnson Jia

A Glance at Iwasawa Theory

I will discuss some general notions, present some known results, and survey some conjectures in Iwasawa theory, following Greenberg's article "Iwasawa Theory for P-adic Representations".

10/25/06

Leo Goldmakher

Pretentious Functions

In a 2005 paper posted on the arXiv, Granville and Soundararajan introduced a notion of distance between arithmetic functions. This gave rise to the theory of pretentious functions - the study of the behavior of a complicated arithmetic function which is 'close' to a simpler function. Using the theory, the authors were able to improve the Polya-Vinogradov inequality (the first fundamental improvement in 87 years). In a subsequent paper, the authors showed how one can view many other classical results of multiplicative number theory through the lens of pretentious functions. In this talk I'll sketch the results and methods of these papers.

11/8/06	Wansu Kim	Faltings's Proof of the Mordell Conjecture: Part 1
		Reducing to Tate's Conjecture Over Number Fields

One of the basic motivations for number theory and arithmetic geometry (at least at the philosophical level) is to find the rational points X(K) of a projective variety X over a number field K.

Assume, furthermore, that X is a smooth algebraic curve over K. It can be seen that rational points of genus 0 or 1 curves behave quite differently from the rest. Faltings's giant result ('83) says that any smooth algebraic curves of genus at least two can have at most finitely many rational points (archaic term: Mordell conjecture).

The proof consists roughly of 4 reduction steps. At each step, the problem transforms to a whole new problem. It is completely impossible to meaningfully cover any technicality so I'll (try to) mainly concentrate on the last step: Tate's conjecture on the endomorphism ring of an abelian variety over a number field.

At the first talk, I will reduce the Mordell conjecture to Tate's conjecture, and I will give a proof of Tate's conjecture at the second talk.

These talks are based on Darmon's lectures at the Clay Math 2006 summer school.

11/15/06	Wansu Kim	Faltings's Proof of the Mordell Conjecture: Part 2
		Tate's Conjecture

It was shown, in the previous talk, that the Mordell conjecture can be reduced to showing that the cardinality of the isogeny classes of abelian varieties with good reduction outside fixed finite set of places is finite.

To show this, we will associate to each isogeny class [A] an "l-adic representation" V_l(A) of Gal(Kbar/K). Then the Mordell conjecture will be proved by the following steps:

1. Find representation-theoretic properties that are satisfied by V_l(A). (Especially, show they are semi-stable Gal(Kbar/K)-modules!!)

2. Show that there are only finitely many representations (up to equivalence) that have the properties identified in the previous step.

3. Show that associating V_l(A) to the isogeny class [A] is finite-to-one (even one-to-one): Tate's conjecture.

The key step is to show that V_l(A) is semi-simple as a Gal(Kbar/K)-module, and this crucially uses Faltings's theorem that in each isogeny class with good reduction outside S, there are only finitely many isomorphism classes with good reduction outside S. This is an interesting interaction between geometry and galois representations.

Part 1 and Part 2 can be thought of as two independent talks. Those who did not attend the first part will be able to follow the second part.

1/24/07

Jonathan Bober

Primality Testing I

The title is not meant to imply that there will be a second talk. It is merely meant to suggest that maybe there could be.

I'll say something about the Miller-Rabin primality test, first as a probabilistic primality test (this part will be "elementary", at least from a mathematical viewpoint, requiring just elementary number theory) then, as time allows, as a GRH based deterministic primality test. This analysis will use ideas similar to the proof of the prime number theorem.

2/14/07

Greg McNulty

Low Lying Zeros of L-Functions

A current area of active research is the connection between random matrix theory and analytic number theory. The general idea is that the statistical behavior of the non-trivial zeros of L-functions can be modeled by that of eigenvalues of random matricies. We will give the necessary background and some motivation for a few of the results in the paper "Linear Statistics of Low Lying Zeros of L-Functions" by C.P. Hughes and Z. Rudnick.

3/21/07

Johnson Jia

How to construct Galois representations from modular forms

I will discuss the Eichler-Shimura correspondence. If time allows, I will also babble about Deligne's construction using etale cohomology.

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