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 Ben Weiss.

OA
Date: Speaker: Title: WebVideo:
9/9/2009 Organizational Meeting None
9/16/09 Ben Weiss A Classical Introduction to the Gamma Function None
9/23/09 Ben Weiss Adelic Fourier Analysis None
9/30/09 Thomas Wright (Johns Hopkins) Diophantine Questions and the Adeles None
10/7/09 Hunter Brooks Iwasawa Theory of Elliptic Curves with Complex Multiplication None
10/14/09 Johnson Jia The Langlands Program and L-functions None
10/21/09 Alex Mueller The Riemann Hypothesis for Function Fields I None
10/28/09 Alex Mueller The Riemann Hypothesis for Function Fields II None
11/4/09 Not Meeting In order to attend Kiran Kedlaya's Talk None
11/11/09 Ari Shnidman Counting Cubic Fields of Bounded Discriminant None
11/18/09 Julian Rosen Gevrey Asymptotics None
11/25/09 Mr. Turkey Gobble Gobble None
12/2/09 Johnson Jia A Brief Introduction to Eigenvarieties None
12/9/09 Hunter Brooks Constructing Fontaine's Rings of p-adic Periods None
1/6/10 Organizational Meeting None
1/13/10 Benjamin Weiss A Quadratic Analogue of Artin's Conjecture YouTube in 4 parts
1/20/10 Ricardo Portilla Parametrizing Nilpotent Orbits in p-adic Symmetric Spaces YouTube in 5 parts
1/27/10 Mike Zieve The number of values of a polynomial over a finite field None
2/3/10 Hunter Brooks Why Is the Tate-Shafarevich Group Like an Ideal Class Group? None
2/10/10 Johnson Jia Modular Forms and Galois Representations (Part I) None
2/17/10 Johnson Jia Modular Forms and Galois Representations (Part II) None
2/24/10 Zachary Scherr An Elementary Proof of Kronecker-Weber YouTube in 4 parts
3/3/10 Spring Break Not Meeting
3/10/10 Alex Mueller TBA YouTube in 5 parts
3/17/10 Not Meeting This Week To Attend the Arizona Winter School
3/24/10 Hunter Brooks Deuring's Theorem
3/31/10 Ari Shnidman Mass Formulae for p-adic Extensions
4/7/10 Julian Rosen TBA
4/14/10 Dan Hernandez TBA


9/9/09 Organizational Meeting
I hope to arrange speakers for the rest of the semester. We have speakers arranged through the end of September, so that people have time to prepare talks. Please come and be ready to sign up for weeks. The dates are available online.


9/16/09 A Classical Introduction to the Gamma Function
We will discuss Barnes' approach to the Gamma function from the solutions to difference equations. We will relate the construction to the Bernoulli Numbers, and various other classical constructions. This talk should be accessible to all.


9/23/09 Adelic Fourier Analysis
This talk will start from first principles, and should be accesible to anyone. We will define the p-Adics and discuss some basic analysis and measure theory. We will then extend to the Adeles (which we will define and introduce to all), and discuss how to compute Fourier transforms on this space, and some applications to number theory.


9/30/09 Diophantine Questions and the Adeles
In this talk, we discuss new developments in adelic methods for Diophantine problems. In particular, we show how they can be used to approximate the number of solutions to Diophantine equations, as well as how they might be used to shed light on questions such as the Goldbach Conjecture or Twin Prime Conjecture.


10/07/09 Iwasawa Theory of Elliptic Curves with Complex Multiplication
Iwasawa theory studies modules over an infinite extension of number fields with Galois group a "p-adic Lie group." Initiated by Iwasawa to study the class groups of cyclotomic fields, the theory was applied by Mazur to the study of rational points on abelian varieties. When applied to elliptic curves with complex multiplication, Iwasawa theory yields important geometric information about the curves and important arithmetic information about imaginary quadratic fields. We will discuss two important examples, one of each type of result.


10/14/09 The Langlands Program and L-functions
I plan to offer a quick introduction to the Langlands program and discuss its role in number theory. The talk will be primarily based on Blasius and Rogawski's paper, "Zeta functions of Shimura varieties".


10/21/09 The Riemann Hypothesis for Function Fields I
For any function field there is a natural analog of the Riemann zeta function. Over two weeks, we will enjoy an elementary proof the Riemann Hypothesis for function fields, which states that all zeros of these zeta functions have real part 1/2. The proof will be relatively self-contained and all are encouraged to attend.


10/28/09 The Riemann Hypothesis for Function Fields II
This is the second part of a two part talk proving the Riemann hypothesis for Function Fields. The talks should be accessible to all.


11/11/09 Counting Cubic Fields of Bounded Discriminant
I will briefly discuss results and conjectures on asymptotics for the number of number fields of degree n having discriminant bounded by X. As an example of some of the modern methods used to answer this type of question, I will present asymptotic results for the number of cubic fields and orders having a fixed quadratic resolvent.


11/18/09 Gevrey Asymptotics
In this talk, we will discuss the following question: given a formal power series, when is this power series the asymptotic expansion of a holomorphic function, and under what circumstances is the holomorphic function determined uniquely by the power series? For a certain class of power series, called Gevrey series, this question has a nice answer. Many functions of arithmetical interest are Gevrey, and we will discuss these as well.


12/2/09 A Brief Introduction to Eigenvarieties
Eigenvarieties are--roughly speaking--rigid analytic geometric spaces parametrizing classical and p-adic Hecke eigenforms. We will try to review the history that led to the (fairly recent) discovery of these interesting gadgets. Some applications will also be discussed.


12/9/09 Constructing Fontaine's Rings of p-adic Periods
The ordinary comparison isomorphism between the de Rham and singular cohomologies of a complex manifold requires transcendental "periods" like 2pii to make sense. To imitate it p-adically, using cohomology theories coming from algebraic geometry, we need analogues of these periods, which turn out to live in somewhat outlandish rings. In this expository talk, we build these rings. The tools used in the construction are classical and of general use in the study of local fields.


1/6/10 Organizational Meeting


1/13/10 A Quadratic Analogue of Artin's Conjecture
I will present a paper of Hans Roskam which states and proves an analogue of Artin's Conjecture for quadratic extensions of the rational numbers; in particular, I will calculate the density of primes where the residue of the fundamental unit is maximal modulo those primes. A brief review of Artin's Conjecture and Hooley's proof will also be presented.


1/20/10 Parametrizing Nilpotent Orbits in p-adic Symmetric Spaces
Let k be a p-adic field, and suppose G is the group of k-rational points of a reductive, linear algebraic group defined over k. If θ is an involution of G, there is a notion of a nilpotent orbit which we view as lying in -1-eigenspace of Lie(G). We will discuss how to parametrize nilpotent H-orbits, where H is equal to the group of fixed points under θ. This will be done using techniques from Bruhat-Tits theory.


1/27/10 The number of values of a polynomial over a finite field
I will discuss the possibilities for #h(Fq), where h(x) is a "low-degree" polynomial over Fq. For instance, I will explain Daqing Wan's result that if #h(Fq)<q then #h(Fq)≤q - (q-1)/deg(h). This and other results exhibit a great deal of structure to the set of numbers occurring as #h(Fq). The methods involved range from the elementary (and clever!) to elliptic curves, the function field analogue of the Chebotarev density theorem, the classification of finite simple groups, etc. After introducing these methods, I will explain how they can be used to prove generalizations to the study of #h(V(Fq)), where h:V→W is a "low-degree" morphism of varieties over Fq.


2/3/10 Why Is the Tate-Shafarevich Group Like an Ideal Class Group?
The following heuristic motivates many results and conjectures in arithmetic geometry: the arithmetic of elliptic curves is a one-dimensional analogue of the zero-dimensional theory of number fields. We'll give an account of one way to make this more precise. We will need basic definitions of algebraic number theory and use a little class field theory toward the end of the talk, but we won't require any background in arithmetic geometry.


2/10/10 Modular Forms and Galois Representations (Part I)
We outline the steps in the classical construction of Galois representations attached to holomorphic modular forms of weight k.


2/17/10 Modular Forms and Galois Representations (Part II)
We discuss (in brief sketches) the works on constructing Galois representations associated to automorphic forms on GSp_4, and time permitting, also the case of U(3).


2/24/10 An Elementary Proof of Kronecker-Weber
In this talk I will sketch a proof of the Kronecker-Weber Theorem using only elementary ideas from algebraic number theory. I will use this proof to motivate and give intuition for Class Field Theory.


3/24/10 Deuring's Theorem
Deuring's theorem equates the L-function of a CM elliptic curve over a number field with the L-function of an associated Hecke character. We will explain what that sentence means, then prove it.


3/31/10 Mass Formulae for p-adic Extensions
Serre's mass formula determines the number of totally ramified extensions of a p-adic field (when counted with the appropriate weights). We will present Serre's proof of this beautiful result and then discuss a recent generalization by Bhargava.

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