Spring 2003 Number Theory Seminar Schedule

Thursdays at 1:20 pm, B223 Van Vleck Hall.

January 30, Matt Papanikolas (Brown Univ.)
Title: Algebraic independence over function fields
Abstract: The connections between analytic quantities arising from geometry and number theory have intrigued mathematicians since Gauss and Euler. In this talk we will investigate how to understand transcendence properties of periods of algebraic groups in terms of the underlying geometry of the group. Using theorems and conjectures about periods over the complex numbers as a starting point, we will present new algebraic independence results for periods of Drinfeld modules in the setting of function fields.
February 6, Dongho Byeon (Seoul National Univ.)
Title: Class numbers of quadratic fields Q(\sqrt{D)} and Q(\sqrt{tD})
Abstract: Let t be a square free integer. First we shall show that there exist infinitely many positive fundamental discriminants D > 0 with a positive density such that the class numbers of quadratic fields Q(\sqrt{D}} and Q(\sqrt{tD}) are both not divisible by 3. Finally we shall discuss its application to rank-one quadratic twist of elliptic curves.
February 13, Ahmad El-Guindy (UW-Madison)
Title: Weierstrass points on X_0(pM) and suprsingular j-invariants
Abstract: Weierstrass points are special points on a Riemann surface that carry a lot of information. Ogg studied such points on X_0(pM) (for M such that g(M)=0) and proved that if R is a Q-rational Weierstrass point on X_0(pM), then it's reduction modulo p is supersingular. Recently, Ono and Ahlgrenn gave precise description of the correspondence between supersingular j-invariants and Weierstrass points for X_(pM) for M=1. In this talk we shall genarlize their result to some other M's, including all primes in the list. Along the way we shall also genarlize an important formula of Rohrlich for computing a certain Wronskian of modular forms.
February 20, Dorian Goldfeld (Columbia University)
Title: On the average number of occurrences of a generator in words in a group
Abstract:We consider an abstract group defined by generators and relations. Every word or element in the group can be expressed as a product of the generators, but the representation is not unique. In certain cases the number of occurrences of a particular generator in an arbitrary word may be a well defined function, and it is then an interesting question to explore the average value. In joint work with C. O'Sullivan, we introduce a new method in analytic number theory to study this question. The main tool is the theory of Eisenstein series twisted by modular symbols.
February 27, Gebhard Boeckle (ETHZ)
Title: Function fields, Galois representations and a conjecture of de Jong (joint with C. Khare)
Abstract: In the talk I shall give a brief introduction to function fields and their fundamental group. Then I plan to present various reformulations and consequences of the conjecture of de Jong as well as known results. For instance, assuming the conjecture, one easily obtains an analogue of Serre's conjecture for function fields. In the end I hope to sketch one approach to proving the conjecture in many cases. It is an adaption of methods used in the modularity proofs by Taylor and Wiles and also uses the recent results of Lafforgue on the Langlands conjecture.
March 6, Jeremy Lovejoy (UW-Madison)
Title: Differences of Partition Functions
Abstract: I will discuss how looking at appropriate differences of partition functions leads to a variety of number theoretic objects, such as theta functions, divisor functions, indefinite binary and ternary quadratic forms, mock theta functions, L-functions, Kloosterman quadratic forms,... I will give some samples of combinatorial facts which can be deduced using number theory and indicate how connections between the difference functions and number theory arise via q-series identities
March 13, Kristin Lauter (Microsoft)
Title: Complex multiplication methods for generating curves over finite fields
Abstract: Elliptic curves with a known number of points over a prime field are often needed for use in cryptography. In the context of primality proving, Atkin and Morain suggested the use of the theory of complex multiplication to construct such curves. One of the steps in this method is the calculation of the Hilbert class polynomial H(X) for a given discriminant D. The usual way is to compute H(X) over the integers and then to find a root modulo the prime. We present a modified version of the Chinese remainder theorem (CRT) to compute H(X) modulo the prime directly from the knowledge of H(X) modulo enough smaller primes. Our complexity analysis suggests that asymptotically our algorithm is an improvement over previously known methods. This is joint work with Amod Agashe and Ramarathnam Venkatesan. I will discuss several possible generalizations of this work.
March 20, Spring Break
March 27, David Goss (Ohio State Univ.)
Title: Modularity in Characteristic p
Abstract: The connection between elliptic curves over the rational numbers $\bf Q$ and modular forms for $SL_2({\bf Z})$ is now very well known. This fundamental relationship both establishes that the L-series of one such elliptic curve has an analytic continuation and functional equation and gives representatives of the isogeny class of the elliptic curve (inside the Jacobians of modular curves). Now let $A:={\bf F}_q[T]$, where ${\bf F}_q$ is the finite field with $q$-elements, and let $k:={\bf F}_q(T)$. Let $K:={\bf F}_q((1/T))$ with associated algebraic closure $\bar{K}$. Mimicking the classical definition of $\bf Z$-lattices inside the complex numbers $\bf C$, one has the notion of ${\bf F}_q[T]$-lattices inside $\bar{K}$. Rank one lattices correspond to analogs of the exponential function and rank two lattices uniformize analogs of elliptic curves. These rank two "Drinfeld modules" give rise to modular curves in exact analogy with elliptic curves. Remarkably, Drinfeld and Zarhin have shown that the Jacobians of such modular curves describe very general isogeny classes of elliptic curves over $k$ in close analogy with the $SL_2({\bf Z})$-theory. But what about the Drinfeld modules themselves? Can a Drinfeld module be modular? In this talk we describe the recent habilitation thesis of Gebhard B\"ockle that makes such modularity actually quite reasonable.
April 3, Farshid Hajir (U Mass)
Title: Shallow Ramification and the Fontaine-Mazur Conjecture
Abstract: Suppose L/K is a Galois extension of number fields and fix a positive integer n. We say a prime P of K is ramified to depth at most n in L if the nth term in the filtration of Gal(L/K) by the upper-numbering ramification groups at P is trivial. If L/K is unramified outside a finite set and ramified to depth at most n everywhere, we say L/K is shallow. I'll discuss how the Fontaine-Mazur conjecture for tame extensions extends to shallow ones. This will allow us to reformulate the Fontaine-Mazur conjecture in terms of root discriminants. This is joint work with C. Maire.
April 10, Juping Wang (Harvard and Fudan Univ.)
Title: Singular Jacobi Forms
Abstract: For a singular Siegel modular form, we have known that it can be determined completely by the weight of associated representation of a general linear group. A similar theory for Jacobi forms has been proposed recently, and one hopes to characterize singular Jacobi forms with higher degree by their weight and index. In this talk we will introduce some ideas and some results about this problem.
April 17, Shou-Wu Zhang (Columbia University)
Title: Periods integrals on Hilbert modular surfaces
Abstract: The integrals of eigen functions on modular subvarieties (such as Shimura curves and CM-points) can be exprssed in terms of special values of L-functiuons. These expressions can be used to construct p-adic L-functions and show some equidistribution properties of modular subvarieties
April 24, Yoshi-hiro Ishikawa (Univ. Maryland and Okayama University)
Title: Fourier-Jacobi expansion, Shintani's integral and the Standard $L$-function of $U(2,1)$
Abstract: Our question is "What kind of special functions will appear in Fourier-Jacobi expansion of automorphic forms, which are not necessarily holomorphic?" First, we recall classical cases. Next, we shall give the answer for arbitrary cohomological representations of $U(2,1)$. This is an Archimedian counterpart of Casselman-Shalika formula. As a bonus, we get the multiplicity one result for the generalized Whittaker model, which detects exactly when the model exists. If time permits, we would give an application of our explicit formula to a zeta integral for the Standard $L$-function. The result is a product of three gamma functions for any discrete series.
May 1, You-Chiang Yi (UW-Madison and UIUC)
Title: Modularity of a Calabi-Yau variety
Abstract: I will prove the modularity of a Calabi-Yau variety by using the method of Wiles.