Number Theory Seminar Spring 2008
Abstract
Frank Thorne (UW-Madison)
Title: Tate's Thesis I, II (student seminar)
Abstract: I will give an overview of Tate's thesis. In his thesis, Tate
defined a broad class of zeta functions as integrals over
the ideles of a general weight function times a character of the idele
class group. These include the Riemann zeta
function and Dirichlet and Hecke L-functions as special cases. He then
presented an elegant proof
of analytic continuation and the functional equation in the general
setting.
I will give some general background on Fourier analysis in number
fields, discuss Tate's proof, and describe how the
classical L-functions and their functional equations fit into the big
picture.
Feb. 14,
Paul Jenkins (UCLA)
Title: Integral traces of singular values of Maass forms
Abstract: In his influential paper, Zagier proved that the
generating functions for traces of singular moduli
associated to polynomials in $j(\tau)$ are weight 3/2 modular forms on
$\Gamma_0(4)$. At the end of his paper, he
suggested a method for generalizing these results to higher
weights. One such generalization was given by Bringmann and
Ono, who give an identity for the traces associated with certain Maass
forms in terms of the Fourier coefficients of
certain half integral weight Poincar\?e series. However, it does
not seem to be known when these traces are integral or
even rational. We give an identity for the traces associated to
an arbitrary weakly holomorphic form $f$ of negative
integral weight on $SL_2(Z)$ in terms of the coefficients of specific
weakly holomorphic forms of half integral weight.
If the coefficients of $f$ are integral, then these traces are integral
as well. We use this correspondence to obtain a
negative weight analogue of the classical Shintani lift.
Feb. 19 and 26.
Rob Rhoades (UW-Madsion)
Title: Class Groups, Quadratic Forms, and Homogenous Spaces
Abstract: I will describe the correspondence between class group
elements of quadratic fields and quadratic forms
of discriminant d. With this correspondence in hand I will show
how to place the class group in a geometric
framework. This construction has many important
applications. For example, this construction allows one to
place rational points on elliptic curves. This construction also
has connections to asymptotics to for class
group sizes. If time permits we will describe some of these
connections. We will also discuss how class groups
of higher degree fields can be placed into a geometric study.
This construction does not require the connection
to quadratic forms.
Ramin Takloo-Bighash (UIC, Feb. 21)
Title: "Gross-Prasad Conjecture for GSp(4)."
Abstract: In this talk I will explain a recent result joint with
Dipendra Prasad on the existence of certain
functionals of arithmetic interest for representations of the
symplectic group of order four over a local field. I
will also explain possible connections between this theorem and
special values of L functions of Siegel modular of
genus two.
Gerhard Boeckle (U. Duisburg-Essen, Germany, March 6)
title: Hecke characters and compatible systems of abelian Galois
representations
abstract: In a way similar to the case of elliptic modular forms, one
can attach
strictly compatible systems (SCS) of Galois representations to Drinfeld
modular forms. Unlike in the classical situations, these are abelian!
Goss
had asked whether they would arise from Hecke characters.
This is indeed the case. Our approach follows work of Khare in the
number
field case. As a consequence one obtains for all global fields a
correspondence between SCS of semisimple abelian mod p Galois
representations and Hecke characters. I will formulate the result and
present parts of the rather elementary proof. If time permits,
I may discuss some open questions regarding the relations of these Hecke
characters to Drinfeld modular forms.
Christian Zorn (Ohio State U,
March 13.)
Title: Explicit computations in the theory of local theta lifts.
Abstract: There is a well understood connection between the
non-vanishing of local theta lifts and the non-vanishing of
certain local doubling integrals. We shall compute these doubling
integrals for all the constituents of the unramified
principal series for the rank two symplectic group and its metaplectic
cover. These computations rely on a choice of
"good test vectors" coming from both the underlying representation as
well as an induced representation on a larger
symplectic (resp. metaplectic group) used to form an Eisenstein series.
Ultimately, we will discuss the choice of "good
test vectors", the explicit computation of the doubling integral, and
how it relates to the theory of local theta lifts
to various orthogonal groups.
Eric Errthum (Winona State U., March
27)
Title: Singular Moduli of Shimura Curves
Abstract: For a genus-0 Shimura curve, a properly normalized coordinate
function evaluated at a complex multiplication (CM) point is algebraic
over the rationals but is in general difficult to explicitly compute.
This talk will demonstrate that the coordinate maps for the Shimura
curves associated to the quaternion algebras with discriminants 6 and 10
are Borcherds lifts of vector-valued modular forms. This property is
then used to explicitly compute the rational norms of singular moduli on
these curves. This method not only verifies conjectural values for the
rational CM points, but also provides a way of algebraically calculating
the norms of irrational CM points with arbitrarily large negative
discriminant on these Shimura curves.
Farshid Hajir (U. Mass., April 3)
Title: Asymptotically Good Families
Abstract: I will describe the recurring theme of asymptotically good
families in the study of codes, graphs, lattices, 3-manifolds, number
fields, curves over finite fields, ... . In many of these
contexts, the
"goodness" of the family is encapsulated in properties of an associated
group. In this expository talk, I will describe many contexts in
which
asymptotically good families arise and give an axiomatization of the
concept, inviting us to look for them in other situations, especially in
the context of pro-p groups.
Wei Zhang (Columbia Univ. April
10)
Title: Gross-Zagier formula for GL(2)
Abstract: I'll describe a new formulation of Gross-Zagier formula based
on Waldspurger's formula. This puts the
classical Gross-Zagier formula into a more representation theoretical
framework and will remove all ramification
hypothesis in the previous related work. This is a joint work with
Xinyi Yuan and Shou-wu Zhang.
Patrick Rault (UW_Madison)
Title:
On uniform bounds for rational points on rational curves and thin sets
Abstract:
We show that for any $\epsilon>0$ the number of rational points of
height less than $B$ on the image of a degree 2 map from $\mathbb P^1$
to $\mathbb P^n$ ($n\geq 1$), under certain conditions, is at most
$CB/|RD|^\delta+C_\epsilon |RD|^\epsilon$, where the point is that the
constants $C$ and $\delta$ are independent of the choice of the map.
$R$ and $D$ are respectively the resultant and discriminant of the
map. In the special case of degree 2 plane curves we prove a
bound of
$CB/|RD|^\delta+4$ which improves on a result of Heath-Brown and
Browning Heath-Brown by establishing an inverse dependence on the
resultant and discriminant. Heath-Brown proved that for any
$\epsilon>0$ the number of rational points of height less than $B$ on
a degree $d$ plane curve is $O_{\epsilon,d}(B^{2/d+\epsilon})$.
Browning and Heath-Brown later proved that this result holds with
$\epsilon=0$ for degree 2 curves. It is known that Heath-Brown's
theorem is sharp apart from the $\epsilon$, and in fact Ellenberg and
Venkatesh have proven that there is some $\delta>0$ (depending only
on
$d$) for which the counting function for any plane curve of positive
genus is $O_d(B^{2/d-\delta})$. It is an open question whether
Heath-Brown's Theorem is true with $\epsilon=0$.
Brian Conrad (U. Michigan and
Stanford, April 17)
Title: Finiteness of class numbers for group varieties over
global function fields
Abstract: Generalized ideal class groups of number fields
can be described adelically in terms of a coset space for the group
GL_1,
and this in turns leads to a notion of "class number" (as the size
of a certain set, if finite!) for an arbitrary smooth affine group
variety
over a global field. Over number fields the finiteness of these
numbers was proved by Borel and Harish-Chandra via reduction theory.
The situation for the analogous finiteness problem
over global function fields was not as well-understood,
due to a variety of reasons, but some cases were known by work of Borel
and G. Prasad.
After reviewing some of the history and motivation, I will explain how
to
use a mixture of arithmetic and geometric techniques to
prove the finiteness of class numbers for all affine group varieties
over global function fields, conditional on a certain structure theorem
for such group varieties. The required structure theorem has been
recently
established in odd characteristic (joint work with G. Prasad),
so I will say something about that part of the story too if time
permits.
Andrew Granville (U. Montreal,
April 22)
Title: Pretentiousness in analytic number theory.
Andrew Granville, University of Montreal
Abstract:
Inspired by the "rough classification" ideas from additive
combinatorics,
Soundararajan and I have recently introduced the notion of
pretentiousness
into analytic number theory. Besides giving a more accessible
description of
the ideas behind the proofs of several well-known difficult results of
analytic number theory, it has allowed us to strengthen several results,
like the Polya-Vinogradov inequality, the prime number theorem, etc. In
this
talk we will introduce these ideas and gave some flavour of these
developments.
Speaker: Yongqiang Zhao
(UW-Madison, April 29, May 6)
Title:
Higher composition laws I: Quadratic generalizations of Gauss
composition.
Higher composition laws II: Cubic analogues of Gauss composition.
Abstract:
In his 2001 Ph.D. Thesis, Bhargava made very beautiful and remarkbale
generalizations of Gauss's composition law on
integral quadratic forms. I will give a brief introduction on some of
his work. In the first talk, we will rivisit
Gauss's composition law from a different of view and introduce other
quadratic composition laws. In the second, we will
discuss cubic composition laws, which are genuine higher analogues of
Gauss composition.
Speaker: Jeremy Rouse (UIUC, May 1)
Title: Bounds for the coefficients of powers of the Delta-function.
Abstract: Let tau_k(n) be the nth coefficient of the kth power of
Ramanujan's classical Delta function. Work of Deligne implies that
|tau_k(n)| <= C_k n^(12k-1)/2 for some constant C_k. We will
show that the constant C_k tends to zero very rapidly. The proof will
use the fact (noticed by Goldfeld, Hoffstein and Lieman) that symmetric
square L-functions associated to newforms have no Siegel zeroes.
Speaker: Ye Tian (Columbia Univ.
and Morningside Center of Math., May 8)
Title: Heegner points over False Tate curve extensions.
Abstract: Let $E$ be an elliptic curve over $Q$, $p$ a prime number,
and $L_\infty$
an (infinite) Galois extension over $\BQ$ with Galois group
a $p$-adic Lie group.
We are interested in how the Mordell-Weil group $E(L)$ varies when $L$
runs over sub number
fields of $L_\infty$. In this talk, we study several examples.
Speaker: Ameya Pitale (U. Oklahoma,
May 29)
Title : Integral Representations of L-functions
Abstract : I will try to give a brief general overview of L-functions -
their applications, construction and importance.
Time permitting, I will talk about my recent work with Ralf Schmidt on
L-functions for GSp(4)xGL(2).