Fall 2003 Number Theory Seminar
Schedule
Thursdays at 2:25 pm, B131 Van
Vleck Hall.
- September 4, 2003, Ahmad El-Guindy (UW-Madison)
Title: Weierstrass Points on X_0(pM) and Supersingular 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 the genus of X_0(M)is 0 and prime p not dividing M) and proved that
the reduction of Weierstrass points on X_0(pM) is supersingular mod p. In
this talk we show that, for square free M on the list, all supersingular
j-invariants are covered this way. Furthermore, In most cases where M is
prime we describe the explicit correspondence between Weierstrass points
and supersingular j-invariants. Along the way we also genarlize a
useful formula of Rohrlich for computing a certain Wronskian of modular
forms modulo p.
- September 11, Nigel Boston (UW-Madison)
Title: Bounds on cyclic codes using arithmetical geometry
Abstract: There are many results on the minimum distance of a cyclic code
of the form that if a certain set T is a subset of the defining set
of the code, then the minimum distance of the code is greater than some
integer t. This includes the BCH, Hartmann-Tzeng, Roos, and
shift bounds and generalizations of these. In this talk,
we define certain projective varieties V(T,t) whose properties determine
wh
ether, if T is in the defining set, the code has minimum distance
exceeding t. Thus our attention shifts to the study of these varie
ties. By investigating them using class field theory and arithmetical
geometry, we will prove various new bounds. It is interesting,
however, to note that there are cases that existing methods handle, that
our methods do not, and vice versa. We end with a number o
f conjectures.
No prior familiarity with codes is assumed.
- September 18, Tonghai Yang (UW-Madison)
Title: The Gross-Keating invariants of a quadratic form
Abstract: For a quadratic form over Q_p with p \ne 2, its Jordan form is unique in some sense. This is not true when p=2.
Gross and Keating define some mysterious invariants in this case, which is magically unifying certain formulas related to quadratic forms, such as local densities. In this talk, we will
introduce these invariants, how to compute them in some cases, and
motivation for study these invariants. The question is wide open in
understanding these invariants.
- September 25, Eric Bach (UW-Madison)
- October 2, Ken Ono (UW-Madison)
Title: Parity of the partition function
- October 9, Brian Conrad (Univ. Michigan at Ann Arbor)
Title: Modular curves and Ramanujan's continued fraction
Abstract:
In Ramanujan's 1916 letter to Hardy, he proposed some remarkable
identities that were a special case of a general q-series continued
fraction that he had discovered and investigated. Ramanujuan's continued
fraction has been the subject of intense scrutiny by many people who have
investigated its many identities and the algebraic properties of special
values. However, none of this work addresses the most important question
of all: is there an underlying principle that explains why Ramanujan's
continued fraction has such properties?
We provide an affirmative answer to this question by proving that
Ramanujan's function (or rather, its reciprocal) is related to the modular
curve X(5) as j is to X(1). Moreover, we prove that this link with X(5)
is extremely well-behaved with respect to two arithmetic models of X(5),
and by means of such a connection with algebraic geometry we are able to
give clean conceptual proofs of results that require a lot of algebraic
effort from the q-series point of view. For example, we give a new proof
that the values of Ramanujan's function at imaginary quadratic points are
algebraic integral units, we propose a new algorithm for rapid computation
of special values, and we use positive-characteristic geometry to develop
an analogue of Kronecker's congruence formula (going far beyond the few
results obtained on a case-by-case basis via the q-series methods).
This is joint work with Bryden Cais.
- October 16, David Manderscheid (U. Iowa)
Title: Waldspurger's Involution and Types.
Abstract: In this talk I will parametrize Waldspurger's involution for supercuspidal
representations in terms of types in the case p odd. This involution is
defined on the discrete series representations of the non-trivial two-fold
cover of SL(2,F) where F is a p-adic field. Waldspurger used this
involution
in his deep study of automorphic forms on the two-fold cover of SL(2) over
a
number field. In general, types give inducing data that contains the sort
of
arithmetic information necessary to study the fundamental role that
supercuspidal representations play in questions in representation theory
of
number-theoretic interest, especially functoriality. In the case at hand,
types provide a rather surprising result in contrast to the case of SL(2).
I will keep prerequisites to a minimum, provide motivation, and try to
make
this talk understandable for graduate students.
- October 23 Two speakers this week.
Yangbo Ye (U. Iowa) at 11am
Title: Selberg's orthogonality for automorphic
L-functions
Abstract:
Selberg's orthogonality conjecture predicts that the coefficients of
automorphic L-functions attached to different cuspidal representations
are orthogonal. Professor Jianya Liu and I first proved a weaker,
weighted
version of this conjecture. As an application, we then proved that if
an L-function can be factored into a product of L-functions of possibly
different GL(m) over Q, then this factorization is unique. In other
words,
we proved the uniqueness of functoriality in this case. In particular,
an
L-function attached to a cuspidal representation of GL(m) over Q cannot
be
factored further. The proofs are unconditional.
Next we proved the original version of Selberg's orthogonality
conjecture
under the generalized Ramanujan conjecture. Our results can be used to
characterize asymptotically whether two cuspidal representations are
equivalent, twisted equivalent, or not twisted equivalent at all. This
proof also allowed us to study statistical correlations between non
trivial
zeros of two automorphic L-functions, under the Ramanujan conjecture.
Our
proofs are unconditional when the representations are corresponding to
holomorphic cusp forms for GL(2).
Jianya Liu (Shandong Univ., China) at 2:25pm
Title: SUBCONVEXITY FOR RANKIN-SELBERG
L-FUNCTIONS OF MAASS FORMS
Abstract: This is a joint work with Yangbo Ye. We prove a subconvexity bound for
Rankin-Selberg L-functions L(s,f x g) associated with a Maass cusp
form f and a fixed cusp form g in the aspect of the Laplace eigenvalue
1/4+k^2 of f, on the critical line Re s=1/2. Using this subconvexity
bound,
we prove the equidistribution conjecture of Rudnick and Sarnak on
quantum
unique ergodicity for dihedral Maass forms. Also proved here is that the
generalized Lindelof hypothesis for the central value of our L-function
is
true on average.
- October 30 Fernando Rodriquez Villegas (U. Texas at Austin)
Title: On the E-polynomial of certain character variety
Abstract:
This is work in progress joint with Tamas Hausel (UT Austin). The
character variety of the title is a twisted form X of the variety
parameterizing the representations of the fundamental group of a
Riemann surface of genus g into GL_n. We expect to obtain a
description of its E-polynomial (a cohomological invariant of X) by
counting the number of points of X over finite fields. Eventually, we
would like to use this technique to prove a mirror symmetry type
conjecture of Hausel and Thaddeus relating the corresponding
character varieties with, respectively, PGL_n and SL_n in place of
GL_n. In this talk I will describe the calculation of a generating
function giving the zeta functions of X for all n over a given finite
field.
- November 6 Jeremy Teitelbaum (UIC)
Title: p-adic Fourier Theory and Lubin-Tate Groups
Abstract: In this talk I will discuss some results (joint
with Peter Schneider) on p-adic integration, generalizing old results of
Amice and Lazard.
Let $L$ be a finite extension of ${\bf Q}_p$. A locally $L$-analytic
function on the ring of integers $o=o_L$ is a function given locally by
a convergent power series in one variable. We study the space of these
functions and its dual, the ring $D(o_L)$ of locally $L$-analytic
distributions. We show that
this ring of distributions is isomorphic to the global functions on a
rigid space $\hat{o}$ parameterizing $L$-analytic characters of $o$.
The space $\hat{o}$ turns out to be quite interesting. If $L=\Qp$, then
Amice and Lazard showed that $\hat{o}$ is the open unit disk (viewed as a
rigid space). We show that, if $L$ is not ${\bf Q}_{p}$,
then $\hat{o}$ is a "twisted form" of the open unit disk; it is
isomorphic over ${\bf C}_{p}$ to the open unit disk, but is
not a disk over any discretely valued extension of $L$.
Our methods rely on Lubin-Tate theory and some
results from Tate's classic paper on p-divisible groups. We will mention
some applications to representation theory and to p-adic L functions.
- November 13 Alex Popa (Princeton University)
Title: Central values of Rankin L-series over real quadratic fields
Abstract:
We consider the L-function of a classical modular form f of weight 2 or 0
over the rational numbers, twisted by a Hecke character of a real
quadratic field. When the sign of the functional equation is +1, we give
an explicit formula for the central value of the L-function in terms of a
"toric integral" of a modular form on a quaternion algebra related to f by
the Jacquet-Langlands correspondence. The proof uses techniques from the
theory of automorphic representations, in particular an adelic version of
the Rankin-Selberg integral, and an explicit computation of the theta
correspondence between representations of GL(2) and representations of
certain similitude groups associated to 2 and 4 dimensional quadratic
spaces.
- November 20, 2003, Jordan Ellenberg (Princeton Univ.)
Title: Arithmetic of towers of algebraic curves
Abstract:
"Let
...-> X_n -> ... -> X_0 = X
be a tower of curves over a fixed number field F, whose Galois group is a
p-adic Lie group (for instance, the tower of Fermat curves x^{p^n} +
y^{p^n} + z^{p^n} = 0, or the tower of modular curves X(p^n)). One is
interested in general in the variation of arithmetic invariants of curves
in such a tower; in particular, we study the variation of the Mordell-Weil
rank of the Jacobian of X_n as n grows. We show that this rank grows at
most linearly in genus(X_n), and describe how to bound the constants in
special cases. The problem has an evident Iwasawa-theoretic flavor, and
indeed we show how to package the Mordell-Weil groups of the X_n into an
Iwasawa module for a certain non-abelian Iwasawa algebra, of the type
recently studied by Coates, Schneider, Sujatha, Howson, Venjakob, and
others."
- November 27 (Thanksgiving, no seminar)
- December 4 Paul Jenkins (UW-Madison)
Title: Traces of Singular Moduli
Abstract: We use the circle method to examine the difference
between the trace T(d) of a singular modulus coming from the ideal
class group of Q(sqrt(-d)) and the estimate given by the first term of the
j-function and the elements of the ideal class group.
Spring 2004 Number Theory Seminar
Schedule
Thursdays at 1:20 pm,
- Jan. 22, Romyar Sharifi (Max-Planck Inst. for Math., Bonn)
- Jan. 29, Ralf Schmidt (U. Minnesota)
Title: "Local newforms for GSp(4)"
I will report on a joint project with Brooks Roberts concerning local
newforms for the p-adic group $GSp(4)$. Let $\pi$ be an irreducible,
admissible generic representation of $PGSp(4,F)$, where $F$ is a p-adic
field. We conjecture that (i) There exists an N such that $\pi$
contains a vector invariant under the paramodular group of level N.
(ii) If N is minimal with this property, then such a vector is unique up
to multiples; we call it a local newform. (iii) The Novodvorski zeta
integral of the local newform computes the L-factor of $\pi$. In other
words, we conjecture that there is a newform theory analogous to the
well-known theory for $GL(2)$. There is another conjecture concerning the
structure of oldforms. I will report on recent progress concerning these
conjectures.
- Feb. 5,
- Feb. 12,
- Feb. 19,
- Feb. 26, Stephen Wainger (UW-Madison)
- Mar. 4, Stephen Wainger (UW-Madison)
- Mar. 11, Rob Benedetto (Amherst College)
- Mar. 18, Spring Break
- Mar. 25,
- Mar. 31
- April 8, 2004, Wenzhi Luo (The Ohio State University)
- Apr. 15, 2004, Ling Long (Iowa State U.)
- Apr. 22, George Pappas (Michigan State U.)
- Apr. 29, Mark Kisin (U. Chicago)
- May 6,