Speaker: Jeremy Rouse (UW-Madison)
Time and Place: Thursday Jan 27 at 1:20pm, VV B211
Title: The Arithmetic of Borcherds Products for Hilbert Modular Forms
Abstract: A certain sequence of weight 1/2 modular forms arises in
the
theory of Borcherds products for modular forms on SL_2(Z). Zagier proved
an identity between the coefficients of these weight 1/2 forms and
a
similar sequence of weight 3/2 modular forms and provided an
interpretation of the coefficients in terms of traces of singular moduli.
We prove analogous results for modular forms arising from Borcherds
products for Hilbert modular forms.
Speaker: Amanda Folsom (UCLA)
Time and Place: Thursday Feb. 3 at 1:20pm, VV B211.
Title:Modular Units and Continued Fractions
Abstract: The classical theory of modular units, units in the field
of
modular functions of fixed level, has primarily been developed by
Kubert and Lang. The modular theory reflects much of the theory
developed in the cyclotomic fields. This relation extends
beyond analogy in ways that are still being discovered. For
example of interest are subgroups of the modular unit group whose
special values give subgroups of the unit group in a number field.
We
investigate these questions, and exhibit a set of modular units that
arise
naturally from a set of q-difference equations that may be regarded
as
higher level analogues to the Rogers-Ramanujan continued fraction.
Further, we discuss how these modular units are in correspondence
with real fundamental cyclotomic units for prime power level.
Feb. 17
Speaker: Karl Mahlburg (UW-Madison)
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Title: Cranks Revisited and Higher Ramanujan Congruences
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Abstract: In 1944, Dyson conjectured the existence of an integer-valued
crank
function that would provide a combinatorial proof of
Ramanujan's congruence $p(11n+6) \equiv 0 \pmod{11}$ by sectioning
the partitions of $11n+6$ into $11$ classes of equal size. Forty
years
later, Andrews and Garvan successfully found such a function, and proved
the celebrated result that $M(m,11,11n+6) = p(11n+6)/11,$
where $M(m,N,n)$ is the number of partitions $n$ whose crank is
equivalent to $m$ modulo $N$.
The new result of this work is that for any prime $l \geq 5$, and any
power $j \geq 1$, there is an arithmetic progression $An + B$ such
that
$N(m,l^j,An+B) \equiv 0 \mod{l^k}$ for each $0 \leq m \leq l^j -1$
and
any power $k$. Summing over all $m$ then provides a combinatorial
proof
that $p(An+B) \equiv 0 \pmod{l^j}$, as the partitions are grouped into
classes whose sizes are all divisible by $l^j$.
The proof uses the theory of modular forms in a manner similar to that
found in Ono and Ahlgren's seminal works on partition congruences.
Speaker: Judy Walker (U. Nebraska)
Time and Place: Thursday Feb. 24 at 1:20pm, VV B211
Title:Shadows of self-dual additive codes over GF(4)
Abstract: Additive GF(4)-codes are interesting in part because of their
connections
to quantum coding theory and unimodular lattices. We will review
these
connections and then discuss bounds on the parameters of these codes.
The bounds come from considering the so-called shadow.
Time and Place: Thursday March 3, 2005 at 1:20pm.
TITLE: A Bound for the 3-Part of Class Numbers of Quadratic Fields
via the
Square Sieve
ABSTRACT:
Since Gauss's publication of Disquisitiones Arithmeticae in 1801,
mathematicians have been interested in the divisibility properties
of
class numbers.
However, still today many of the properties of class numbers remain
mysterious.
In this talk we investigate the divisibility by 3 of class numbers
of
quadratic fields.
It is conjectured that the 3-part of the class number of the
quadratic field Q(sqrt{D})
may be bounded above by an arbitrarily small power of |D|. However,
until recently,
the only known bound was the trivial bound O(|D|^{1/2 + epsilon}).
We use a variant of the square sieve and the q-analogue of van der
Corput's method to count the number of squares of the form 4x^3 - dz^2,
where d is a
square-free positive integer and x and z lie in the ranges x <<
d^{1/2}, z<<d^{1/4}.
As a result, we show that the 3-part of the class number of the quadratic
field Q(sqrt{D})
may be bounded by O(D^{27/56 + epsilon}). This gives a corresponding
bound forthe number of elliptic curves over the rationals with conductor
N.
March 10,
Speaker: Paul Jenkins (UW-Madison)
Title:Recent results on traces
of singular moduli
Abstract:We give a new proof
of some identities of Zagier relating traces of
singular moduli to the coefficients of certain half integral weight
modular
forms.These results imply a new
proof of the infinite product isomorphism
announced by Borcherds in his 1994 ICM lecture.In
addition, we derive a
simple expression for writing twisted traces as an infinite series,
and
discuss p-adic properties of traces and the congruences that follow.
March 18,
Bill Duke (UCLA)
Title: On traces of singular moduli
Abstract: I will explain a recent application of the uniform distribution
of CM points
to the asymptotics of traces of singular moduli,
which are certain sums of values of the classical j-function.
Related results of Bruinier, Jenkins and Ono will be
highlighte
March 28,
Speaker: Rafe Jones (Brown U.)
Title: The "Mandelbrot Set" of a Finite Field
Abstract: The famous complex Mandelbrot set has interesting
analogues over other fields. The case of finite fields is of
particular
interest, as many questions about the Mandelbrot set in this context
lead naturally to number-theoretic ideas over function fields.
I will
define the finite-field version of this set (which requires a bit of
tweaking from the usual definition -- hence the quotes in the title),
then
pose a natural question about its size. I will give an outline
of the
solution of this question, which requires ideas from number theory,
group
theory, and probability theory.
MArch 31, 2005
Antun Milas (SUNY at Albany)
Title: Modular Forms and Infinite-dimensional Lie Algebras
Abstract: In studies of infinite-dimensional Lie algebras we
encounter modular forms as graded dimensions (i.e., characters) of
certain irreducible representations. Interestingly, several irreducible
characters often give rise to SL(2,Z)--modules. These modules are subject
to various constraints (e.g., the Verlinde formula).
After reviewing a few classical constructions in the theory
of modular
forms, I will make a detour into the infinite-dimensional Lie theory.
I will introduce, at least at the intuitive level, the notion of vertex
algebra (after Borcherds) and explain how modular forms and modular
invariance arise in this setup. This will allows us to say things about
certain Wronskian modular forms by using the representation theory.
Finally, as a byproduct of our analysis, I will present new proofs
of
several nontrivial q-series identities (various Ramanujan's formulas,
Dyson-Macdonald's identities and generalizations).
April 7, 2005
Freydoon Shahidi(Purdue)
"On Sato-Tate conjecture"
Abstract: Sato-Tate conjecture demands that Hecke eigenvalues of most
modular forms be equidistributed with respect to the measure on
conjugacy classes of SU(2). In this talk we will define these notions
in simple terms and provide some evidence for the conjecture using
recent developements in the theory of automorphic forms and particularly
properties of symmetric power L-functions attached to these forms,
following an idea of Serre. Presentation will be done through examples.
This is joint work with Henry Kim.
April 14,
Eyal Goren (McGill Univ.)
Title: Higher analogues of elliptic units.
Abstract: The problem of explicitly constructing units in number fields
can be viewed as part of explicit class field theory but, more
specifically, is motivated by applications to Stark's conjectures on
special values of L-functions. Until recently little was known beyond
the theory of cyclotomic and elliptic units, which deal with,
respectively, abelian extensions of Q and of a quadratic imaginary
field.
I shall discuss joint work
with Ehud de Shalit, Kristin Lauter,
and others on construction of units in abelian extensions of a CM field
K of degree 4 over Q. Our main achievements is the construction of
algebraic numbers ("invariants") with interesting algebraic properties,
which are S units for a set S that is effectively and easily determined
in terms of the field K. Moreover, the speaker has recently been able
to
bound the powers to which primes in S may appear in our invariants.
Time
allowing, I shall mention current projects (of theoretical and
computational nature) with various collaborators and connections with
some beautiful new results by Jan Bruinier and Tonghai Yang.
April 21, A., Raghuram (U. Iowa)
Title: Special values of L-functions.
. Abstract: This talk will be an introduction to Deligne's conjectures
on
the special values of the symmetric power L-functions associated to
a
holomorphic cuspform. The latter half of the talk will be an account
of
some work in progress, which is joint work with Freydoon Shahidi, toward
the special values of the symmetric fourth power L-functions.
April 22, Eki Ghate (Tata and IAS)
Title: On the local behaviour of ordinary Galois representations
Abstract:
Let $f$ be a $p$-ordinary primitive cusp form of weight at least 2.
Then the
restriction to a decomposition group at $p$ of the two dimensional
$p$-adic
Galois representation attached to $f$ is upper-triangular. When $f$
has
complex multiplication it is not hard to see that it is even diagonal.
In this
talk we will provide some evidence which shows that the converse also
holds
April 28, John Gossey (UIUC)
Title: Galois 2-groups unramified outside 2
Abstract: We study those quadratic and biquadratic number
fields
for which the Galois group of their maximal pro-2 extension unramified
outside 2 and infinity is virtually free. In the case when the
above group
has a free subgroup of index 2, we explicitly describe the group
using
a result of Herfort-Ribes-Zalesskii on virtually free pro-p
groups.
May 5, You-Chiang Yi (UIUC)
Title: On the Hilbert modularity of a quintic threefold.
Abstract: In this talk, I will prove a Calabi-Yau threefold with
4-dimensional middle cohomology is Hilbert modular.