Title: Vanishing and Non-Vanishing of Traces of Hecke Operators.

Abstract: Using a reformulation of the Eichler-Selberg trace formula, due
to Frechette, Ono, and Papanikolas, we consider the problem of the
vanishing (resp. non-vanishing) of traces of Hecke operators of even
weight cusp forms with trivial Nebentypus character. For example, we show
that for a fixed operator and weight, the set of levels for which the
trace vanishes is effectively computable. Also, for a fixed operator the
set of weights for which the trace vanishes (for any level) is finite.
These results motivate the generalized Lehmer conjecture,'' that the
trace does not vanish for even weights 2k >= 16 or 2k = 12.

Title: "The Andrews-Stanley partition function and p(n)"

Abstract: In a recent paper, G.E. Andrews formulated a new partition function t(n) based

on a condition of R. Stanley, and showed that t(n) satisfies the Ramanujan

congruence mod 5 for the usual partition function p(n).This prompted the

investigation of what properties these two functions share.We discuss two

specific aspects of this question: asymptotics and congruence properties.

Title: On Fourier coefficients of Siegel cusp forms with

small weight

Abstract:

Let Sk(_g) be a Siegel cusp form of degree and weight on _g, the Siegel

modular group. We want to prove the following

Theorem 0.1 Let_be an integer and suppose that g+1; let Sk(_g)

with Fourier coefficients a(T), where is a positive definite symmetric halfintegral

× matrix. Then we have

a(T__,F (det T)k/2-1/(2g)-(1-1/g)_g+(_ > 0),

where _-1

:= 4(1) + 4 _g-1

_2

g+2 and where the constant implied in __,F

only depends on and F.

For the proof one can use the Fourier-Jacobi decomposition of F

F(Z) =Xm>0

Pm(_, z)e2_i tr(m_0),

were the summation extends over all positive definite symmetric half-integral

(1) × (1) matrices. Then the coefficients Pm(_, z) are Jacobi cusp forms.

The case k > g+1 is treated byB¨ocherer and Kohnen. They use certain Poincar´e

series for the Jacobi group and develop a kind of Petersson coefficient formula.

Unfortunately these series fail to converge absolutely in the case + 1.

Therefore we use the so-called Hecke trick and multiply every summand of the

Poincar´e series with a factor depending on a complex variable s, such that the

new series Pk,m;(n,r),s is again absolutely convergent for =Re(s) sufficiently large.

Moreover this factor is chosen such that the new series is again invariant under

the slash operation of the Jacobi group. Now the method is the following one:

we compute the Fourier expansion of the Poincar´e seriesPk,m;(n,r),s, show that

it is even absolutely and locally uniformly convergent in a larger domain of C,

that contains the point = 0 if + 2 and take it as a new definition for the

Poincar´e series in this larger domain. What is left to show is that these series

are Jacobi cusp forms and that the Petersson coefficient formula is still valid.

Afterwards we can show very easily the desired estimate.

Oct. 14, Matt Boylan (UIUC)

Title : "Half-integral weight modular forms with few non-vanishing coefficients"

Abstract: Half-integral weight cusp forms whose reduction modulo a prime l has few

non-vanishing coefficients have been studied recently by Bruinier, Ono,

and Skinner.  In this talk, I will describe some conditions

necessary for such forms to exist and discuss applications of this work to

central critical values of modular L-functions and orders of

Tate-Shafarevich groups.

Oct. 19, Steve Miller (Rutgers Univ. and Hebrew Univ.)

Title: Automorphic Distributions

Abstract: I will speak on the notion of automorphic distributions, which

in the simplest setting of SL(2,R) can be thought of as the boundary

values of a modular form on the upper half plane as one approaches the

x-axis.These have applications to topics in analytic number theory such

as:

1) Summation formulas

2) L-functions

3) Riemman's Non-differentiable function'' $\sum_{n\ge 1}\sin(n^2 x)/n^2$

I will focus mainly on a recent

application, which is to produce new examples of the full holomorphic

continuation of the exterior square L-function on GL(n).Much of this

represents joint work with Wilfried Schmid.

Title: A New Proof of the Ramanujan Congruences for the Partition  Function

Abstract: Let $p(n)$ denote the number of partitions of $n$.  Recall

Ramanujan's three congruences for the partition function,

\begin{eqnarray*}

p(5n + 4) &\equiv& 0 \pmod{5}   \\

p(7n + 5) &\equiv& 0 \pmod{7}   \\

p(11n + 6) &\equiv& 0 \pmod{11}.

\end{eqnarray*}

These congruences have been proven via $q$-series identities,

combinatorial arguments, and the theory of Hecke operators.  We

present a new proof which does not rely on any specialized identities

or combinatorial constructions, nor does it necessitate introducing

Hecke operators.  Instead, our proof follows from simple congruences

between the coefficients of modular forms, basic properties of Klein's

modular $j$-function, and the chain rule for differentiation.

Furthermore, this proof naturally encompasses all three congruences in

a single argument.

Title:  Kloosterman sums and 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.  In

addition, we derive a simple expression for writing twisted traces as an

infinite series.

Nov 11, 2004,   Tom Haines (U. Maryland)

Title:  Local L-factors of simple Shimura varieties at primes of

Abstract: This talk will concern certain Shimura varieties attached to

"fake" unitary groups, and their reduction modulo a prime where the level

structure is given by a parahoric subgroup. For these varieties it is

possible to express the semi-simple local factor of the Hasse-Weil zeta

function explicitly in terms of semi-simple automorphic L-functions, as

predicted by Langlands and proved in the case of good reduction by

Kottwitz.  After an overview of the general problem, this talk will

discuss the difficulties created by bad reduction, and how these

difficulties are overcome (joint work with Bao Chau Ngo).

Nov. 18, Keith Conrad (U. Connecticut)

Title: Elliptic curves with elevated rank

Abstract:  An elliptic curve over Q(T) is said to have elevated rank if

its rank over Q(T) is less

than the rank over Q of all but finitely many of its specializations to

a rational elliptic curve.

All known examples of elevated rank depend on the parity conjecture,

and the first such example is due to Cassels and Schinzel: 7(1+T^4)y^2

= x^3 - x has rank 0 over Q(T) but, for every rational number t, the

elliptic curve 7(1+t^4)y^2 = x^3-x has odd (and thus positive) rank

under the parity conjecture.

This example is isotrivial, so from a geometric point of view the

example is less than ideal.  Are there non-isotrivial elliptic curves

over Q(T) with elevated rank?  Plausible conjectures imply the answer

is NO, but one of these conjectures breaks down in characteristic p.

Does that mean there are non-isotrivial examples of elevated rank in

characteristic p?  YES, if p > 2 and we assume the parity conjecture.

This is joint work with B. Conrad and H. Helfgott.

Dec. 2   Ye Tian (McGill Univ.)

Title: Twisted Fermat curves over totally real fields.

The Abstract: In this talk, we will discuss a Diaphantine application of Euler systems

of Shimura curves and double L-series. It is based on a joint work with

Diaconu and one with Zhang.

Dec. 9,  Minhyong Kim (U. Arizona)

Title:The non-abelian method of Chabauty
Abstract: The classical method of Chabauty gives a quick proof of
the Mordell conjecture for a curve if its Jacobian has
Mordell-Weil rank less than the genus. We outline a program for
removing this restriction using the motivic fundamental
group of the curve.