**Sept. 9---Jeremy Rouse** (UW-Madison)

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.

**Sept.30---Holly Swisher (UW-Madison)**

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.

**Oct. 7 **Kathrin Bringmann (UW-Madison)

Title: On Fourier coefficients
of Siegel cusp forms with

small weight

Let f 2 Sk(_g) be a Siegel cusp form of degree g and
weight k on _g, the Siegel

modular
group. We want to prove the following

Theorem
0.1 Letg _2 be an integer and suppose that k _ g+1; let F 2 Sk(_g)

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

g × g matrix.
Then we have

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

where _-1

g := 4(g - 1) +
4 _g-1

2 _+ 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

(g - 1) × (g - 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 k = g + 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 s = 0
if k = g + 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.

Oct 28, **Sam Lachterman
(UW-Madison, undergraduate)**

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.

Nov. 4**, Paul Jenkins
(UW-Madison)**

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

bad reduction".

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)

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.