links to: Fall 2000 | Spring 2001 | Fall 2001 |

Thursdays, 11:15-12:05, 103 McAllister

*Periods of Drinfeld modules
with complex multiplication*

We investigate transcendence properties
of periods of Drinfeld modules with complex multiplication. In particular
we show that if such Drinfeld modules have different CM fields then their
fundamental periods are algebraically independent over the algebraic numbers.
Joint work with Dale Brownawell.

**January 25
Robert Vaughan (PSU)**

*Waring's problem: A Survey*

Recent work on* G(k)
*in
Waring's problem, jointly with T. D. Wooley, will be described and placed
in a historical context. Some speculations will be made about future
directions in the Hardy-Littlewood method.

**February 1
No seminar: Peter Sarnak (Princeton University and the Institute
for Advanced study) is giving the Russell Marker Lectures in Mathematics**

*L-Functions, Arithmetic and
Semiclassics*

Monday, January 29, 8:00 p.m., 112
Osmond Laboratory, *Hilbert's Eleventh Problem*

Tuesday, January 30, 4:30 p.m., 112
Osmond Laboratory, *Lp Norms of Eigenfunctions*

Wednesday, January 31, 4:30 p.m.,
112 Osmond Laboratory, *Quantum Unique Ergodicity*

Thursday, February 1, 4:30 p.m.,
112 Osmond Laboratory, *Families of L-Functions and Symmetry*

**February 8
Sanju Velani (Queen Mary Westfield College, London), 103 McAllister**

*On simultaneously badly approximable
pairs*

For any pair $i,j \geq 0$ with $i+j
=1$ let $\Bad(i,j)$ denote the set of pairs $(\a,\b) \in \R^2$ for
which $ \max \{ ||q\a||^{1/i}\, ||q\b||^{1/j} \} > c/q $ for all $ q \in
\N $. Here $c = c(\a,\b)$ is a positive constant. If $i=0$ we identify
the set $\Bad(0,1)$ with $\R \times \Bad $ where $\Bad$ is the set of badly
approximable numbers. That is, $\Bad(0,1)$ consists of pairs $(\a,\b)$
with $\a \in \R$ and $\b \in \Bad$. If $j=0$ the roles of $\a$ and $\b$
are reversed. We prove that the set $\Bad(1,0)\cap \Bad(0,1) \cap \Bad(i,j)$
has Hausdorff dimension 2, i.e. full dimension. The method easily generalizes
to give analogous statements in higher dimensions.

**February 15
Mark Watkins (PSU), 10:10, 103 Mcallister: NOTE earlier time.**

*Special values of L-functions
and modular parametrisations of elliptic curves*

Formulae which relate L-values to arithmetic information can be viewed
in two directions: you can first compute the arithmetic information to
determine the size of the L-value, or conversely you can compute the special
L-value by a separate method, thus gaining arithmetic information.
The generic method for computing special L-values (actually any L-value)
goes back to Cohen and Zagier in the 1970s, but only recently has it appeared
in full generality in print (appendix of Cohen's latest book). We
describe how their method works, and then use it in a specific example,
namely the computation of the degree of modular parametrisation of an elliptic
curve. In fact, we give data from a large-scale project to compute modular
degrees, with over 40000 curves considered.

**February 22
Edward Formanek (PSU)**

*A relation between the Bezoutian
and the Jacobian*

**March 1
Trevor Wooley (University of Michigan) : Colloquium**

*Slim exceptional sets in Waring's
problem*

A result of Hua from 1938 shows that
the expected asymptotic formula holds in Waring's problem for sums of four
squares of primes for almost all integers in the expected residue classes,
in the sense that the number of exceptions up to N is O(N(log N)^{-A}).
This estimate was recently improved by Liu and Liu to O(N^{13/15+epsilon}).
From a naive viewpoint, both conclusions are surprisingly weak, in the
sense that a similar conclusion holds already for sums of three squares
of primes, and the excess square of a prime brings a negligible improvement
in the estimate for the exceptional set. This phenomenon permeates
the subject, especially when the variables under consideration are from
such exotic sets as the prime numbers or integers possessing only small
prime factors. We present a method for better exploiting excessive variables,
especially exotic variables, and thereby slim down the available estimates
for associated exceptional sets in various problems of Waring type. By
way of illustration, we establish that the above exponent 13/15 may now
be replaced by 13/30. As with the best miracle diet plans, this slimming
process

involves almost no effort.

**March 15
Tonghai Yang (University of Wisconsin)**

*Taylor expansion of an Eisenstein
series*

In this talk, we will give an analogue
of the well-known Kronecker limit formula for a classical Eisenstein series
(with character). In this case, the Eisenstein series is holomorphic at
its center and its central value is given by theta functions via the Siegel-Weil
formula. We will give an explicit formula for its central derivative. We
will also use the formula to compute the central derivative of certain
Hecke L-functions, which are related to CM elliptic curves.

**March 22
Joel Anderson (PSU)**

**March 29
Scott Parsell (Texas A&M)**

*Pairs of additive equations
and inequalities*

We will discuss recent progress on
obtaining upper bounds for the number of variables required to ensure that
a pair of diagonal forms of differing degree, satisfying appropriate local
solubility conditions, has a non-trivial integral zero. The arguments
are based on the Hardy-Littlewood method, and in particular on the iterative
methods of Vaughan and Wooley for estimating mean values of exponential
sums over smooth numbers. Our estimates can also be applied to the
corresponding problem for inequalities, in which one tries to show that
two forms with real coefficients assume arbitrarily small values simultaneously
at integral points.

**April 5
David Farmer (Bucknell)**

*Deformation of Maass forms*

Phillips and Sarnak conjecture that Maass forms on cofinite subgroups
of SL(2,R) are destroyed by almost all deformations of the group.
Some calculations will be described which indicate that "almost all" cannot
be replaced by "all." Time permitting, the dynamics of the
motion of the Maass forms under deformation will also be discussed.

**April 12
William Stein (Harvard)**

*Visibility of Mordell-Weil groups*

I will introduce the notions of visibility and modularity of Mordell-Weil
groups of abelian varieties. My notion of visibility is analogous
and dual to Barry Mazur's notion of visibility of Shafarevich-Tate groups.
In my talk, I will make conjectures about visibility of Mordell-Weil groups,
prove that Mordell-Weil groups of certain elliptic curves are visible in
an appropriate restriction of scalars, and give some explicit examples.
If time permits, I will discuss connections with the Birch and Swinnerton-Dyer
conjecture for elliptic curves of analytic rank greater than one.

**April 19
Wenzhi Luo (Ohio State University)**

*Equidistribution of Hecke eigenforms
on modular surface*

For the holomorphic Hecke eigenforms of weight 2k, one can associate with it naturally a probability measure $mu _{k}$ on the modular surface X. We show that

\mu _{k}(A) = \mu (A) + O(k^{-1/2})

holds uniformly for any set A on
X as k tends to infinity, where $\mu$ is the invariant measure associated
to the Poincare metric. Moreover the above decay rate is sharp. This equidistribution
property of Hecke eigenforms can be regarded as an analogue of ergodicity
of Laplacian eigenfunctions.

**April 26
David Boyd (University of British Columbia): Colloquium**

*Mahler's measure and the Bloch group*