




Number Theory / Théorie des nombres (Rajiv Gupta and Nike Vatsal, Organizers)
 MICHAEL BENNETT, University of Illinois at UrbanaChampaign, Illinois, USA
Applications of the hypergeometric method to
polynomialexponential equations

We generalize and refine the socalled hypergeometric method in
Diophantine approximation, to apply it to certain
polynomialexponential equations. Sharpening work of Beukers, we
deduce sharp bounds on the number of solutions to families of
generalized RamanujanNagell equations, including those of the form
x^{2}+D = p^{n}, for a fixed prime p and nonzero integer D. This is
joint work with Mark Bauer (also University of Illinois).
 DAVID BOYD, University of British Columbia, Vancouver, British Columbia
Mahler's measure and the dilogarithm

The Mahler measure m(P) of a polynomial P(x,y) is its geometic mean
over the 2torus. We will describe various situations in which
m(P) can be evaluated as a sum of BlochWigner dilogarithms of
algebraic numbers. This leads to infinitely many examples in which
m(P) is a rational multiple of d^{1/2}z_{F}(2)/p^{2n1}, where
F is a number field of degree n and discriminant d with exactly
one nonreal embedding into C. These formulas can be
regarded as generalizing Smyth's well known evaluation of m(1+x+y) in
terms of z_{F}(2), where F = Q(Ö3), and are
manifestations of the theme ``period = Lvalue''.
 JOHN FRIEDLANDER, University of Toronto, Toronto, Ontario
Class group Lfunctions

We discuss joint work with W. Duke and H. Iwaniec, on the title topic,
extending over several years and just recently reaching culmination.
 EYAL GOREN, McGill University, Montreal, Quebec H3A 2K6
Hilbert modular varieties over ramified primes

We shall report on a joint work with F. Andreatta on the geometry and
arithmetic of Hilbert modular varieties over ramified primes. In
particular, we discuss certain strata and the theory of modular forms.
This complements previous works of the speaker (partly joint with Oort
and Bachmat) in the unramified case, and we shall compare the ramified
with the unramified situation. Furthermore, we shall indicate
applications to congruences for abelian L functions and to filtration
on qexpansions.
 DAVID MCKINNON, Tufts University, Medford, Massachusetts 02155, USA
Vojta's conjectures and rational points

In this talk, I will discuss how weakened versions of Vojta's
Conjectures imply results on how many rational points there are on
curves. I will then discuss the proof of Vojta's Main Conjecture for
certain surfaces and how this implies unconditional quantitative
results on the numbers of points on curves on these surfaces.
 KUMAR MURTY, University of Toronto, Toronto, Ontario M5S 3G3
Discrete logs on Elliptic curves and rank one liftings

We discuss the Elliptic curve discrete logarithm problem and its
relation to lifting of points to curves of rank one.
 RAM MURTY, Queen's University, Kingston, Ontario K7L 3N6
The Euclidean algorithm

We will discuss some recent joint work with Malcolm Harper on the
classification of Euclidean rings of integers of number fields. The
generalized Riemann hypothesis predicts that the ring of integers of an
algebraic number field whose unit group is infinite and which is a PID
is necessarily Euclidean (though not necessarily for the norm
mapping). We will describe our recent unconditional treatment of this
prediction.
 KEN ONO, University of Wisconsin, Wisconsin, USA
qseries identities and values of Lfunction

Recently, Zagier proved a qseries identity related to Dedekind's
etafunction. This identity was important in his work on Vassiliev
invariants in knot theory. Here we present several infinite families
of such identities. These identities are then used to produce
generating functions for the values of certain Lfunctions at
negative integers. This is joint work with G. E. Andrews and
J. JimenezUrroz.
 CHRISTOPHER SKINNER, University of Michigan, Ann Arbor, Michigan, USA
Denominators of some Eisenstein classes for GL(3)

Let S_{3} be the symmetric space associated to the reductive group
GL(3). We define some classes in the singular cohomology of S_{3}
associated to Eisenstein series built up from cusp forms on GL(2).
We relate the denominators of these Eisenstein classes (which measures
their failure to be integral) to certain special values of the
Lfunctions of the cusp forms.
 CAMERON STEWART, University of Waterloo, Waterloo, Ontario N2L 3G1
On intervals with few prime numbers

In this talk we shall discuss some recent joint work with H. Maier.
Our objective is to prove that there exist relatively long intervals
which contain few primes. Our results interpolate between and include
the result of Rankin from 1938 on large gaps between consecutive primes
and the result of Maier from 1985 on intervals starting at x of
length a power of logx which have fewer than the expected number of
primes.
 SIMAN WONG, University of Massachusetts, Amherst, Massachusetts, USA
On the rank of Jacobian Fibrations (progess report)

We apply Tate's conjecture on algebraic cycles to study the
NeronSeveri groups of fiber products of fibered varieties. This is
inspired by the work of Rosen and Silverman, who carry out such an
analysis to derive a formula for the rank of the group of sections of
an elliptic surface. Using a Shioda Tate formula for fibered surfaces
plus an analysis of the precise form of the local zetafunctions of the
singular fibers, we propose, under Tate's conjecture, to give an
formula for the rank of the Jacobian fibration associated to a fibered
surface, and to give a conjectural description of the NéronSeveri
group of Kuga fiber varieties.

