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Number Theory / Théorie des nombres
(Rajiv Gupta and Nike Vatsal, Organizers)

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MICHAEL BENNETT, University of Illinois at Urbana-Champaign, Illinois, USA
Applications of the hypergeometric method to polynomial-exponential equations

We generalize and refine the so-called hypergeometric method in Diophantine approximation, to apply it to certain polynomial-exponential equations. Sharpening work of Beukers, we deduce sharp bounds on the number of solutions to families of generalized Ramanujan-Nagell equations, including those of the form x²+D = pⁿ, 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 2-torus. We will describe various situations in which m(P) can be evaluated as a sum of Bloch-Wigner dilogarithms of algebraic numbers. This leads to infinitely many examples in which m(P) is a rational multiple of d^1/2z_F(2)/p^2n-1, where F is a number field of degree n and discriminant d with exactly one non-real 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 = L-value''.

JOHN FRIEDLANDER, University of Toronto, Toronto, Ontario
Class group L-functions

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 q-expansions.

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
q-series identities and values of L-function

Recently, Zagier proved a q-series identity related to Dedekind's eta-function. 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 L-functions at negative integers. This is joint work with G. E. Andrews and J. Jimenez-Urroz.

CHRISTOPHER SKINNER, University of Michigan, Ann Arbor, Michigan, USA
Denominators of some Eisenstein classes for GL(3)

Let S₃ be the symmetric space associated to the reductive group GL(3). We define some classes in the singular cohomology of S₃ 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 L-functions 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 Neron-Severi 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 zeta-functions 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 Néron-Severi group of Kuga fiber varieties.