Number Theory at UBC

SFU/UBC Number Theory Seminar
Fall Semester 2005

Thursday, September 29, 2005
UBC Campus, Room WMAX 110
3:00-3:50 Chris Sinclair (PIMS/SFU/UBC)
Counting reciprocal polynomials with integer coefficients

Abstract: I'll introduce a method which will allow us to produce asymptotic estimates for the number of reciprocal polynomials with integer coefficients, bounded degree and bounded Mahler measure. This method can be extended to other multiplicative measures of complexity of polynomials. Time permitting, I will demonstrate a connection between these topics and physics.

3:50-4:10 tea break
4:10-5:00 Mike Bennett (UBC)
Integer points on congruent number curves

Abstract: An integer N is called a congruent number if there exists a right-angled triangle with area N, all three of whose sides have rational lengths. Equivalently, N is a congruent number if the elliptic curve y2 = x3 - N2x has positive Mordell-Weil rank over the rational numbers. In this talk, we consider the question of finding integer points on these elliptic curves and discuss a number of related issues.

Thursday, October 13, 2005
SFU Campus, Room ASB 10900
3:00-3:50 Vishwa Dumir (Panjab University)
View obstruction problems and their generalizations

Abstract: In the view obstruction problem, congruent closed convex bodies centered at the points (½, ..., ½) + Zn in Rn are expanded uniformly until they block all rays from the origin into the open positive cone. The central problem is to determine the minimal blocking size. We shall trace the history of the view obstruction problems. Then we shall study a more general problem in which rays are replaced by subspaces of dimension d. This leads to a nice method for solving the view obstruction problems and obtaining isolation results for spheres. A new generalization is obtained by replacing subspaces by flats. This is related to Schoenberg's billiard ball motion problem. The view obstruction problem for boxes is equivalent to the Lonely Runner Problem, which has been solved for n ≤ 5 by combinatorial methods.

3:50-4:10 tea break
4:10-5:00 Jason Bell (SFU)
A generalized Skolem-Mahler-Lech theorem for affine varieties

Abstract: The Skolem-Mahler-Lech theorem says that if f(n) is a linear recurrence over a field of characteristic 0, then the set of n such that f(n)=0 is a finite union of arithmetic progressions possibly augmented by a finite set. In this talk, we prove the following generalization of this result: Let X be an affine variety over a field of characteristic 0 with an automorphism σ and subvariety Y. If x is a point in X, then the set of integers n such that σn(x) ∈ Y is a finite union of two-way arithmetic progressions possibly augmented by a finite set. We discuss how this is a generalization of the classical Skolem-Mahler-Lech theorem and discuss recent progress in obtaining a positive characteristic analogue of this result.

Thursday, October 27, 2005
UBC Campus, Room WMAX 110
3:00-3:50 Nigel Boston (University of Wisconsin)
Galois groups of p-extensions and applications

Abstract: Unlike the ramified-at-p case, Galois groups of p-extensions unramified at p are poorly understood, partly because they should have only finite-image p-adic representations. We present an analogous theory of arboreal representations, a nonabelian Jugendtraum, and nonabelian Cohen-Lenstra heuristics. Applications of this theory to root-discriminant problems, algebraic topology, and group theory have emerged.

3:50-4:10 tea break
4:10-5:00 Patrick Ingram (UBC)
Primitive divisors in elliptic divisibility sequences

Abstract: A divisibility sequence is a sequence of intregers {an} with the property that an divides am whenever n divides m. We will survey the literature on divisibility sequences and consider various problems relating to elliptic divisibility sequences, a certain class of sequences arising from the study of elliptic curves. In particular, we will ask whether the terms in such a sequence must eventually all have primitive divisors, that is, divisors that divide no previous term in the sequence.

Thursday, November 10, 2005
SFU Campus, Room ASB 10900
3:00-3:50 Greg Martin (UBC)
Smooth numbers, primes, and Egyptian fractions

Abstract: This will be an expository talk about smooth numbers (numbers without large prime factors) and their connection to the distribution of prime numbers, particularly prime values of polynomials. An application to questions on Egyptian fractions (sums of reciprocals of distinct positive integers) will ensue, mainly as an excuse to bring in some of the speaker's old research. The entire talk will be accessible to students and non-experts.

3:50-4:10 tea break
4:10-5:00 Peter Borwein (SFU)
The Riemann Hypothesis - not necessarily for experts

Abstract: to be announced

Thursday, November 24, 2005
UBC Campus, Room WMAX 110
3:00-3:50 Yoonjin Lee (SFU)
The structure of the class groups of global function fields of any unit rank

Abstract: The problem of determining the structure of the class group dates back to Gauss. In this talk we discuss the structure of the class groups of global function fields. Let F be a finite field and T a transcendental element over F. We show an explicit method of constructing, for positive integers m, n, and r with 0 ≤ r ≤ m-1, infinitely many global function fields K of degree m over F(T) such that K has a given unit rank r and the ideal class group of K contains a subgroup isomorphic to (Z/nZ)m-r.

3:50-4:10 tea break
4:10-5:00 Stephen Choi (SFU)
Why Lehmer's conjecture is so difficult

Abstract: We compute asymptotic formulas for mean values of Mahler's measure and the Lp-norms of several classes of polynomials with restricted coefficients and bounded degree. We study the unimodular polynomials with complex coefficients of modulus 1, the Littlewood polynomials with {-1,1} coefficients, the height-1 polynomials with {-1,0,1} coefficients, and their restrictions to reciprocal polynomials. We show for instance that both the geometric mean and the arithmetic mean of Mahler's measure of the unimodular polynomials with degree n-1 approach e-γ/2n1/2 as n grows large, and that this same result holds for the Littlewood polynomials. Moreover, we prove that the normalized measure or Lp-norm of a polynomial in one of these families lies arbitrarily close to the corresponding mean value with probability approaching 1 as the degree grows large.

Thursday, December 8, 2005
SFU Campus, Room ASB 10900
3:00-3:50 Chris Sinclair (PIMS/SFU/UBC)
Determinants, Pfaffians, and volumes of polynomials

Abstract: I will outline a method for computing certain volumes of polynomials of interest to number theorists, random matrix theorists, and physicists.

3:50-4:10 tea break
4:10-5:00 Mat Rogers (UBC)
Multivariable Mahler measures and related integrals

Abstract: In this talk, we will discuss several new formulas for three-variable Mahler measures. This work generalizes some recent formulas of Condon and Lalin. The identities are proved by relating three-variable Mahler measures to hypergeometric functions.