Number Theory at UBC

SFU/UBC Number Theory Seminar
Fall Semester 2006

Thursday, September 21, 2006
SFU Campus, Room ASB 10900 (IRMACS)
3:00-3:50 Nils Bruin (SFU)
Deciding the existence of rational points on curves

Abstract: While it is known that Hilbert's 10th problem - deciding whether a polynomial equation has integral solutions - has no automatic solution, one can still hope that for subclasses of polynomial equations and for rational solutions, such an algorithm might exist. Recently, experiments and theoretical work inspired by these experiments have provided some quite convincing evidence that for rational points on projective curves, such an algorithm does indeed exist and that we in fact already know the algorithm. I will outline this algorithm and indicate the heuristics that indicate it is correct.

3:50-4:10 tea break
4:10-5:00 Imin Chen (SFU)
Diophantine equations via Galois representations

Abstract: Recently, the use of Galois representations attached to elliptic curves has been used to resolve several cases of the generalized Fermat equation. In this talk, I will discuss the method and some further cases which can be analyzed at least partially, including the equation a2 + b2p = cr, where r = 3 or 5. Although a complete resolution is not yet possible, a computational criterion can be obtained for r = 3, based on previous work by Bennett-Skinner and Kraus. For r = 5, I outline a possible strategy using a combination of quadratic Q-curves and elliptic curves over Q.

Thursday, October 5, 2006
UBC Campus, Room WMAX 110 (PIMS)
3:00-3:50 Igor Pritsker (Oklahoma State University)
Polynomial inequalities, Mahler's measure, and multipliers

Abstract: We shall discuss polynomial inequalities for integral norms defined by the contour and the area integrals over the unit circle. The common feature of these inequalities is that they are obtained by using coefficient multipliers. Special attention will be devoted to Mahler's measure.

3:50-4:10 tea break
4:10-5:00 Matilde Lalin (PIMS/SFU/UBC)
Functional equations for Mahler measures of genus-one curves
(part of a joint work with Mat Rogers)

Abstract: The Mahler measure of an n-variable polynomial P is the integral of log |P| over the n-dimensional unit torus Tn with the Haar measure. Consider a family of two-variable polynomials whose coefficients depend on one parameter. Then the Mahler measure is a function of that parameter. Mat Rogers has discovered several examples for which this function satisfies functional equations. They all correspond to families of elliptic curves. We may deduce these functional equations from modularity properties or evaluations of elliptic regulators following works by Rodriguez-Villegas, Zagier, Deninger, etc.

Thursday, October 19, 2006
SFU Campus, Room ASB 10900 (IRMACS)
3:00-3:50 Jason Bell (SFU)
Christol's theorem and quasi-automatic functions

Abstract: A theorem of Christol states that a power series expansion f(t) ∈ Fq[[t]], with q a power of a prime p, is algebraic over Fq[t] if and only if the sequence of coefficients of f(x) is “p-automatic”, that is, there is a finite state machine which inputs the base p-expansion of a number n and outputs the coefficient of tn in f(t). Kedlaya pointed out that it is more natural to work in the ring of Hahn power series Fq((tQ)), since Fq[[t]] is not algebraically closed. We will discuss his analogue of Christol's theorem for Hahn power series in terms of quasi-automatic functions. Furthermore, we give a quasi-automatic analogue of a theorem of Cobham which states that if a sequence is k- and l-automatic and k and l are multiplicatively independent then the sequence is eventually periodic. We show that if f is k-quasi-automatic and l-quasi-automatic and k and l are multiplicatively independent, then f is “quasi-periodic”, a property which is very similar to being periodic. This is joint work with Boris Adamczewski.

3:50-4:10 tea break
4:10-5:00 Michael Coons (SFU)
General moment theorems for non-discinct unrestricted partitions
(joint work with Klaus Kirsten, Baylor University)

Abstract: A well-known result from Hardy and Ramanujan gives an asymptotic expression for the number of possible ways to write an integer as the sum of smaller integers. In this vein, we consider the general partitioning problem of writing an integer n as a sum of summands from a given sequence L of non-decreasing integers. Under suitable assumptions on the sequence L, we obtain results using the associated zeta-function and saddle-point techniques. We also calculate higher moments of the sequence L as well as the expected number of summands and the variance. Then applications are made to various sequences, including those of Barnes and Epstein types.

Thursday, November 2, 2006
UBC Campus, Room WMAX 110 (PIMS)
3:00-3:50 Chris Sinclair (PIMS/SFU/UBC)
Heights of polynomials and random matrix theory

Abstract: We will discuss a method for producing asymptotic estimates for the number of integer polynomials of degree N with bounded (but large) Mahler's measure. This method also produces a closed form for averages of class functions over ensembles of asymmetric random matrices. In this talk I will explain why this is important, and its potential for resolving some open problems surrounding certain ensembles of random matrices.

3:50-4:10 tea break
4:10-5:00 Shabnam Akhtari (UBC)
The Diophantine equation aX4 - bY2 = 1

Abstract: In a series of papers over nearly forty years, Ljunggren derived remarkably sharp bounds for the number of solutions to various quartic Diophantine equations, particularly those of the shape aX4 - bY2 = ±1, typically via a sophisticated application of Skolem's p-adic method. More recent results along these lines are well surveyed in a paper of Walsh. For general a and b, however, there is no absolute upper bound for the number of integral solutions to aX4 - bY2 = 1 available in the literature. Computations and assorted heuristics suggest the following conjecture of Walsh: For any positive integers a and b, the equation aX4 - bY2 = 1 has at most two solutions in positive integers X and Y. In this talk, we will appeal to a classical result of Thue from the theory of Diophantine approximation to deduce the following result: For any positive integers a and b, the equation aX4 - bY2 = 1 has at most three solutions in positive integers X and Y.

Thursday, November 9, 2006
SFU Campus, Room ASB 10900 (IRMACS)
3:00-3:50 Yoonjin Lee (SFU)
Construction of cubic function fields from quadratic infrastructure

Abstract: We present an efficient method for generating non-conjugate cubic function fields of a given squarefree discriminant, using the infrastructure of the dual real function field assisociated with the hyperelliptic field of the same discriminant. This method was first proposed by Shanks for number fields in an unpublished manuscript from the 1970s.

3:50-4:10 tea break
4:10-5:00 Karl Dilcher (Dalhousie)
A Pascal-type triangle characterizing twin primes

Abstract: It is a well-known property of Pascal's triangle that the entries of the kth row, without the initial and final entries 1, are all divisible by k if and only if k is prime. In this talk I will present a triangular array similar to Pascal's that characterizes twin prime pairs in a similar fashion. The proof involves generating function techniques. Connections with orthogonal polynomials, in particular Chebyshev polynomials, will also be discussed. If time allows, I will talk about another triangle, the so-called Stern-Brocot tree, and about some recent work on related number and polynomial sequences.

Thursday, November 23, 2006
UBC Campus, Room MATH 100
3:00-3:50 Luis Goddyn (SFU)
Two lower bounds for subset sums

Abstract: I present two new results in additive number theory. First, let A be a finite subset of an abelian group G, and let S(A) denote the set of all group elements representable as a sum of a subset of A. We derive the following quadratic lower bound on the size of S(A): |S(A)| ≥ |H| + |A-H|2/64. Here H = {g ∈ G : S(A)+g = S(A)} is the stabilizer of S(A). It may be possible to improve our constant 1/64, but not beyond 1/4. This implies a result of Erdös/Heilbronn, and improves a difficult theorem of Van Vu regarding the integers modulo n. Second, let A = (A1, A2, ..., An) be a sequence of subsets of an abelian group, and let Sk(A) denote the set of group elements representable as a sum of k elements taken from distinct sets in A. Let H be the stabilizer of Sk(A). We prove the following generalization of Kneser's theorem: |Sk(A)| ≥ |H| (1 - k + ΣQ min{k, r(A,Q)}). Here the sum runs over all H-cosets Q, and r(A,Q) is the number of indices i for which Ai contains an element of Q. This result is a very special case of a little-known conjecture of Schrijver and Seymour regarding group-weighted matroids. However it is already strong enough to imply a large number of results and conjectures by Cao, Gao, Grynkiewicz, Hamidoune, Bollobás, and Leader, mostly in the spirit of the following classic result of Erdös, Ginsberg, and Ziv: Every set of 2n-1 integers contains an n-subset which sums to 0 modulo n. This is joint work with Matt DeVos, Bojan Mohar, and Robert Šámal.

3:50-4:00 tea break
4:00-4:50 Lior Silberman (Harvard)
Arithmetic quantum chaos in the higher-rank case

Abstract: I shall discuss joint work with Akshay Venkatesh on the quantum unique ergodicity conjecture for locally symmetric spaces. In the case of a (co-compact) lattice in PGL3(R) associated to an order in a division algebra of degree 3 over Q, we show that any non-degenerate sequence of Hecke-Maass eigenforms becomes equidistributed in the measure-theoretic sense. We first reduce the problem to showing the equidistribution of a limit measure on the homogeous space of the lattice which is invariant under the action of a Cartan subgroup. By recent measure rigidity results it then suffices to show that elements of the Cartan subgroup act with “positive entropy”. I will describe this property and how we establish it using harmonic analysis on the building and a (global) diophantine argument on the group.

Thursday, December 7, 2006
SFU Campus, Room ASB 10900 (IRMACS)
1:00-1:50 David Boyd (UBC)
Mahler's measure and L-functions of elliptic curves evaluated at s = 3

Abstract: The logarithmic Mahler measure m(P) of a polynomial P in n variables is the average of log |P| over the product of n circles. A few years ago, we conjectured infinitely many formulas evaluating the Mahler measure of certain two-variable polynomials as rational multiples of L(E,2)/π2, where L(E,s) is the L-function of a suitable elliptic curve. A finite number of these formulas have now been proved by Rodriguez-Villegas and more recently by Brunault. In this talk, we will present some experimentally discovered conjectural formulas for some three-variable polynomials as rational linear combinations of L(E,3)/π4, ζ(3)/π2, and various classical dilogarithms. For example, to 40-decimal-place accuracy, m((x-1)3 + (x+1)(y+z)) = (21/2) L(E,3)/π4 = 6 L'(E,-1), where E is an elliptic curve of conductor 14. So far, none of these formulas have been proved.

1:50-2:10 tea break
2:10-3:00 Erick Wong (UBC)
Combinatorial properties of {x2+ky2}

Abstract: For a fixed k > 0, consider S(k), the set of integers representable as x2+ky2. Answering a question of M. Rosenfeld, we consider the problem of determining the maximum number of consecutive terms of this sequence, as well as the spacing between terms. We will show that there are infinitely many k for which the set S(k) contains infinitely many 5-tuples of consecutive integers, and that this length is best possible. Similarly, we show that for all k > 0 not divisible by 4, the set S(k) contains infinitely many pairs of consecutive elements exactly d apart for every d > 0.