Number Theory at UBC

SFU/UBC Number Theory Seminar
Fall Semester 2004

Thursday, September 2, 2004
UBC Campus, Room WMAX 240 (PIMS)
3:00-3:50 Igor Shparlinski (Macquarie University)
Spherical configurations, exponential sums, and quantum computation

Abstract: We describe two types of vector systems on the n-dimensional sphere over C, which are useful for quantum computation. For one type, such configurations can be obtained from Gaussian sums for every prime n. Configurations of the other type are not known to exist for infinitely many n. We show that using bounds of exponential sums with polynomials one can achieve certain approximate solutions. The results are based on both the Weil and Weyl bounds and also the result of Baker-Harman-Pintz about gaps between consecutive primes.

Thursday, September 16, 2004
UBC Campus, Room WMAX 110 (PIMS)
3:00-3:50 Stephen Choi (SFU)
On linear independence of certain multivariate infinite products

Abstract: For integers q, m, M > 1 and positive rational numbers r1, ..., rm, we consider the following infinite products:

Fk := Πj≥0 (1 + q-Mj-kr1 + q-2Mj-2kr2 + ... + q-mMj-mkrm)   (0 ≤ k ≤ M-1).

In this talk, we show how to obtain a lower bound for the dimension of the vector space generated by F0, F1, ..., FM-1 over Q. In particular, if M = m = 2, we show that at least one of the two infinite products

Πj≥0 (1+q-jr+q-2js)   and   Πj≥0 (1+q-2jr+q-4js)

is irrational. This is joint work with Ping Zhou.

3:50-4:10 tea break
4:10-5:00 Greg Martin (UBC)
Lots of digits, lots of decimal places

Abstract: This will be a very accessible talk about two unrelated and elementary problems. The first problem is to find a number n satisfying the inequality φ(30n+1) < φ(30n), where φ is the usual Euler totient function. The second problem is to construct a number whose decimal-type expansion to any base is “abnormal&rdquo, in the sense that the various base-b digits are not equally frequent in the limit. This should be the best talk of the year, among those that display the first twenty-three-billion-odd decimal places of a real number.

Thursday, September 30, 2004
SFU Campus, Room K9509
3:00-3:50 Friedrich Littman (PIMS/SFU/UBC)
Tauberian theorems and an extremal problem in Fourier analysis

Abstract: A Tauberian theorem deduces statements about the behavior of the summatoric function of a non-negative sequence (a_n) from knowledge about the Dirichlet series of (a_n) in a (right) halfplane. Let f(x) = exp(-rx) for x>0 and f(x) = 0 for x<0. In this talk I will speak about a theorem of Graham and Vaaler in which they used approximations to f(x) by bandlimited functions as a tool to treat the case of Dirichlet series for which only the behavior in a horizontal half strip is known, and I will discuss a way to generalize it. (A function is called bandlimited if its Fourier transform has bounded support.)

3:50-4:10 tea break
4:10-5:00 Nils Bruin (SFU)
The arithmetic of Prym varieties in genus 3

Abstract: The theory of Prym-varieties for hyperelliptic curves can be approached via Kummer theory. In combination with Chabauty-techniques it has given very practical methods to bound the number of rational points on hyperelliptic curves. The simplest non-hyperelliptic curves are of genus 3. We will discuss how the theory of Prym-varieties can be made effective for non-hyperelliptic curves of genus 3 and how this can be applied to a variety of problems. As an example, we will determine the rational points on a curve of genus 3 with no specific geometric properties, without computing the Mordell-Weil group of its Jacobian. We will also give an example of a curve of genus 3 and a curve of genus 5 that violate the Hasse-principle and we will show how one can compute part of the Brauer-Manin obstruction of a genus 5 curve embedded in an Abelian surface.

Thursday, October 14, 2004
UBC Campus, Room WMAX 216 (PIMS)
3:00-3:50 Frank Chu (UBC)
An old conjecture of Erdős and Turán on additive bases

Abstract: There is a 1941 conjecture of Erdős and Turán, on what are now called additive bases, that is stated as follows: Suppose that 0 = δ0 < δ1 < δ2 < ... is an increasing sequence of integers, and set s(z) = Σi≥0 zδi. Define the sequence {bn} by s2(z) = Σi≥0 bizi. If bi > 0 for all i, then {bn} is (conjecturally) unbounded. In joint work with Peter Borwein and Stephen Choi, we show that {bn} cannot be bounded by 8. There is a surprisingly simple, though computationally very intensive, algorithm that establishes this.

3:50-4:10 tea break
4:10-5:00 Idris Mercer (SFU)
Norms of zero-one polynomials and the ubiquity of Sidon sets

Abstract: There is a rich literature that concerns the norms of polynomials with restricted coefficients. One of many questions one can ask on this theme is: What is the expected value of the fourth power of the four-norm of a polynomial whose coefficients are each 0 or 1? We show that for the natural ways to interpret this question, the expected values can be calculated explicitly. One of the expressions thus calculated yields a new proof of the following, which is a special case of results due to Nathanson and others. If m = o(n1/4), then the probability that a randomly chosen m-subset of {1,2,...,n} is a Sidon set approaches 1 as n approaches infinity. (A Sidon set is one where all differences of pairs of elements are distinct.)

Thursday, October 28, 2004
SFU Campus, Room K9509
3:00-3:50 Chia-Fu Yu (Academia Sinica, Taiwan)
Basic points in moduli spaces of PEL-type

Abstract: In this talk we will introduce the moduli spaces of PEL-type and the notion of basic points in the moduli spaces modulo a prime p according to Kottwitz. Then we sketch a proof of the existence of basic points and describe a connection with Hecke orbit problems.

3:50-4:10 tea break
4:10-5:00 Mike Bennett (UBC)
Generalized Fermat equations

Abstract: Following the groundbreaking work of Wiles proving Fermat's Last Theorem, there has been an upswing of interest in Diophantine equations. In this talk, I will survey the “state of play” in the field and highlight some recent results ofinterest.

Thursday, December 2, 2004
UBC Campus, Room WMAX 216
3:00-3:50 Imin Chen (SFU)
Quartic Q-derived polynomials with distinct roots

Abstract: Buchholz and MacDougall classified all polynomials over Q such that they and all of their derivatives have their roots in Q, but subject to a conjecture that there are no quartic polynomials with distinct roots having this property (the other required conjecture concerning quintics with a triple root was solved by Victor Flynn). In this expository talk, I will give a brief introduction to this problem and explore some elementary aspects of the geometry behind the surfaces arising from the quartic distinct root case.

3:50-4:10 tea break
4:10-5:00 Jozsef Solymosi (UBC)
Affine cubes of integers