SFU/UBC Number Theory Seminar 
Thursday, September 2, 2004 UBC Campus, Room WMAX 240 (PIMS) 

3:003:50  Igor Shparlinski (Macquarie University) Spherical configurations, exponential sums, and quantum computation Abstract: We describe two types of vector systems on the ndimensional 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 BakerHarmanPintz about gaps between consecutive primes. 
Thursday, September 16, 2004 UBC Campus, Room WMAX 110 (PIMS) 

3:003:50  Stephen Choi (SFU) On linear independence of certain multivariate infinite products Abstract: For integers q, m, M > 1 and positive rational numbers r_{1}, ..., r_{m}, we consider the following infinite products: F_{k} := Π_{j≥0} (1 + q^{Mjk}r_{1} + q^{2Mj2k}r_{2} + ... + q^{mMjmk}r_{m}) (0 ≤ k ≤ M1). In this talk, we show how to obtain a lower bound for the dimension of the vector space generated by F_{0}, F_{1}, ..., F_{M1} over Q. In particular, if M = m = 2, we show that at least one of the two infinite products Π_{j≥0} (1+q^{j}r+q^{2j}s) and Π_{j≥0} (1+q^{2j}r+q^{4j}s) is irrational. This is joint work with Ping Zhou. 
3:504:10  tea break 
4:105:00  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 decimaltype expansion to any base is “abnormal&rdquo, in the sense that the various baseb digits are not equally frequent in the limit. This should be the best talk of the year, among those that display the first twentythreebillionodd decimal places of a real number. 
Thursday, September 30, 2004 SFU Campus, Room K9509 

3:003:50  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 nonnegative 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.) 
3:504:10  tea break 
4:105:00  Nils Bruin (SFU) The arithmetic of Prym varieties in genus 3 Abstract: The theory of Prymvarieties for hyperelliptic curves can be approached via Kummer theory. In combination with Chabautytechniques it has given very practical methods to bound the number of rational points on hyperelliptic curves. The simplest nonhyperelliptic curves are of genus 3. We will discuss how the theory of Prymvarieties can be made effective for nonhyperelliptic 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 MordellWeil 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 Hasseprinciple and we will show how one can compute part of the BrauerManin obstruction of a genus 5 curve embedded in an Abelian surface. 
Thursday, October 14, 2004 UBC Campus, Room WMAX 216 (PIMS) 

3:003:50  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 = δ_{0} < δ_{1} < δ_{2} < ... is an increasing sequence of integers, and set s(z) = Σ_{i≥0} z^{δi}. Define the sequence {b_{n}} by s^{2}(z) = Σ_{i≥0} b_{i}z^{i}. If b_{i} > 0 for all i, then {b_{n}} is (conjecturally) unbounded. In joint work with Peter Borwein and Stephen Choi, we show that {b_{n}} cannot be bounded by 8. There is a surprisingly simple, though computationally very intensive, algorithm that establishes this. 
3:504:10  tea break 
4:105:00  Idris Mercer (SFU) Norms of zeroone 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 fournorm 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(n^{1/4}), then the probability that a randomly chosen msubset 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 

3:003:50  ChiaFu Yu (Academia Sinica, Taiwan) Basic points in moduli spaces of PELtype Abstract: In this talk we will introduce the moduli spaces of PELtype 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. 
3:504:10  tea break 
4:105:00  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 

3:003:50  Imin Chen (SFU) Quartic Qderived 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. 
3:504:10  tea break 
4:105:00  Jozsef Solymosi (UBC) Affine cubes of integers
