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Number Theory at UBC

## SFU/UBC Number Theory Seminar Fall Semester 2003

 Thursday, September 4, 2003UBC Campus, MATX 1102 (Math Annex) 3:00-3:50 Chris Skinner (University of Michigan) p-adic L-functions for Unitary Groups and Stable/Endoscopic Congruences 3:50-4:10 tea break 4:10-5:00 Chris Rowe (PIMS/SFU/UBC) Coates-Wiles Towers for CM-Abelian Varieties Abstract: Let A be an abelian variety with complex multiplication. When A is an elliptic curve over the rationals of positive Mordell-Weil-rank, Coates and Wiles showed that the Hasse-Weil L-series L(A,s) vanishes at s=1. At the time, this was the best theoretical evidence for the Birch and Swinnerton-Dyer conjecture. An alternate proof was given by Gupta, by computing conductors in what he called the Coates-Wiles tower. We extend Gupta's conductor calculations to higher dimensions. But due to the lack of a suitable theory of "abelian units", a higher dimensional Coates-Wiles theorem remains unattainable. However, these conductor calculations lead to congruence relations on units, which relate to a recent conjecture of de Shalit. Thursday, September 18, 2003SFU Campus, Room K9509 (Shrum Science Building) 3:00-3:50 Friedrich Littman (PIMS/SFU/UBC) An extremal problem in Fourier analysis Abstract: This talk deals with the problem of approximating sgn(x)xn by entire functions of finite exponential type (these are essentially functions with Fourier transforms that have bounded support) and applications of this problem to analytic number theory. We will discuss Selberg's proof of a version of the large sieve inequality and - if time permits - a quantitative version of the Wiener-Ikehara Tauberian theorem due to Graham and Vaaler. Some basic knowledge of Fourier analysis on the real line is desirable. 3:50-4:10 tea break 4:10-5:00 Ben Green (PIMS/SFU/UBC) Roth's theorem in the primes Abstract: It has been known for nearly 70 years that the primes contain infinitely many arithmetic progressions of length 3, like (11,17,23). It is still unknown whether there are infinitely many progressions of length 4, let alone whether there are progressions of arbitrary length. Recently it was shown that any subset of the primes with positive density (so, for example, any set containing 0.1 percent of the primes) contains infinitely many 3-term arithmetic progressions. My aim will be to discuss this and related results. Thursday, October 2, 2003UBC Campus, MATX 1102 (Math Annex) 3:00-3:50 Jozsef Solymosi (UBC) Sums, products, and convexity Abstract: In this talk we prove results related to a conjecture of Erdos and Szemeredi. They conjectured that if A is a finite set of integers then either the sum-set A+A or the product-set AA should be large, namely |A+A| + |AA| > c|A|2-ε. If time permits, we will discuss some recent results of Bourgain, Chang, and Konyagin. 3:50-4:10 tea break 4:10-5:00 Michael Bennett (UBC) Euler's equation xy=yx revisited Abstract: A classic result of Euler characterizes the rational numbers x and y for which the equation of the title is satisfied. In this talk, we consider some generalizations of this work, from a somewhat more modern perspective, with an emphasis on (perhaps slightly surprising) connections to Transcendental Number Theory. Thursday, October 16, 2003SFU Campus, Room K9509 (Shrum Science Building) 3:00-3:50 Idris Mercer (SFU) Probabilistic methods for unimodular polynomials Abstract: Unimodular polynomials are those whose coefficients are complex numbers of unit modulus. An important special case is the Littlewood polynomials, whose coefficients are plus or minus one. If we let Un (resp. Ln) denote the set of unimodular (resp. Littlewood) polynomials of length n, then Un and Ln can be regarded as sample spaces, in which case probabilistic techniques yield information about the "flatness" of polynomials from Un or Ln on the unit circle. 3:50-4:05 tea break 4:05-4:20 Peter Borwein (SFU) The Identify function in Maple IX: What is 8.5397342226? (computer demonstration) 4:20-5:00 Stephen Choi (SFU) Quadratic equations with five prime unknowns Abstract: Let $n$ be an integer and let $b_1, \dots, b_5$ be non-zero integers. In this talk, we consider the quadratic equation $$\label{1.1} b_1p_1^2 + \cdots + b_5p_5^2 = n,$$ where $p_1, \dots, p_5$ are prime unknowns. Our goal is to prove the existence of solutions of \eqref{1.1} that do not grow too rapidly as $\max \{ |b_1|, \dots, |b_5| \} \to \infty$. The main result is \begin{theorem}\label{thm1.1} Let $b_1, \dots, b_5$ be non-zero integers satisfying $|b_1| \ge \dots \ge |b_5|,$ If $b_1, \ldots, b_5$ are not all of the same sign, then \eqref{1.1} is soluble in primes $p_1, \dots, p_5$ satisfying $$\label{1.5} p_j \ll \sqrt{|n|} + |b_1 \cdots b_4|^{2 + \ep}.$$ If $b_1, \ldots, b_5$ are all positive, the above conclusion holds when $$\label{1.6} n \gg b_5(b_1 \cdots b_4)^{4 + \ep}.$$ The implied constants in \eqref{1.5} and \eqref{1.6} depend only on $\ep$. \end{theorem} This result improves my previous result with J. Liu. This is a joint work with Angel Kumchev. Thursday, October 30, 2003UBC Campus, MATX 1102 (Math Annex) 3:00-3:50 Alexa van der Waall A demonstration of computations with differential rings and operators in Magma Abstract: After a short introduction to indicate the analogue between "conventional" Galois theory and differential Galois theory, and why it is interesting, we introduce the concepts of differential rings, fields and differential operators. Computer structures of these mathematical objects have been designed for and implemented in the computer package Magma. We give a live demonstration of some of these routines in Magma, amongst which are computing the rational solutions of an operator and constructing symmetric powers. 3:50-4:10 tea break 4:10-5:00 Nils Bruin (SFU) Arithmetic Geometry with Magma Abstract: In this talk I will outline the recent developments in the computer algebra system Magma related to arithmetic geometry. I will do so by demonstrating some worked examples and a real-time demonstration. Of special noteworthyness is the fact that the language one uses to instruct Magma to perform certain computations is particularly close to the notation and the concepts that mathematicians are used to. Thursday, November 13, 2003SFU Campus, Room K9509 (Shrum Science Building) 3:00-3:50 Patrick Ingram (UBC) Diophantine approximation and rational points on elliptic curves Abstract: In a recent paper, Wieczorek claimed that for any integers A and B satisfying 0 < |B| < A the (rational) torsion group of the elliptic curve y2 = x3+Ax+B is either trivial, Z/3Z, or Z/9Z. The talk will present our joint work with Michael Bennett, constructing (probably) infinitely many counterexamples, and the nearest possible true statement, namely that for any ε > 0 there are only finitely many integers 0 < |B|1+&epsilon < A such that the torsion group of the above curve is not the trivial group or Z/3Z. The talk should be accessible to anyone who knows what a group is. 3:50-4:10 tea break 4:10-5:00 Keshav Mukunda (SFU) Pisot numbers that are roots of Littlewood polynomials Thursday, November 27, 2003UBC Campus, MATX 1102 (Math Annex) 3:00-3:50 Ron Ferguson (MITACS/SFU/UBC) Zeros of finite Dirichlet sums Abstract: The zeros of a finite Dirichlet sum of the form $$1+1/n_1^s+1/n_2^s+\ldots+1/n_k^s,$$ where $n_1 < n_2 < \ldots < n_k$ are integers, has approximately $$T \ln(n_k)/2\pi$$ zeros in the strip $0 < Im(s) < T$. Thus, the zeros occur somewhat regularly as we move up the imaginary axis, but still their actual positions are not regular. Reducing modulo imaginary periods involving the primes dividing the $n_j's$, brings a dramatic regularity. For example, if there are only two primes, $p$ and $q$, involved, by reducing the zeros modulo a period of the form $$2\pi i/(e_p \ln(p)+e_q \ln(q))$$ with integral $e_p, e_q$, the zeros are found to lie on a curve. In general, we find that these zeros reduced into an appropriate period are dense in the image of a smooth curve. This enables us to use calculus to find bounding curves for these reduced zeros, and, in some cases, precise asymptotics for the extreme real values of zeros of the truncated zeta function. 3:50-4:10 tea break 4:10-5:00 Imin Chen (SFU) Diophantine equations and elliptic curves Abstract: I will describe some of the mechanisms behind the construction of elliptic curves used to resolve various diophantine equations. The talk will largely be expository.