Web Page of the Ottawa-Carleton Number Theory Seminar

Year 2009-2010

Speaker: Gary Walsh, CSE and University of Ottawa
Title: On a Diophantine Problem of Bennett
Abstract: In Richard Guy's book on Unsolved Problems in Number Theory, M.A. Bennett poses the problem of solving the Diophantine equation (x^2-1)/(y^2-1)=(z^2-1)^2, which is an equation derived from a certain K3 surface whose rational points are dense. We will present a solution to this and variants of Bennett's problem, and also discuss a very difficult related unsolved problem that arises from considering such variants.

Speaker: Rajender Adibhatla (Carleton University)
Title: Constructing minimally ramified characteristic 0 lifts
Abstract: I will show how one might use the deformation theory of Galois representations to construct characteristic 0 lifts of residual Galois representations. This construction usually comes at the cost of additional ramification but under certain circumstances the lift is only as ramified as the residual representation.

Speaker: Damien Roy, University of Ottawa
Title: Measures of approximation to Markoff extremal numbers
Abstract: Let xi be a real number which is neither rational nor quadratic over Q. Based on work of Davenport and Schmidt, Bugeaud and Laurent have shown that, for any real number theta, there exist a constant c>0 and infinitely many non-zero polynomials P with integer coefficients and degree at most 2 such that |theta-P(xi)| is at most c |P|^{-gamma} where gamma=(1+sqrt{5})/2 denotes the golden ratio and where the norm |P| of P stands for the largest absolute value of its coefficients. In this talk, we show conversely that there exists a class of transcendental numbers xi for which the above estimates are optimal up to the value of the constant c when one takes theta to be of the form R(xi) for a fixed polynomial R with integer coefficients of degree d=3, 4, or 5 but curiously not for degree d=6, even with theta=xi6. This is joint work with Dmitrij Zelo.

Speaker: Manfred Kolster, McMaster University
Title: Special values of L-functions
Abstract: Classically, the special value of the Dedekind zeta-function of a number field at 1 (or 0) is related to the order of the class group and logarithms of units. The values at negative integers have a similar relation to algebraic K-groups and motivic cohomology groups, at least conjecturally. In the talk I will try to explain this relationship, the tools involved in proving some of the results, and discuss related questions.

Speaker: Francesco Sica
Title: Towards a New Subexponential Factoring Algorithm
Abstract: We will outline, along the lines already developed last year, an analytic number theory approach towards factoring. The idea is to find an approximation to a arithmetic function related to the sum of divisors of N=pq in order to find p and q. We then use their generating functions, which are products of zeta functions, in order to find express this quantity in terms of a multiple series involving only calculus transcendental functions. Some of these series can be evaluated, while for the remaining ones, we will show possible avenues for their quick computation. Feedback from applied mathematicians with a knowledge of the state of the art in the calculation of multiple series would be welcome.

Speaker: Michael Dewar (University of Illinois at Urbana-Champaign)
Title: Combinatorics via modular forms
Abstract: We show how to answer some fundamental questions in combinatorial number theory using modular forms. Modular forms are analytic functions which play a central role in modern number theory. We describe a beautiful application to the theory of partitions. Ramanujan famously proved congruences modulo 5, 7, and 11 for the partition-counting function (for example, he showed that p(5n+4) = 0 modulo 5). He speculated that there were no other such congruences, and in 2003 Ahlgren and Boylan proved that this was indeed the case. We provide a broad generalization of this phenomenon. We illustrate with several examples and place this phenomenon in context by giving the exact probability of having a "Ramanujan Congruence".

Speaker:Kenneth S. Williams (Carleton University)
Title: Liouville's elementary method in number theory
Abstract: Liouville's elementary method for proving arithmetic results will be described. Speculation on where Liouville got his ideas from will be made. Several applications of Liouville's method will be given.

Speaker: Ramesh Sreekantan (Indian Statistical Institute)
Title: Conjectures on special values of L-functions
Abstract: Most mathematicians at some point or another have come across the equation zeta(2) = 1 + 1/2^2 + 1/3^2 + ... = pi^2/6 where zeta(s) = 1 + 1/2^s + 1/3^s + ... is the Riemann zeta function. At first it seems like a numerical curiosity, but it turns out to have a deeper meaning - it can be `explained' in terms of certain algebraic invariants of the rational number field Q. Beilinson, building on the work of several people before him, formulated conjectures which attempt to explain integer values of the zeta function and its generalization to L-functions of algebraic varieties over number fields. However, these conjectures have been proved in only a handful of cases. In this talk we will describe these conjectures and some generalization of these conjectures to varieties over the function field over a finite field as well as describe some of the cases where they have been resolved.

Speaker: Hugh C. Williams (U. of Calgary/Carleton U.)
Title: A Problem Concerning Divisibility Sequences
Abstract: A sequence of rational integers {A_n} is said to be a divisibility sequence if A_m | A_n whenever m | n. If the divisibility sequence {A_n} also satisfies a linear recurrence relation, it is said to be a linear divisibility sequence. The best known example of a linear divisibility sequence is the Lucas sequence {u_n(p,q)}, one particular instance of which is the famous Fibonacci sequence. One way to generalize the Lucas sequence is to consider linear divisibility sequences which have a characteristic polynomial of even degree 2k, and distinct zeros with the property that k pairs of these zeros have the same integral product. The case of k=1 is, of course, the Lucas sequence. In this talk I will discuss the case of k=2. This deceptively simple sounding investigation turns up some rather difficult problems.

Speaker: Theo Garefalakis (University of Crete)
Title: Self-reciprocal irreducible polynomials with prescribed coefficients
Abstract: We prove estimates for the number of self-reciprocal monic irreducible polynomials over a finite field of odd characteristic, that have their lowest degree coefficients fixed to given values. The estimates imply that one may specify up to m/2 - log_q(2m) values for a self-reciprocal monic irreducible polynomial of degree 2m to exist with its lowest degree coefficients fixed to those values.

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