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Number Theory at UBC

## SFU/UBC Number Theory Seminar Winter Semester 2004

 Thursday, January 15, 2004SFU Campus, Room K9509 (Shrum Science Building) 4:00-4:50 Mark Watkins (Penn State) Solving Systems of Polynomial Equations via Multidimensional p-adic Newton iteration Abstract: The most common methods to solve systems of polynomial equations are (multi)resultants and Groebner bases. Both of these quickly become tedious. Solving systems of equations can be done with a multidimensional Newton method, though the region of convergence is poor, partially due to the norm. I shall describe how to find rational and algebraic solutions to polynomial systems using p-adic techniques, and explain a few motivating problems (suggested by Noam Elkies) that led to my investigation of this subject. Thursday, February 12, 2004SFU Campus, Room K9509 (Shrum Science Building) 3:00-3:50 Igor Shparlinski (Macquarie University, Sydney, Australia) Character sums and congruences with n! Abstract: We estimate sums of multiplicative characters and the number of solutions of various congruences modulo a prime p involving n!. In particular we show that there is n = O(p1/2 + ε) such that n! is a primitive root modulo p. We also discuss several issues related to the size of the value set { n! (mod p) : n ≤ N }. (joint work with Moubariz Garaev and Florian Luca) 3:50-4:10 tea break 4:10-5:00 Kevin O'Bryant (University of California, San Diego) A one-dimensional tiling problem Abstract: A tiling is a finite collection of disjoint sets whose union is Z. A nice exercise is to show that in any tiling with congruence classes, two of the moduli are equal. I present work (joint with R. Graham and G. Martin) on tilings with sets of the form { [ nx+y ] : n ∈ Z }, where [ . ] is the floor function and x, y are real numbers. Friday, February 27, 2004SFU Campus, Room K9509 (Shrum Science Building) 10:30-11:45 Andy Pollington (Brigham Young University) to be announced Thursday, March 11, 2004UBC Campus, MATH 104 (Math Building) 3:00-3:50 Peter Borwein (SFU) The Mahler measure of polynomials with odd coefficients Abstract: We resolve an old conjecture of Schinzel and Zassenhaus for the class of polynomials with no cyclotomic factors whose coefficients are all odd. More generally, we solve the problems of Lehmer for irreducible polynomials in the above class by showing that the Mahler measure of such polynomials is bounded away from 1. (joint work with Edward Dobrowolski, Kevin G. Hare, and Michael J. Mossinghoff) 3:50-4:10 tea break 4:10-5:00 Nicholas Ramsey (Harvard University) Geometric and p-adic modular forms of half-integral weight Abstract: I'll discuss a geometric formalism for modular forms of half-integral weight, with particular emphasis on the Hecke action. Then I'll introduce a straghtforward modification to a p-adic theory a la Katz and Coleman, complete with completely continuous Up opetrator. Thursday, March 25, 2004UBC Campus, MATH 104 (Math Building) 3:00-3:50 David Grant (University of Colorado, Boulder) On almost rational torsion points Abstract: Following Ribet, if G is a commutative algebraic group defined over a field k, we say a point P on G is almost rational over k, if for any g,h in the absolute galois group of k, gP+hP = [2]P implies that gP = hP = P. Ribet proved that if G is an abelian variety and k is a number field, then there are only finitely many almost rational torsion points of G over k, and showed how this result yields a new proof of the Manin-Mumford conjecture. I will discuss joint work with John Boxall, and present results bounding the number of almost rational torsion points on tori and CM abelian varieties, and describing the nature of almost rational torsion points on commutative algebraic groups over local and finite fields. 3:50-4:10 tea break 4:10-5:00 Ravi Ramakrishna (Princeton University) Constructions of Semisimple p-adic Galois Representations with prescribed properties Abstract: In this talk we show that we can lift two dimensional mod p Galois representations satisfying mild hypotheses to p-adic Galois representations ramified at infinitely many primes while guaranteeing that the characteristic polynomials of Frobenius at a density one set of unramified primes are defined over the rational integers. In particular we show that often one can "density one compatibly" lift mod p and mod q Galois representations. This is joint work with C. Khare and M. Larsen. Tuesday, April 13, 2004SFU Campus, Room AQ 4130 (note room change) 3:00-3:50 Greg Martin (UBC) Dimensions of spaces of newforms Abstract: A formula for the dimension of the space of cuspidal modular forms on Γ0(N) of weight k (k≥2 even) has been known for several decades.  More recent but still well-known is the Atkin-Lehner decomposition of this space of cusp forms into subspaces corresponding to newforms on Γ0(d) of weight k, as d runs over the divisors of N.  A recursive algorithm for computing the dimensions of these spaces of newforms follows from the combination of these two results, but it would be desirable to have a formula in closed form for these dimensions.  In this paper we establish such a closed-form formula, not only for these dimensions, but also for the corresponding dimensions of spaces of newforms on Γ1(N) of weight k (k≥2).  This formula is much more amenable to analysis and to computation.  For example, we derive asymptotically sharp upper and lower bounds for these dimensions, and we compute their average orders; even for the dimensions of spaces of cusp forms, these results are new.  We also establish sharp inequalities for the special case of weight-2 newforms on Γ0(N), and we report on extensive computations of these dimensions: we find the complete list of all N such that the dimension of the space of weight-2 newforms on Γ0(N) is less than or equal to 100 (previous such results had only gone up to 3). 3:50-4:10 tea break 4:10-5:00 David Boyd (UBC) The arithmetic of Turk's Head knots Abstract: The Turk's Head knots are a family of highly symmetric knots that are beloved of sailors, boy scouts and designers of jewellery. The symmetry group of the 2m-crossing knot is the dihedral group D2m which seems to be the largest possible symmetry group for a 2m-crossing knot. The A-polynomial (not Alexander polynomial) of a knot is a two variable polynomial invariant of the knot which is notoriously difficult to compute. It was a major computational challenge to compute the A-polynomial for even the simplest of these Turk's Head knots, the 8-crossing knot 818. We have recently developed a new algorithm for computing A-polynomials that handles this with ease and have used it to compute further polynomials in the sequence (the polynomial for the 14-crossing knot is of degree 12 in x and 84 in y). By experiment, we discovered a remarkable factorization of the polynomial A(xm,y) over the cyclotomic field Q(ζ2m) into a product of 4x4 polynomials G(w,x,y) depending on a parameter w. We explain the meaning of G(w,x,y) as a generalization of the A-polynomial and show that π m(G(ωm,x,y)) is the volume of the 2m-crossing knot, where ωm = 2 cos(π/m) and m denotes the logarithmic Mahler measure. The talk will be as elementary as possible. No experience at tying the Wood Badge woggle is necessary.