Number Theory at UBC

SFU/UBC Number Theory Seminar
Winter Semester 2006

Thursday, January 26, 2006
SFU Campus, Room ASB 10900
3:00-3:50 Andrew Odlyzko (Digital Technology Center, University of Minnesota)
Zeros of the Riemann zeta function: Computations and implications

3:50-4:10 tea break
4:10-5:00 Stephen Choi (SFU)
On the maximum modulus of multivariate polynomials: Preliminary report

Thursday, February 9, 2006
UBC Campus, Room WMAX 110
3:00-3:50 Matilde Lalin (Institute for Advanced Study)
Mahler measure as values of regulators

Abstract: Regulators allows us, sometimes, to explain and compute examples of Mahler measure formulas for multivariate polynomials. I will sketch some ideas of how to use regulators for computations and show some old and new examples.

3:50-4:10 tea break
4:10-5:00 Nike Vatsal (UBC)
Special values of L-functions modulo p

Abstract: It has been known since Euler that the values of the Riemann zeta function at negative integers are certain rational numbers, namely the Bernoulli numbers Bk. Similarly, the values of Dirichlet L-functions at s=0 are related to class numbers of certain number fields. These are simple instances of a common phenomenon, namely that the values of L-functions at critical points are algebraic, up to a simple factor, and that these algebraic numbers are related to algebraic quantities such as class numbers and Selmer groups. The present talk will be a survey talk on the algebraicity of special values of L-functions and their divisibility properties modulo primes.

Thursday, March 9, 2006
UBC Campus, Room WMAX 110
3:00-3:50 Kate Petersen (Queen's University)
Cusps and congruence subgroups of PSL(2,OK)

Abstract: For a number field K, the groups PSL(2,OK) have markedly different characteristics depending on whether the ring of integers OK has infinitely many units or not. We'll discuss how this difference manifests itself in terms of the topology of certain quotients and connect it to a generalization of Artin's Primitive Root Conjecture.

3:50-4:10 tea break
4:10-5:00 Ching-Li Chai (University of Pennsylvania)
Niven Lecture - Canonical coordinates for leaves of p-divisible groups

Abstract: Let p be a prime number and g be a positive integer. Let M be the moduli space of abelian varieties of PEL type. A leaf in M is the locus corresponding to a fixed isomorphism class of polarized p-divisible group with prescribed endomorphisms. Although a leaf is defined by a “pointwise” condition, it turns out that the formal completion (or jet space) of a leaf at a point has a rigid structure: It is built up from a finite collection of p-divisible formal groups via a family of fibrations. This structural description can be regarded as a generalization of the Serre-Tate coordinates of the local deformation space of an ordinary abelian variety. We also explain a local rigidity result related to the action of the local stabilizer subgroup on the canonical coordinates. (more about the Niven lectures)

Thursday, March 23, 2006
SFU Campus, Room ASB 10900
3:00-3:50 Renate Scheidler (Centre of Information Security and Cryptography, University of Calgary)
The real model of a hyperelliptic curve

Abstract: Arithmetic geometers and cryptographers are familiar with what we call for our purposes the “imaginary model” of a hyperelliptic curve. Another less familiar description of such a curve is the so-called “real model”; the terminology stems from the analogy to real and imaginary quadratic number fields. Structurally and arithmetically, the real model behaves quite differently from its imaginary counterpart. While divisor addition with subsequent reduction (“giant steps”) is still essentially the same, the real representation no longer allows for unique representation of elements in the Jacobian by their reduced representatives. However, degree-zero divisors in the real model exhibit a so-called infrastructure, with an additional, much faster operation (“baby steps”). We present the real model of a hyperelliptic curve and its two-fold baby-step-giant-step divisor arithmetic. We go on to illustrate how to use these algorithms in the principal infrastructure for efficient cryptographic applications.

3:50-4:10 tea break
4:10-5:00 Peter Borwein (SFU)
Littlewood's 22nd problem

Abstract: Littlewood, in his 1968 monograph Some Problems in Real and Complex Analysis, poses the following research problem, which appears to still be open: “If the nm are integral and all different, what is the lower bound on the number of real zeros of Σ1≤mn cos (nmθ)? Possibly N−1, or not much less.” No progress appears to have been made on this in the last half century.

Thursday, April 6, 2006
SFU Campus, Room K9509
(note venue change!)
3:00-3:50 Denis Charles (Microsoft)
Some applications of the graph of supersingular elliptic curves over a finite field

Abstract: The graph of supersingular elliptic curves over a finite field connected by isogenies has many applications in computational number theory. In this talk we look at some old (in number theory) and new (in cryptography) applications of these graphs. In particular, we discuss new constructions of secure hash functions and pseudorandom number generators from these graphs. We will also study the interesting graph-theoretic properties of this graph. If time permits, I will sketch a generalization of these graphs to graphs of superspecial abelian varieties. The new results in this talk are from joint work with Eyal Goren and Kristin Lauter.

3:50-4:10 tea break
4:10-5:00 Patrick Ingram (UBC)
In my defense: Integral points on elliptic curves

Abstract: In a talk not completely unrelated to my thesis defense, I will examine the question “For which elliptic curves E, which integral points P on E, and which integers n is nP also an integral point?” Most (if not all) of the attention will be devoted to congruent number curves.