Melanie Matchett Wood (AIM & University of Wisconsin)

The probability that a curve over a finite field is smooth

Given a fixed surface over a finite field, we ask what proportion of curves in that surface are smooth. Poonen's work on Bertini theorems over finite fields answers this question for certain families of curves in the surface. In this case the probability of smoothness is predicted by a simple heuristic assuming smoothness is independent at different points in the surface. In joint work with Erman, we consider this question for other families of curves in **P**^{1} ⨉ **P**^{1} and Hirzebruch surfaces. Here the simple heuristic of independence fails, but the answer can still be determined and follows from a richer heuristic that predicts at which points smoothness is independent and at which points it is dependent. abstract notes

Bianca Viray (Brown University)

Descent on elliptic surfaces and transcendental Brauer elements

Transcendental elements in the Brauer group are notoriously difficult to compute. Wittenberg and Ieronymou have worked out explicit representatives for 2-torsion elements of elliptic surfaces, in the case that the Jacobian fibration has rational 2-torsion. We use ideas from descent to develop techniques to study the 2-torsion elements of elliptic surfaces without an assumption on the 2-torsion of the Jacobian. abstract notes

Jennifer Balakrishnan (Harvard University)

Computations with Coleman integrals

The Coleman integral is a *p*-adic line integral that can encapsulate valuable information about the arithmetic and geometry of curves and abelian varieties.
For example, certain integrals allow us to find rational points or torsion points; certain others give us *p*-adic height pairings. I will present a brief overview of the theory, describe algorithms to calculate some of these integrals, and illustrate these techniques with numerical examples computed using Sage. abstract notes

Kristin Lauter (Microsoft Research)

Constructing genus 2 curves for cryptography

Jacobians of genus 2 curves can be used in cryptography, but constructing curves which are appropriate to use requires deep methods from number theory, including the theory of complex multiplication. This talk will explain some of these methods, including Igusa class polynomials and the conjecture of Bruinier and Yang. Bruinier and Yang conjectured a formula for an intersection number on the arithmetic Hilbert modular surface, CM(K)⋅Tm, where CM(K) is the zero-cycle of points corresponding to abelian surfaces with CM by a primitive quartic CM field K, and Tm is the Hirzebruch-Zagier divisors parameterizing products of elliptic curves with an m-isogeny between them. In this talk, we examine fields not covered by Yang's proof of the conjecture. We give numerical evidence to support the conjecture and point to some interesting anomalies. We compare the conjecture to both the denominators of Igusa class polynomials and the number of solutions to the embedding problem stated by Goren and Lauter. This project was initiated at the WIN workshop in Banff in November 2008, and is joint work with: Bianca Viray, Jennifer Johnson-Leung, Adriana Salerno, Erika Frugoni and Helen Grundman. abstract notes

Renate Scheidler (University of Calgary)

Classification and Symmetries of a Family of Continued Fractions With Bounded Period Length

It is well-known that the continued fraction expansion of a quadratic irrational is symmetric about its centre; we refer to this symmetry as horizontal. However, an additional vertical symmetry is exhibited by the continued fraction expansions arising from a certain one-parameter family of positive integers known as Schinzel sleepers. This talk investigates the period lengths as well as both the horizontal and vertical symmetries of this family. We also outline a method for generating all Schinzel sleepers. This is joint work with Kell Cheng, Richard Guy and Hugh Williams. abstract notes

Cristina Ballantine (College of the Holy Cross)

Ramanujan bigraphs associated with SU(3) over a p-adic field

We use the representation theory of the quasisplit form G of SU(3) over a *p*-adic field to investigate whether certain quotients of the Bruhat-Tits tree associated to this form are Ramanujan bigraphs. We show that a quotient of the tree associated with G (which is a biregular bigraph) is Ramanujan if and only if G satisfies a Ramanujan type conjecture. This result is analogous to the seminal case of PGL_{2}(**Q**_{p}) considered by Lubotzky, Phillips and Sarnak. As a consequence, Rogawski's classification of the automorphic spectrum of the unitary group in three variables implies the existence of certain infinite families of Ramanujan bigraphs.
abstract notes

Brooke Feigon (City College of New York)

L-functions and periods

I will describe recent work relating L-functions to periods. abstract notes

Marie-France Vignéras (Institut de Mathématiques de Jussieu)

From p-adic representations of the Galois group of Qp to sheaves on flag varieties

Colmez's functor from finite dimensional *p*-adic representations of the Galois group of **Q**_{p} to GL(2,**Q**_{p})-equivariant sheaves on **P**^{1}(**Q**_{p}) (the flag variety of GL(2,**Q**_{p})) can be generalized. In a common work with Peter Schneider and Gergely Zábrádi, we construct functors from *p*-adic Galois representations to GL(n,**Q**_{p})-equivariant sheaves on the flag variety of GL(n,**Q**_{p}) for any integer n ≥ 2. abstract notes