Thursday, September 13, 2007
UBC Campus, Room WMAX 110 (PIMS)
4:30-5:20 Ronald van Luijk (PIMS/SFU/UBC)
Cubic points on cubic curves and the Brauer-Manin obstruction on K3 surfaces

Abstract: It is well-known that not all varieties over Q satisfy the Hasse principle. The famous Selmer curve given by 3x3 + 4y3 + 5z3 = 0 in P2, for instance, indeed has points over every completion of Q, but no points over Q itself. Though it is trivial to find points over some cubic field, it is a priori not obvious whether there are points over a cubic field that is galois. We will see that such points do exist. K3 surfaces do not satisfy the Hasse principle either, which in some cases can be explained by the so-called Brauer-Manin obstruction. It is not known whether this obstruction is the only obstruction to the existence of rational points on K3 surfaces. We relate the two problems by sketching a proof of the following fact. If there exists a smooth curve over Q given by ax3 + by3 + cz3 = 0 that is locally solvable everywhere, and that has no points over any cubic galois extension of Q, then the algebraic part of the Brauer-Manin obstruction is not the only one for K3 surfaces. No previous knowledge about Brauer-Manin obstructions or K3 surfaces will be necessary.

Thursday, September 27, 2007
UBC Campus, Room WMAX 110 (PIMS)
3:00-3:50 Greg Martin (UBC)
λ(λ(n)): A case study in analytic number theory

Abstract: A 2005 result of Carl Pomerance and myself identifies the normal order (that is, the asymptotic size for 100% of integers) of the twice-iterated Carmichael lambda-function λ(λ(n)), a function that arises when considering an exponential pseudorandom number generator xk+1xkC (mod n). Often I suppress most of the technical details of the multi-stage proof in talks on this topic; however, today I will use the result as an excuse to point out the techniques involved, techniques that might be labeled “Erdős mathematics&rdquo — elementary yet involved and (in his case) inspired calculations.

3:50-4:10 tea break
4:10-5:00 Michael Bennett
Powers in progression, Chebotarev, and Hilbert class polynomials

Abstract: I will sketch some rather odd connections between ternary Diophantine equations, the Chebotarev Density Theorem and heights of Hilbert class polynomials evaluated at rational arguments.

Thursday, October 18, 2007
SFU Campus, Room ASB 10900 (IRMACS)
simulcast on UBC Campus, Room WMAX 110 (PIMS)
3:00-3:50 Tony Shaska (Oakland University)
Genus two curves which admit a degree 5 map to an elliptic curve

Abstract: We study genus 2 curves C that admit a cover C → E to a genus 1 curve E of prime degree n. These curves C form an irreducible 2-dimensional subvariety Ln of the moduli space M2 of genus 2 curves. Here we study the case n=5. This extends earlier work for degree 2 and 3, aimed at illuminating the theory for general n. We compute a normal form for the curves in the locus L5 and its three distinguished subloci. Further, we compute the equation of the elliptic subcover in all cases, give a birational parametrization of the subloci of L5 as subvarieties of M2 and classify all curves in these loci which have extra automorphisms.

3:50-4:10 tea break
4:10-5:00 Mat Rogers (UBC)
An extension of a p-adic algorithm due to Boyd, and consequences

Abstract: I will talk about a computational problem that I recently encountered. Suppose that p is an odd prime and y is rational, then define Jp(y) = {n : Σ1≤jn yj/j ≡ 0 (mod p)} . I have shown that it is possible to obtain results such as |J13(9)| = 18763, |J47(8)| = 27024, and |J7(2)| = 0. I will mention ways to apply my method to sums involving Fibonacci numbers, connections to classical problems, and a possible connection with mirror maps.

Thursday, October 25, 2007
UBC Campus, Room WMAX 110 (PIMS)

simulcast on SFU Campus, Room ASB 10900 (IRMACS)
3:00-3:50 Youness Lamzouri (Université de Montréal)
The two-dimensional distribution of values of ζ(1+it)

Abstract: In 1928, Littlewood proved that |ζ(1+it)| < (2eγ+o(1)) log log t assuming the Riemann Hypothesis, and conjectured that maxtT |ζ(1+it)| ∼ eγ log log T. Recently, Granville and Soundararajan computed the distribution function of |ζ(1+it)|, giving strong evidence for Littlewood's Conjecture. In this talk, we present several results on the joint distribution function of arg ζ(1+it) and |ζ(1+it)|. One consequence of our work is that almost all values ζ(1+it) with large norm (close to the conjectured maximum) are concentrated near the positive real axis. Indeed, we prove that the larger the arguments, the more it becomes rare to find values with large norm. Also for t ∈ [T,2T] and τ < (eγ+o(1)) log log T, we show that arg ζ(1+it) with |ζ(1+it)| ≈ τ is normally distributed with mean zero and variance depending on τ. Finally we prove similar results in the case of L(1,χ), where χ varies over non-principal characters modulo a large prime q.

Thursday, November 8, 2007
SFU Campus, Room ASB 10900 (IRMACS)

simulcast on UBC Campus, Room WMAX 110 (PIMS)
3:00-3:50 Patrick Ingram (University of Toronto)
Diophantine approximation in arithmetic dynamics

Abstract: In 1978, Lang conjectured a lower bound on the canonical height of a non-torsion point on an elliptic curve which depends on various data related to the curve. This conjecture remains open (although there are several partial results). Much more recently, Silverman posited a more general conjecture about lower bounds on canonical heights in “arithmetic dynamical systems” (that is, systems defined by a morphism mapping a variety to itself), which at least roughly reduces to Lang's Conjecture when the underlying system is that defined by the multiplication-by-n map on an elliptic curve (n > 1). We'll discuss the first result towards Silverman's conjecture in which the underlying structure is not that of an abelian variety. Some other problems related to heights and diophantine approximation in arithmetic dynamics will come up along the way.

3:50-4:10 tea break
4:10-5:00 O-Yeat Chan (Dalhousie University)
Effective computation of Bessel functions

Abstract: Bessel functions are some of the most important functions in mathematical physics and the theory of special functions, and the ability to compute their values is equally important. The standard method of evaluating the Bessel functions has been to use an ascending series for small argument, and the asymptotic (but divergent) series for large argument. In this talk, we describe a new series that is geometrically convergent in the number of summands, with explicitly computable error estimates for the tails. This is joint work with David Borwein and Jonathan Borwein.

Thursday, November 22, 2007
UBC Campus, Room WMAX 110 (PIMS)
2:00-2:50 Soroosh Yazdani (McMaster University)
Level lowering and Szpiro's conjecture

Abstract: Let E/Q be an elliptic curve over the rationals. One can associate two rational integers that measure the ramification of this elliptic curve over various primes, the conductor NE and the minimal discriminant ΔE. The Szpiro's conjecture states that for any ε>0 there exists a constant Cε>0 such that |ΔE| < Cε (NE)6+ε. This conjecture is equivalent to the ABC-conjecture and, if true, would imply solutions to many Diophantine equations. A consequence of Szpiro's conjecture is that we can bound vpE) for the largest prime p dividing ΔE. In this talk I will show how a generalization of Ken's level-lowering result can be used to bound vpE).

2:50-3:00 tea break
3:00-3:50 Sander Dahmen (Universiteit Utrecht)
Modular methods applied to Diophantine equations

Abstract: Deep results about elliptic curves, modular forms and Galois representations have successfully been applied to solve FLT and other Diophantine equations. Most of such applications broadly proceed along the following lines. To a hypothetical solution is associated a certain elliptic curve, called a Frey curve, and it is shown that the mod l Galois representation ρl associated to the l-torsion points of the Frey curve (with l, say, an odd prime occurring as exponent in the Diophantine equation) is irreducible. Then by modularity and level lowering, one obtains that ρl is modular of some explicitly known level (weight 2 and trivial character). Finally, the modular forms of this level provide (possibly in a non trivial way) information about the original Diophantine equation. In this talk I will first describe the above mentioned method in some more detail and then focus on the problem of finding Frey curves and proving irreducibility of ρl for small primes l.

3:50-4:10 tea break
4:10-5:00 Paul Mezo (Carleton University)
A Paley-Wiener theorem and Arthur's trace formula

Abstract: Modular forms may be recast and generalized as automorphic representations, which are objects of abstract harmonic analysis. The trace formula is a theorem in harmonic analysis which allows one to compare automorphic representations. The Paley-Wiener theorem is also a theorem in harmonic analysis. It characterizes the Fourier transform of smooth compactly-supported functions, and is essential in the proof of Arthur's trace formula. We shall expand on each of these statements, highlighting new results. This is joint work with P. Delorme.