
Thursday, September 13, 2007 UBC Campus, Room WMAX 110 (PIMS)

4:305:20 
Ronald van Luijk (PIMS/SFU/UBC)
Cubic points on cubic curves and the BrauerManin obstruction on K3 surfaces
Abstract:
It is wellknown that not all varieties over Q satisfy
the Hasse principle. The famous Selmer curve given by 3x^{3} + 4y^{3} + 5z^{3} = 0
in P^{2}, 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 socalled BrauerManin 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 ax^{3} + by^{3} + cz^{3} = 0 that is locally solvable everywhere, and
that has no points over any cubic galois extension of Q,
then the algebraic part of the BrauerManin obstruction is not the only one
for K3 surfaces.
No previous knowledge about BrauerManin obstructions or K3 surfaces
will be necessary.



Thursday, September 27, 2007 UBC Campus, Room WMAX 110 (PIMS)

3:003:50 
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 twiceiterated Carmichael lambdafunction λ(λ(n)), a function that arises when considering an exponential pseudorandom number generator x_{k+1} ≡ x_{k}^{C} (mod n). Often I suppress most of the technical details of the multistage 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.

3:504:10 
tea break 
4:105:00 
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)

3:003:50 
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 2dimensional subvariety L_{n} of the moduli space M_{2} 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 L_{5} 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 L_{5} as subvarieties of M_{2} and classify all curves in these loci which have extra automorphisms.

3:504:10 
tea break 
4:105:00 
Mat Rogers (UBC)
An extension of a padic 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 J_{p}(y) = {n : Σ_{1≤j≤n} y^{j}/j ≡ 0 (mod p)} . I have shown that it is possible to obtain results such as J_{13}(9) = 18763, J_{47}(8) = 27024, and J_{7}(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)

3:003:50 
Youness Lamzouri (Université de Montréal)
The twodimensional distribution of values of ζ(1+it)
Abstract:
In 1928, Littlewood proved that ζ(1+it) < (2e^{γ}+o(1)) log log t assuming the Riemann Hypothesis, and conjectured that max_{t≤T} ζ(1+it) ∼ e^{γ} 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 < (e^{γ}+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 nonprincipal characters modulo a large prime q.



Thursday, November 8, 2007 SFU Campus, Room ASB 10900 (IRMACS)
simulcast on UBC Campus, Room WMAX 110 (PIMS)

3:003:50 
Patrick Ingram (University of Toronto)
Diophantine approximation in arithmetic dynamics
Abstract:
In 1978, Lang conjectured a lower bound on the canonical height of a nontorsion 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 multiplicationbyn 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.

3:504:10 
tea break 
4:105:00 
OYeat 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)

2:002:50 
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 N_{E} and the minimal discriminant
Δ_{E}. The Szpiro's conjecture states that for any ε>0
there exists a constant C_{ε}>0 such that
Δ_{E} < C_{ε} (N_{E})^{6+ε}.
This conjecture is equivalent to the ABCconjecture and, if true,
would imply solutions to many Diophantine equations. A consequence of
Szpiro's conjecture is that we can bound v_{p}(Δ_{E}) for the largest
prime p dividing Δ_{E}. In this talk I will show how a generalization of
Ken's levellowering result can be used to bound v_{p}(Δ_{E}).

2:503:00 
tea break 
3:003:50 
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 ρ_{l} associated to the ltorsion 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 ρ_{l} 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 ρ_{l} for small primes l.

3:504:10 
tea break 
4:105:00 
Paul Mezo (Carleton University)
A PaleyWiener 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 PaleyWiener theorem is also a
theorem in harmonic analysis. It characterizes the Fourier transform of
smooth compactlysupported 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.

