
Tuesday, January 29, 2008 UBC Campus, Room WMAX 110 (PIMS)

3:304:20 
Sujatha Ramdorai (Tata Institute)
Root numbers and Selmer groups of elliptic curves
Abstract:
The theory of root numbers predicts when elliptic curves over number
fields have rational points of infinite order. In this talk, we shall discuss
results which bring together the root numbers and noncommutative Iwasawa
theory. It is joint work with Coates, Fukaya and Kato, and has connections to
some recent work of Rohrlich and T. Dokchitser and V. Dokchitser.

4:204:40 
tea break 
4:405:30 
Katherine Stange (Brown University)
Elliptic nets
Abstract:
Elliptic divisibility sequences are integer recurrence sequences, each
of which is associated to an elliptic curve over the rationals
together with a rational point on that curve. I'll give the background
on these and present a higherdimensional analogue over arbitrary
fields. Suppose E is an elliptic curve over a field K, and P_{1}, ..., P_{n}
are points on E defined over K. To this information we associate an
ndimensional array of values of K satisfying a complicated nonlinear
recurrence relation. These are called elliptic nets. All elliptic nets
arise from elliptic curves in this manner. I'll explore some of the
properties of elliptic nets and the geometric information they
contain, including a connection to generalised Jacobians and the Tate
and Weil pairings on the elliptic curve.



Thursday, February 7, 2008 SFU Campus, Room ASB 10900 (IRMACS)

3:003:50 
Stephen Choi (SFU)
An extension to the BrunTitchmarsh theorem
Abstract:
The SiegelWalfisz theorem states that for any B>0, we have Σ_{p≤x, p≅d (mod v)} 1 ∼ x/φ(v) log(x) for v ≤ log^{B}(x) and (v,d)=1. This only gives an asymptotic formula for the number of primes in an arithmetic progression for quite a small modulus v compared to x. However, if we are concerned only with an upper bound, the BrunTitchmarsh theorem says that for any 1≤v≤x, we have Σ_{p≤x, p≅d (mod v)} 1 << x/φ(v) log(x). In this talk, we will discuss an extension to the BrunTitichmarsh theorem that concerns the number of integers with exactly s distinct prime factors in an arithmetic progression. This is joint work with Kai Man Tsang and Tsz Ho Chan.

3:504:10 
tea break 
4:105:00 
Ronald van Luijk (PIMS/SFU/UBC)
Manin conjectures for K3 surfaces
Abstract:
The Manin conjectures describe for geometrically easy varieties how
the number of their rational points of bounded height should grow as
the height bound varies. In this talk I will describe recent computations
that suggest a similar statement for K3 surfaces, which are geometrically
more complicated. Part of the talk will focus on how to count the number
of points in a specific example, using a variation of an algorithm by
Noam Elkies.



Thursday, February 28, 2008 UBC Campus, Room WMAX 110 (PIMS)

3:003:50 
Kate Petersen (Queen's University)
Primitive roots and the Euclidean algorithm
Abstract:
An integer s is called a primitive root modulo a prime p if the
multiplicative set generated by s surjects onto all nonzero residue classes
modulo p. Artin's primitive root conjecture states that all integers s
other than 1 or squares are primitive roots modulo infinitely many
primes. I'll discuss a generalization of Artin's primitive root
conjecture to number fields and connections this has to the Euclidean
Algorithm problem. This is joint work with R. Murty.

3:504:10 
tea break 
4:105:00 
Erick Wong (UBC)
Eigenvalues of random matrices and not the Riemann Hypothesis
Abstract:
Random matrix theory has been a hot topic in number theory, particularly
since the Rudnick and Sarnak landmark work on the spacing of consecutive zeros
of Lfunctions. This highly accessible talk has a far more elementary
flavour, focusing on eigenvalues of random integer matrices instead of the
Gaussian Unitary Ensemble.
For a fixed n, consider a random n×n integer matrix with entries
bounded by the parameter k. I'll give a simple proof that such a matrix
almost certainly has no rational eigenvalues (as k increases). Then we'll
delve into more detail on the exact eigenvalue distribution of the 2×2
case. Along the way we'll rediscover a forgotten determinant identity and
tackle some quadruple sums. This is joint work with Greg Martin.



Thursday, March 13, 2008 SFU Campus

3:003:50 
Cecilia Busuioc (Boston University)
The Steinberg symbol and special values of Lfunctions
Room K9509
Abstract:
In this talk, we will describe a construction of Eisenstein classes in the parabolic cohomology of Γ_{0}(M) with values in Milnor's K_{2}group of the ring of Sintegers of the cyclotomic extension Q(μ_{M}). In the case where p is an irregular prime, we will explain a congruence between certain projections of Steinberg symbols of punits and Lvalues of level1 cusp forms congruent to an Eisenstein series mod p, a result that was predicted by a conjecture of R. Sharifi.

3:504:10 
tea break 
4:105:00 
Melissa Macasieb (UBC)
Character varieties of hyperbolic knot complements
Room ASB 10900 (IRMACS)
Abstract:
To every hyperbolic 3manifold M with nonempty boundary, one can associate a pair of related algebraic varieties X(M) and Y(M) called the character varieties of M. These varieties carry much topological information about M, but are in general difficult to compute. In the case that M is a knot complement, X(M) and Y(M) are defined over Q. In this talk, I will discuss how properties of these varieties reflect the topology of M in the case M is a hyperbolic knot complement. I will also show how to obtain explicit equations for the the character varieties associated to a biinfinite family of hyperbolic knots K(m,n) and discuss consequences of these results related to the existence of integral points on these curves. This is joint work with Kate Petersen and Ronald van Luijk.



Thursday, March 27, 2008 UBC Campus, Room WMAX 110 (PIMS)

3:003:50 
Ram Murty (Queen's University)
Effective equidistribution of Hecke eigenvalues
This talk is one of the 2008 Niven Lectures.
Abstract:
For a fixed prime p, we consider the space S(N,k) of cusp forms
of weight k and level N, with N coprime to p. In 1995, J.P. Serre
proved the existence of a measure u_{p} with respect to
which the eigenvalues of the pth Hecke operator acting on S(N,k)
are equidistributed as k+N tends to infinity. We will derive
an effective version of Serre's theorem and apply it to study
the factorization of J_{0}(N) into simple abelian varieties. Our
methods can also be applied to study the variation of eigenvalues
of the Frobenius automorphism acting on a family of curves mod p
and the variation of eigenvalues of adjacency matrices of regular
graphs. (This is joint work with Kaneenika Sinha.)

3:504:10 
tea break 
4:105:00 
Vishaal Kapoor (UBC)
Characters and things
Abstract:
In this expository talk, I'll be discussing the work of Granville and Soundararajan on sums of multiplicative functions in arithmetic progressions.



Thursday, April 10, 2008 SFU Campus, Room K9509

3:003:50 
Nathan Ng (University of Lethbridge)
Lower bounds for discrete moments of the Riemann zeta function
Abstract:
Assuming the Riemann Hypothesis, we establish lower bounds for moments of
the derivative of the Riemann zetafunction averaged over the nontrivial
zeros of zeta. Our proof is based upon a method of Rudnick and
Soundararajan that provides analogous bounds for moments of Lfunctions at
the central point. This is joint work with Milinovich.

3:504:10 
tea break 
4:105:00 
Habiba Kadiri (University of Lethbridge)
A bound for the least prime ideal in the Chebotarev density problem
Abstract:
A classical theorem due to Linnik gives a bound for the least prime number
in an arithmetic progression. Lagarias, Montgomery and Odlyzko gave a
generalization of this result to any number field.
Their proof relies on some results on the distribution of the zeros of the
Dedekind zeta function (zerofree regions, DeuringHeilbronn phenomenon).
In this talk, I will present some new results about these zeros.
As a consequence, we are able to prove an effective version of the theorem
of Lagarias et al.
