Vanishing and non-vanishing critical values of elliptic L-functions

Speaker: 

Hershy Kisilevsky

Institution: 

Concordia University

Time: 

Thursday, February 21, 2008 - 3:00pm

Location: 

MSTB 254

I will discuss some results on the vanishing and non-vanishing of
critical values of L-functions and their derivatives, both experimental and
theoretical. I will present an example of a computational "elliptic Stark
point" in a cyclic quintic extension of the rationals.

Drinfeld modular forms and Hecke characters

Speaker: 

Gebhard Boeckle

Institution: 

Essen

Time: 

Thursday, December 6, 2007 - 3:00pm

Location: 

MSTB 254

In a similar way as in the case of elliptic modular forms, one can attach
strictly compatible systems (SCS) of Galois representations to Drinfeld
modular forms. Unlike in the classical situations, these are abelian. Goss
had asked whether they would arise from Hecke characters. Adapting to the
function field setting a correspondence of Khare between SCS of mod p
Galois representations and Hecke characters, this can indeed be shown to
be the case. If time permits, I shall also give some examples and discuss
some open questions regarding these Hecke characters.

Dynamic modular curves

Speaker: 

Michelle Manes

Institution: 

USC

Time: 

Thursday, November 15, 2007 - 3:00pm

Location: 

MSTB 254

Consider a rational map φ on the projective line, from which we form a (discrete) dynamical system via iteration, and let K be a number field. A fundamental question in arithmetic dynamics is the uniform boundedness conjecture of Morton and Silverman, which states that there is a constant independent of φ (depending only on its degree) giving an upper bound for the number of K-rational preperiodic points of φ. This is a deep conjecture, and no specific case of it is known. I have proposed a specific version of the conjecture: that in the case of a degree-2 rational map and K = Q, the upper bound is 12.

In this talk, which assumes no previous knowledge of arithmetic dynamics, I will describe why this question is so difficult and sketch work that has been done to date, including giving justification for my refined uniform boundedness conjecture. The techniques used so far, which have clear limitations, involve constructing algebraic curves parameterizing maps $\phi$ together with points of period n for varying n (so-called dynamic modular curves).

Number Theoretical Problems From Coding Theory

Speaker: 

Professor Daqing Wan

Institution: 

UCI

Time: 

Thursday, October 25, 2007 - 3:00pm

Location: 

MSTB 254

This is an essentially self-contained introductory talk.
We shall discuss several fundamental coding theoretical problems
and reformulate them in terms of the basic number theoretical problems
about rational points, zeta functions and L-functions on curves/higher
dimensional varieties over finite fields.

Stark units and Gras-type Conjectures

Speaker: 

Kazim Buyukboduk

Institution: 

Stanford University

Time: 

Tuesday, April 17, 2007 - 2:00pm

Location: 

MSTB 254

B. Howard, B. Mazur and K. Rubin proved that the existence of Kolyvagin systems relies on a cohomological invariant, what they call the core Selmer rank. When the core Selmer rank is one, they determine the structure of the Selmer group completely in terms of a Kolyvagin system. However, when the Selmer core rank is greater than one such a precision could not be achieved. In fact, one does not expect a similiar result for the structure of the Selmer group in general, as a reflection of the fact that Bloch-Kato conjectures do not in general predict the existence of special elements, but a regulator, to compute the relevant L-values.

An example of a core rank greater than one situation arises if one attempts to utilize the Euler system that would come from the Stark elements (whose existence were predicted by K. Rubin) over a totally real number field. This is what I will discuss in this talk. I will explain how to construct, using Stark elements, Kolyvagin systems for certain modified Selmer structures (that are adjusted to have core rank one) and relate them to appropriate ideal class groups, following the machinery of Kolyvagin systems and prove a Gras-type conjecture.

On Deciding Deep Holes of Reed-Solomon Codes

Speaker: 

Professor Qi Cheng

Institution: 

University of Oklahoma

Time: 

Thursday, May 10, 2007 - 2:00pm

Location: 

MSTB 254

For generalized Reed-Solomon codes, it has been proved
that the problem of determining if a
received word is a deep hole is co-NP-complete.
The reduction relies on the fact that
the evaluation set of the code can be exponential
in the length of the code --
a property that practical codes do not usually possess.
In this talk, we first present a much simpler proof of
the same result. We then consider the problem for standard
Reed-Solomon codes, i.e. the evaluation set consists of
all the nonzero elements in the field.
We reduce the problem of identifying deep holes to
deciding whether an absolutely irreducible
hypersurface over a finite field
contains a rational point whose coordinates
are pairwise distinct and nonzero.
By applying Cafure-Matera estimation of rational points
on algebraic varieties, we prove that
the received vector $(f(\alpha))_{\alpha \in \F_p}$
for the Reed-Solomon $[p-1,k]_p$, $k < p^{1/4 - \epsilon}$,
cannot be a deep hole, whenever $f(x)$ is a polynomial
of degree $k+d$ for $1\leq d < p^{3/13 -\epsilon}$.
This is a joint work with Elizabeth Murray.

Grobner Bases and Linear Codes

Speaker: 

Professor Shuhong Gao

Institution: 

Clemson University

Time: 

Tuesday, May 8, 2007 - 2:00pm

Location: 

MSTB 254

Abstract: We show how Grobner basis theory can be used in coding
theory, especially in the construction and decoding of linear codes.
A new method is given for construction of a large class of linear codes
that has a natural decoding algorithm. It works for any finite field
and any block length. The codes constructed include as special cases
many of the well known codes such as Reed-Solomon codes, Hermitian
codes and, more generally, all one-point algebraic geometry codes.
This method also allows us to construct random codes for which
our decoding algorithm performs reasonably well. Joint work with
Jeffrey B. Farr.

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