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.

Mumford curves parameterizing hyperelliptic curves

Speaker: 

Samuel Kadziela

Institution: 

University of Illinois at Urbana-Champaign

Time: 

Tuesday, June 5, 2007 - 2:00pm

Location: 

MSTB 254

Tate's work on Rigid Analytic Spaces can be used to obtain the
$p$-adic uniformization of a curve. In this talk, I will describe a
criterion determining which hyperelliptic curves admit this type of
uniformization. Then, we will discuss Mumford curves, which are the
uniformizing spaces, and explain how to approximate the $p$-adic
uniformization of a given totally split hyperelliptic curve.

Attacking the ABC Conjecture with elliptic curves

Speaker: 

Lily Khadjavi

Institution: 

Loyola Marymount University

Time: 

Tuesday, May 1, 2007 - 2:00pm

Location: 

MSTB 254

This talk will investigate the ABC Conjecture, an open problem with a surprising number of implications, viewed by some as a "holy grail" of number theory. We'll describe the conjecture and then consider an idea of Noam Elkies' which exploits special maps from curves to the projective line. Exploiting the group structure of elliptic curves along with these maps, we make progress towards a weak ABC Conjecture. This is joint work with Victor Scharaschkin.

ell-torsion of Abelian Varieties

Speaker: 

Chris Hall

Institution: 

University of Texas

Time: 

Tuesday, April 24, 2007 - 2:00pm

Location: 

MSTB 254

Let $K$ be a number field and $E/K$ an elliptic curve without
complex multiplication. A well-known theorem of Serre asserts that the
Galois group of $K(E[\ell])/K$ is as all of ${\rm GL}_2(\Z/\ell)$ for any
sufficiently large prime $\ell$. If we replace $E/K$ by a polarized abelian
variety $A/K$ with trivial endomorphism ring, then Serre later showed
that the Galois group of $K(A[\ell])/K$ is also as large as possible, for
all sufficiently large $\ell$, provided $\dim(A)$ is 2,6 or odd. We will
show how to prove a similar result for `most' $A$ and without any
restriction on $\dim(A)$.

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