# Number Theoretical Problems From Coding Theory

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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.

# Non-commutative Iwasawa theory II

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# Stark units and Gras-type Conjectures

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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

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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

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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

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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.

# Arithmetic of abelian varieties over large fields

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# Attacking the ABC Conjecture with elliptic curves

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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

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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 $\dim(A)$.