This will be an introduction to the speaker's joint
FOCS (04) paper with Q. Cheng. I will outline the
main ideas linking the decoding of Reed-Solomon codes
to discrete logarithms and character sums over finite
fields.

The minimum distance is one of the most important combinatorial
characterizations of a code. The maximum likelihood decoding problem is
one of the most important algorithmic problems of a code. While these
problems are known to be hard for general linear codes, the techniques
used to prove their hardness often rely on the construction of artificial
codes. In general, much less is known about the hardness of the specific
classes of natural linear codes. In this paper, we show that both problems
are NP-hard for algebraic geometry codes. We achieve this by reducing a
well-known NP-complete problem to these problems using a randomized
algorithm. The family of codes in the reductions have positive rates, but
the alphabet sizes are exponential in the block lengths.

This is an introduction to the important aspects of
covers of the integers by residue classes and covers of groups
by cosets or subgroups. The field is connected with number theory,
combinatorics, algebra and analysis. It is quite fascinating, and
also very difficult (but the results can be easily understood).
Many problems and conjectures remain open, some nice theorems and
applications will be introduced.

Suppose E is an elliptic curve defined over a number
field K, and p is a prime where E has good ordinary reduction.
We wish to study the Selmer groups of E over all finite extensions
L of K contained in the maximal Z_p-power extension of K, along
with their p-adic height pairings and a Cassels pairings.
Our goal is to produce a single free Iwasawa module of finite
rank, with a skew-Hermitian pairing, from which we can recover
all of this data. Using recent work of Nekovar we can show that
(under mild hypotheses) such an `organizing module' exists, and we
will give some examples.
This work is joint with Barry Mazur.