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.

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.

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.