We discuss a refinement of the Rubin-Stark Conjecture for abelian L-functions of arbitrary order of vanishing at s=0. This generalizes Gross's v-adic refinement of the abelian, order of vanishing 1, integral Stark Conjecture and predicts a link between special values of derivatives of p-adic and global L-functions. Time permitting, we will also show how our refinement relates to a recent strengthening of Gross's Conjecture due to Tate.
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.
This will be an introduction to the Iwasawa theory
of elliptic curves, including all the relevant Iwasawa modules,
p-adic L-functions, and conjectures about them. The aim is
to provide the necessary background for the lecture of 11/2.
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.
Let X be a projective variety over a finite field
with function field K(X). Let Y be a projective variety
over K(X). We may associate to this a height zeta
function. In this talk, we will recall some facts
about these functions and provide some new results
and research directions.