# Gross-Rubin-Stark-Tate

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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.

# Problems and Results on Covering Systems

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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.

# Introduction to Iwasawa theory of elliptic curves

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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.

# Selmer groups and skew-Hermitian matrices

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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.

# Introduction to Iwasawa theory of elliptic curves, II

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# l-adic cohomology and exponential sums, II

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# On the function field height zeta function

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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.