D. Benois

Title:  Iwasawa theory of crystalline representations


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M. Bertolini

Title: Hida families and rational points on elliptic curves

Abstract: I will discuss work in progress with Henri Darmon,
which relates the second derivative of a Hida p-adic L-function
of an elliptic curve E, having split multiplicative reduction at p, to the square of the logarithm of a rational point on E.
 

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 W. Bley 

abstract.dvi
abstract.ps

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D. Burns 

Title : Equivariant Tamagawa numbers and Iwasawa theory

Abstract: We discuss the application of (non-commutative) Iwasawa-theoretic
techniques to the study of a natural equivariant refinement of the Tamagawa number
conjecture formulated by Bloch and Kato (and later reworked by Fontaine and Perrin-Piou).

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W. Mc Callum

 Title and Abstract

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J. Coates

Title : Some aspects of non-commutative Iwasawa theory


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D. Delbourgo

Title: Tate-Shafarevich groups in ordinary families

Abstract: For an elliptic curve E defined over Q, the Birch and Swinnerton-Dyer
 formula provides a (conjectural) link between its L-function
 and arithmetic invariants. Assume that E is ordinary at a prime
 number p greater than 3. We can deform these arithmetic invariants
 along the critical line s=1, about a small p-adic ball in weight-space
 centred at k=2. We then prove a formula for the p-part of III(E)
 in the rank zero case by using 2-variable zeta-elements.
 In particular, if E has no CM and p is greater than 7, then
 the p-part of III(E) is controlled by the Hida deformation.

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R. Greenberg

Title: p=37

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Y. Hachimori

Title: On non pseudo-nullity of certain Iwasawa modules
in Iwasawa theory of p-adic Lie extensions.

Abstract: The notion of pseudo-null modules was generalized
to non-commutative Iwasawa algebras by Venjakob.
We discuss the (non) pseudo-nullity of the classical Iwasawa modules,
which means the Galois groups of the unramified extensions,
over certain p-adic Lie extensions.
This is a joint work with Romyar Sharifi.

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A. Hayward

Explicit units and a refinement of the Rubin-Stark conjecture

In global fields with at most one infinite place, there exist systems of
analytically-constructed units which give rise to Stark units in the sense
of the Rubin-Stark conjecture.  It is possible to axiomatise these systems,
giving a unified approach to cyclotomic units, elliptic units and division
points on Drinfeld modules.  I will describe this with particular reference
to the elliptic units.  This axiomatisation may be applied to proving
Rubin-Stark, and a recent strengthening of it in the spirit of a conjecture
of Gross (due to Burns), in these fields, with the infinite place designated
as splitting.  The proof is by combinatorics involving the distribution
relations for the units.

slides

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A. Huber

Title : From the Tamagawa number conjecture to the main conjecture

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A. Iovita
 

Title: p-Adic families of automorphic forms on quaternion algebras and
the Mazur-Tate-Teitelbaum conjecture

Abstract: Jointly with H.Darmon we were able to find a new proof
of the relationship between the derivative of the p-th Fourier coefficient
of a p-adic familiy containing our split multiplicative modular form and
its L-invariant. This is an essential ingredient in Glenn Stevens' proof
of the conjecture.

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G. Kings

Title : The modular polylog


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S. Kobayashi

Title: The generating function of the two variable $p$-adic $L$-function of CM elliptic curves

Abstract: We discuss an algebraic characterization of the generating function
of the two variable $p$-adic $L$-function of CM elliptic curves. We also give its applications.

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M. Kurihara

Title : On the structure of Iwasawa modules

Abstract :We discuss the structure of Selmer groups, especially
class groups and the Iwasawa modules of CM fields,
describing the higher Fitting ideals, which is regarded as
a refinement of usual main conjectures.

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M. Le Floc'h

Title : On capitulation cokernels in Iwasawa theory

Abstract: For a number field $F$ and an odd prime $p$, we study the "capitulation cokernels"
$coker (A'_n \rightarrow A'_{\infty}^{\Gamma_n})$ associated with the $(p)$-class groups
of the cyclotomic $\Z_p$-extension of $F$. We prove that these cokernels stabilize and we
characterize their direct limit in Iwasawa theoretic terms, thus generalizing previous partial
results obtained by H. Ichimura. This problem is intimately related to Greenberg's Conjecture.

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K. Matsuno

Title: On 2-adic Iwasawa invariants of ordinary elliptic curves

Abstract: In this talk, we will discuss how some results known for odd
primes p on Iwasawa invariants associated to an (ordinary)
elliptic curve E defined over a number field K can be extended
to the case p=2. More precisely, we will describe the behavior
of the lambda-invariant of E in finite Galois 2-extensions L/K
of the base field (analogue of Kida's formula), and describe the
difference of the lambda-invariants of two elliptic curves with
Galois-isomorphic 2-torsion points (due to Greenberg and Vatsal
for odd p). The results and the proofs should be modified
because of the influence of the real places.
We will also discuss some consequences of the above results
on p=2, unboundedness of lambda, behavior under the quadratic
twists, and the coincidence of invariants defined in two ways,
algebraically and analytically.


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J. Nekovar

Title : "Parity of ranks of Selmer groups associated to Hilbert modular forms"

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T. Ochiai
 

Title:  On the two-variable Iwasawa theory

Abstract: In this talk, the two-variable Iwasawa theory for
nearly ordinary Hida deformation are discussed.
We introduce our result on one of the divisibility of
the Iwasawa main conjecture for this case.
The key tool is Beilinson-Kato elements in the continuous
Galois cohomology with coefficients in Hida deformations.
If time permits, we also discuss other related results, examples
and perspective on the Iwasawa theory for
more general Galois deformation cases.

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M. Ozaki

Title: Iwasawa's formula for non-abelian unramified p-extensions
of a Z_p-extension

Abstract: In the classical theory of Z_p-extensions,
Iwasawa gave the cerebrated formula which describes
the order of the Galois group of the maximal unramified
abelian p-extension over the n-th layer k_n of a Z_p-extension K/k.
In this talk, I will give a similar formula for the maximal
unramified p-extension of a fixed nilpotent class i(>=1)
over k_n.
In this non-abelian situation, the formula involves
a certain invariant associated to the structure
of the Galois group of the maximal unramified p-extension
(not necessary abelian) over K.
 

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R. Pollack

 

Title: Efficient computations of p-adic L-functions via overconvergent
modular symbols

Abstract: The p-adic L-function of a modular form is defined as a certain
distribution on Z_p. Approximations to this L-function can thus be
obtained by Riemann sums. However, by using this direct approach, to get
"n digits" of p-adic accuracy one must take a sum over p^n discs.
Because of the exponential nature of this algorithm one simply cannot use
it to compute p-adic L-functions to high levels of accuracy.
In this talk, we will present an alternate approach (joint with Glenn
Stevens) to computing p-adic L-functions using overconvergent modular
symbols. The key idea is that by repeatedly applying the U_p operator to
an overconvergent version of the classical symbol attached to a modular
form one obtains better and better approximations to the p-adic L-function
of this form. Indeed, each application of U_p adds one p-adic digit of
accuracy and thus yields a polynomial time algorithm to computing such
L-functions.

If time allows we will describe a joint project with Henri Darmon where
these ideas are used to efficiently compute Stark-Heegner points which are
(conjecturally) global points over ray class fields of real quadratic
fields.

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C. Popescu

Title: 1-Motives, Tate sequences and L-functions

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J. Ritter & A. Weiss

(2 talks)

 Title: Toward nonabelian equivariant Iwasawa theory; 1,2

 Abstract: Let K/k be a Galois extension of totally real number fields, with
k/Q finite and with K finite over the cyclotomic l-extension of k, where l
is an odd prime number. In these talks a nonabelian "equivariant main
conjecture" of Iwasawa theory is formulated and verified in what one might
regard as the maximal order case. Moreover, it is shown that a complete
proof would follow from the existence of a nonabelian pseudomeasure, i.e.,
an element in the l-completion of the completed group ring of G(K/k) over
the l-adic integers, whose reduced norm is determined by the l-adic Artin
L-functions of the characters of G(K/k) with open kernel.

A. Weiss' slides

J. Ritter's talk

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K. Rubin

Title:  Organizing the arithmetic of elliptic curves.

Abstract:  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 ${\bf Z}_p$-power extension of
$K$.  Each of these Selmer groups comes equipped with a $p$-adic
height pairing and a Cassels pairing.
    Our goal is to produce a single free Iwasawa module of finite rank
with a skew-Hermitian pairing that packages all of this data.
This "organizing" module should give rise to all of the intermediate
groups and pairings.  Using recent work of Nekov{\'a}r we can show
that, under mild hypotheses, such an organizing module exists.
    This work is joint with Barry Mazur.
 

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R. Sharifi

Title: The Eisenstein ideal and cup products

Abstract: I will discuss relationships between the Eisenstein ideal in Hida's
ordinary Hecke algebra, Iwasawa modules over nonabelian extensions of
the rationals, and cup products in Galois cohomology with restricted
ramification.  In particular, I will give a condition for nontriviality
of certain cup products, which I have verified for p < 1000, proving a
conjecture of McCallum and myself for such primes.  I will also discuss
applications to the structure of Selmer groups of modular
representations.
 
 

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D. Solomon

Francais:   La fonction $\Theta_{K/k}(s)$ en $s=1$

Anglais:   The function $\Theta_{K/k}(s)$ at $s=1$

slides  

English resume:

Thanks to the functional equation, Stark's basic
conjecture for Artin L-functions of $K/k$ can be formulated equivalently
at s=1 or s=0. In the case where $K/k$ is \emph{abelian}, Stark, Brumer, Rubin,
Popescu et. al. have made refined conjectures at $s=0$ for the group-ring-valued
function $\Theta_{K/k}(s)$. This, however, satisfies no simple functional
equation.
  In this talk, therefore, we study (the minus part of) $\Theta_{K/k}(1)$.
Combined with a $p$-adic regulator, it give rise to a new invariant,
${\frak S}_{K/k}$, an analogue at $s=1$ of the generalised Stickelberger ideal.
When $p$ splits in $k$, we use twisted zeta-functions to prove an analogue of
Deligne/Ribet and Cassou-Nogu\`es integrality result. We also make some
conjectures including a congruence linking elements of ${\frak S}_{K/k}$ to
Rubin's conjecture via the Hilbert Symbol.

Resum\'e en francais:

Gr\^ace \`a l'\'equation fonctionnelle des fonctions $L$ d'Artin de $K/k$, la
conjecture de base due a Stark se lit indiff\'eremment en $s=1$ ou en $s=0$.
Au cas o\`u $K/k$ est abelienne,  Stark, Brumer, Rubin, Popescu ont donn\'e des
raffinements  concernant la fonction $\Theta_{K/k}(s)$ (\`a valeurs dans
l'anneau de groupe) en $s=0$.
  Cette derni\`ere ne poss\'edant pas d\'equation fonctionnelle simple,
on est amen\'e \`a une \'etude distincte de (la partie moins de)
$\Theta_{K/k}(1)$. En combinant celle-ci avec un r\'egulateur $p$-adique
j'obtiens une nouvelle invariante ${\frak S}_{K/k}$, analogue en $s=1$ de
l'd\'eal de Stickelberger
generalis\'e. Si $p$ est d\'ecompos\'e dans $k$ je me  sers des fonctions-zeta
tordues pour d\'emontrer un analogue du th\'eor\`eme d'int\'egralit\'e d\^u \`a
Deligne/Ribet and Cassou-Nogu\`es. J'\'enoncerai \'egalement de nouvelles
conjectures dont certaines congruences reliant les \'el\'ements de
${\frak S}_{K/k}$ \`a la conjecture de Rubin par moyen du  symbole d'Hilbert.
 

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E. Urban
 

Title : "$p$-adic deformations of Eisenstein serie, Selmer groups and critical L-values"

Abstract:  The aime of this talk is to discuss on the intimate relations between the three topics of
the title and how this can lead to proving Greenberg-Iwasawa main conjectures. I will go over some important examples for which one can construct lower bound for some important  Selmer groups. If time allows it, I will explain how this intimate relation can also be used to prove finiteness results and even construct upper bounds for these Selmer groups.

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O. Venjakob

Title : The GL_2 main conjecture for elliptic curves without complex multiplication.

Abstract : For studying the mysterious relationship between purely arithmetic
problems and the special values of complex L-functions, typified by the
conjecture of Birch and Swinnerton-Dyer and its generalizations, the main conjectures of Iwaswa theory are of particular interest.
The goal of this talk, which reports on a joint work with Coates, Fukaya, Kato and Sujatha, is to provide  algebraic techniques which enable one to formulate a precise version of  such a   main conjecture for motives over a large  class of p-adic Lie extensions of number fields. At the end of the talk we will formulate and briefly discuss the main conjecture for an elliptic curve E over the rationals Q over the field generated by the coordinates of its p-power division points, where p is a prime greater than 3 of good ordinary reduction for E.