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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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
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Title : Some aspects of non-commutative Iwasawa theory
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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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Title: p=37
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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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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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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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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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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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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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Title : "Parity of ranks of Selmer groups associated to Hilbert modular forms"
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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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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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Title: 1-Motives, Tate sequences and L-functions
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(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.
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 $s=1$
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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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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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.