## Eisenstein cohomology, Bloch-Kato conjecture for Hecke characters, and Fontaine-Mazur conjecture for imaginary quadratic fields

### by Tobias Berger

Abstract: For certain algebraic Hecke characters c of an imaginary quadratic field F we define an Eisenstein ideal in a Hecke algebra acting on cuspidal automorphic forms of GL2/F. We prove a bound for its index in terms of the special L-value L(0,c) by finding a congruence between an Eisenstein cohomology class (in the sense of G. Harder) and a cuspidal cohomology class in the cohomology of a symmetric space associated to GL2/F.

Using the work of R. Taylor et al. on attaching Galois representations to cuspforms of GL2/F we obtain from this a lower bound for the size of the Selmer group of a Galois character coinciding with the value given by the Bloch-Kato conjecture.

We further show, following work by Skinner and Wiles for Q, how the Eisenstein ideal bound can be used to prove certain instances of the Fontaine-Mazur conjecture for imaginary quadratic fields. The latter is joint work in progress with Kris Klosin.

## Algebraic p-adic L-functions and main conjectures in non-commutative Iwasawa theory

### by David Burns

Abstract: We define a notion of "algebraic $p$-adic $L$-function" in non-commutative Iwasawa theory. We use this notion to prove that, modulo the existence of analytic $p$-adic $L$-functions, the main conjecture of non-commutative Iwasawa theory (as formulated by Coates, Fukaya, Kato, Sujatha and Venjakob) can be reduced to consideration of abelian extension.

## Root numbers and Iwasawa theory I and II

### by John Coates and Ramdorai Sujatha

Abstract: We will discuss joint work with T. Fukaya, K.Kato on the connections between root numbers of complex L-functions and non-abelian Iwasawa theory.

## Sur la correspondance de Langlands locale p-adique

### par Pierre Colmez

Abstract: Nous essayerons de faire le point sur la correspondance de Langlands locale p-adique pour GL2.

## L-functions of elliptic curves in the false Tate curve extension

Abstract: A false Tate curve'' extension of $Q$ is generated by the $p^n$-th roots of a fixed integer $m$ and by the $p^n$-th roots of unity, where $n$ runs over the natural numbers and where $p$ is a fixed odd prime. It appears in non-commutative Iwasawa theory as the simplest example of a genuinely non-commutative $p$-adic Lie extension.

The subject of this talk will be the L-functions $L(E/K,s)$, where $E$ is an elliptic curve initially defined over $Q$, and where $K$ is a number field contained in a false Tate curve extension. Although $K/Q$ need not be abelian or even Galois, such L-functions can be shown to possess an analytic continuation to the complex plane. A root number computation predicts several interesting arithmetic phenomena in this setting. In particular, there are examples of elliptic curves whose rank (conjecturally) grows at least as fast as p^n in the false Tate curve tower, and of elliptic curves with rank 0 over $Q$ that should have points of infinite order over every cubic extension of the form $Q(m^{1/3})$.

Finally, if time allows, I will also indicate how one can prove that the special value $L(E/K,1)$, modified as in the Birch-Swinnerton-Dyer conjecture, is a rational number (joint work with Thanasis Bouganis).

## Congruences in non-commutative Iwasawa theory

### by Kazuya Kato

Abstract: This is a joint work with K. Oshima. We compute K1 group of the completed group ring of a compact p-adic Lie group of a special type. This is related to non-commutative Iwasawa theory.

## On a conjecture concerning minus parts in the style of Gross

### by Radan Kucera joint work with Cornelius Greither

Abstract: In the eighties, B.~Gross introduced a conjecture which is close to Stark's conjectures inasmuch as it postulates a link between L-values and regulators, but which differs from Stark's conjectures and already from Dirichlet's class number formula in a very important aspect: the regulators are not complex or $p$-adic numbers, arising as determinants of logarithms of certain algebraic numbers, but they lie in an appropriate quotient of the augmentation filtration of $\Z[G]$, where $G$ is the Galois group of the abelian field extension $K/F$ under consideration, and they are obtained as determinants of matrices made from certain local Artin symbols.

Recently D.~Burns discovered a rather general conjectural framework welding together Stark-type and Gross-type conjectures. For a real abelian extension $K/\Q$ this applies in two steps. First there is a Stark unit $\eta_K$ (whose existence is proven in this case, not just a conjecture), and then there is a statement concerning the position of $\eta_K$ within the whole group ${\cal O}_K^*$'' in terms of a Gross regulator. At the first stage, a classical regulator is used, namely a determinant involving the logarithms of the conjugates of $\eta_K$. At the second stage, the Gross-style regulator is used to obtain a conjectural congruence in $\Z[G]$ modulo a high power of the augmentation ideal. (The whole setup is generalized to any finite abelian extension of global fields, a real abelian extension of $\Q$ was considered here just to keep things as simple as possible.)

In subsequent work of A.~Hayward, where Burns's conjectures are discussed and in some cases proved, another conjecture comes into play which may be considered as the minus part'' of Burns's conjecture for extensions $K/F$ where $F$ is an imaginary quadratic field and $K$ is absolutely abelian. This minus conjecture'' equates, up to constant factors, the leading term of a Stickelberger element and a regulator made up from $S$-units in the minus part. (In fact, this minus conjecture'' is a very special case of what is called the conjecture of Gross on tori'', on which nothing much seems to be known.) In particular, the minus conjecture'' gives what should be obtainable, roughly speaking, by dividing a conjectural equation for $K/F$ by the corresponding equation for $K^+/\Q$, where $K^+$ is the maximal real subfield of $K$. However, this division process often does not make sense (all quantities involved may be zero). So we are interested in direct proofs of the minus conjecture'' which do not use the conjectural equations for either $K/F$ or $K^+/\Q$. This talk presents some (very partial) results in this direction.

## On unramified pro-p Galois groups over cyclotomic Zp-extensions

### by Yasushi Mizusawa

Abstract: A main object of this talk is the Galois group G(kcyc) of the maximal unramified pro-p-extension over the cyclotomic Zp-extension kcyc of an algebraic number field k, which is conjectured to be finitely generated as a pro-p group (i.e., " μ=0").

In "Iwasawa 2004", M. Ozaki gave a certain Iwasawa type formula (analogous to " λn+μpn") for the nilpotent unramified p-extensions along the Zp-extension with μ=0 by considering the lower central series of G(kcyc) with the action of Γ=Gal(kcyc/k). On the other hand, G(kcyc) has been studied in the theory of "pro-p Γ-operator Galois groups". We also expect that G(kcyc) would give good information on the structure of the Galois groups of "p-class field towers''.

In this talk, I will first give an overview of such topics on G(kcyc), which can be regarded as an approach to non-abelianization of Iwasawa theory, and will talk on several results on the explicit presentation of some G(kcyc), which I recently got.

## On a weak form of Greenberg's conjecture

### by Thong Nguyen Quang Do

Abstract: Let p be an odd prime number. For any CM number field K containing a primitive p-th root ok unity, class field theory and Kummer theory put together yield the well-known reflexion inequality $\lambda^+$ <= $\lambda^-$ between the "plus" and "minus" parts of the $\lambda$-invariant of K. Greenberg's classical conjecture predicts the vanishing of $\lambda^+$. We propose a weak form of this conjecture: $\lambda^+=\lambda^-$ if and only if $\lambda^+=\lambda^-=0$. Note that the two forms (weak and strong) are equivalent if the usual Iwasawa series is irreducible. We study the discrepancy $\lambda^- - \lambda^+$ and we prove the weak conjecture in the case where K+ is abelian, p is totally split in K+, and certain conditions on the cohomology of circular units are satisfied (e.g. in the semi-simple case).

## On the Selmer groups of abelian varieties over function fields of characteristic p (joint with Fabien Trihan)

Abstract: Iwasawa theory over a non commutative p-adic Lie extension is initiated by John Coates and has been studied actively for elliptic curves (and abelian varieties) over number fields.
Especially, the algebraic side of the theory has progressed a lot through work on Selmer group and on non-commutative ring theory by various people. In this talk, we study a geometric analogue of non-commutative Iwasawa theory for abelian varieties over a non-commutative p-adic Lie extension of a global function field of characteristic p.
Along a formulation of finiteness" for Selmer groups due to Venjakob and others, we prove a geometric analogue for Selmer group of abelian varieties over function fields. If time allows, we will also talk about other results such as observation on the mu-invariants.

## Growth of Selmer groups in dihedral extensions

### by Karl Rubin

Abstract: In joint work with Barry Mazur, we obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields.

If F/k is a dihedral extension of number fields of degree 2n with n odd, and E is an elliptic curve over k that has odd rank over the quadratic extension K of k in F, then standard conjectures (and a root number calculation) predict that E(F) has rank at least n. The only case where one can presently prove anything close to this bound is when K is imaginary quadratic, and E(F) contains Heegner points.

Mazur and I prove unconditionally that if n is a power of an odd prime p, F/K is unramified at all primes where E has bad reduction, all primes above p split in K/k, and the p-Selmer corank of E/K is odd, then the p-Selmer corank of E/F is at least n. This provides a large class of examples of Zpd-extensions where the Selmer module is not a torsion Iwasawa module.

## A reciprocity map and the two-variable p-adic L-function

### by Romyar Sharifi

Abstract: I will discuss work-in-progress on a conjectural relationship between two objects. The first is a "reciprocity map" arising as a coboundary in a sequence of cohomology groups of cyclotomic fields with restricted ramification at p. The second is the two-variable p-adic L-function of Kitagawa-Mazur modulo an Eisenstein ideal, when viewed in the appropriate space. This has consequences for the nontriviality of cup products studied by McCallum and myself as well as to the structure of Selmer groups of cusp forms satisfying congruences with Eisenstein series.

## Heegner points over false Tate curve extension

### by Ye Tian

Abstract: Let $E$ be an elliptic curve defined over $\BQ$ and let $p$ and $q$ be two odd primes. In the Heegner situation w.r.t $p$ and $q$, we compute the rank of $E(L)$ as $L$ runs over subfield of the False Tate curve extension associated to $p$ and $q$. This talk is based a joint work with Henri Darmon.

## Théorie de Hida et représentations galoisiennes de bas poids

### par Jacques Tilouine

Abstract: Les algèbres de Hecke universelles ordinaires introduites par Hida puis generalisées permettent de faire varier les poids de formes automorphes p$p$-ordinaires pour différents groupes. Ceci permet de construire des formes p-adiques de bas poids munies d'une représentation galoisienne qui peut coincider avec la représentation associée à une variété abélienne ou une représentation d'Artin. Nous discuterons de tels cas pour le groupe $GSp(4)$.

## On a weak version of Greenberg's generalized conjecture

### by David Vauclair

Abstract: Fix p an odd prime and F a number field containing a pth root of unity. Greenberg's generalized conjecture (GG) predicts the pseudo-nullity of the Iwasawa module attached to the p-class groups in the compositum $\tilde F$ of all Zp-extensions of F. The purpose of this talk is to prove, under restrictive assumptions, the validity of a weak form of (GG). The point is to study the capitulation of certain cohomology group into $\tilde F$, which is done here with the help of a cup product.

## On the main conjecture of equivariant Iwasawa theory in the case of pro-l groups of nilpotency class 2

### by Alfred Weiss

Abstract: This talk provides a translation of what we have been calling a main conjecture of equivariant Iwasawa theory into hypothetical congruences between Iwasawa L-functions. The translation emerges from a logarithmic approach by which the conjecture becomes equivalent to recognizing the L-functions as determinants from K_1 of a localization of an Iwasawa algebra. Part of this will already have been explained in Ritter's talk in Besancon. An explicit example will illuminate what is left in verifying the conjecture in the 'smallest' non-abelian case.

## Sur la conjecture de modularité de Serre

### par Jean-Pierre Wintenberger

Abstract: La conjecture de Serre dit qu'une représentation impaire irréductible du groupe de Galois de Q dans un espace vectoriel de dimension 2 sur la clôture algébrique du corps à p éléments provient d'une forme modulaire. Nous exposerons les grandes lignes de la preuve de la conjecture, lorsque p est différent de 2 et la représentations non ramifiée en 2, obtenue en collaboration avec C. Khare.

## Bounds for the dimensions of the p-adic multiple zeta value (L-value) spaces

### by Go Yamashita

Abstract: We show the upper bounds of $p$-adic multiple zeta value (resp. $L$-value) spaces. The bounds are related to the algebraic $K$-theory. It is the $p$-adic analogue of the theorem of Goncharov, Terasoma, Deligne-Goncharov (resp. Deligne-Goncharov). In the $p$-adic multiple $L$-value case, the bounds are not best possible in general. The gap between the true dimensions and the bounds related algebraic $K$-theory is related to spaces of modular forms, by the similar way as complex multiple $L$-values. We also formulate the $p$-adic analogue of Grothendieck's conjecture about an element of motivic Galois group of the category of mixed Tate motives. It seems related to "Cebotarev density" of the motivic Galois group.