# Titles and Abstracts

**Konstantin Ardakov** (QMUL)

*Equivariant $\widehat{\mathcal{D}}$-modules on rigid analytic spaces*

We introduce a sheaf $\widehat{\mathcal{D}}$ of locally Frechet-Stein algebras on smooth rigid analytic spaces that is morally a ``rigid analytic quantisation'' of the cotangent bundle. Using coadmissible equivariant $\widehat{\mathcal{D}}$-modules on rigid analytic flag varieties, we give an approach to the localisation of admissible locally analytic representations of semisimple compact $p$-adic Lie groups.

** Denis Benois ** (Bordeaux)

* On the derivatives of $p$-adic L-functions*

We discuss the conjectural behavior of $p$-adic L-functions at near central points in the case of a non-trivial regulator map.

** Laurent Berger ** (Lyon)

*Sen theory and Lubin-Tate extensions*

The classical theorems of Sen concern an abelian extension $L/K$ whose Galois group is a $p$-adic Lie group of dimension 1. I will propose some extensions of these results to the case when $L$ is generated over $K$ by the torsion points of a Lubin-Tate group.

** Thanasis Bouganis ** (Heidelberg)

*$p$-adic measures for Hermitian modular forms and the Rankin-Selberg method*

We will lecture on the construction of $p$-adic measures for Hermitian modular forms (modular forms associated to unitary groups). For the construction of such measures there exist two general strategies. The first is an ongoing project of Eischen, Harris, Li and Skinner and it is based on the so-called "Doubling-Method". The second is based on Rankin-Selberg integrals involving theta series. In this talk we will report on our work on developing this second strategy. If time allows we will also discuss the similarities and differences to the Siegel modular forms situation.

** Kevin Buzzard ** (Imperial)

* What does the mod $p$ local Langlands correspondence look like?*

What does the mod $p$ local Langlands correspondence look like? No-one is quite sure in general, which is a bit embarassing really. It's all well and good people proving profound results for $\text{GL}(2,\mathbb{Q}_p)$ but as far as I know we're even short of a statement of a conjecture for $\text{GL}(3,\mathbb{Q}_p)$ (although perhaps someone reading this abstract will put me right on this). I'll explain what I think the picture is and what little we know. Don't expect theorems.

**Xavier Caruso** (Rennes)

*What is the analogue of $(\phi,\Gamma)$-modules for the Breuil-Kisin extension?*

Let $K$ be a complete discrete valuation field of mixed characteristic. In the last decade, Breuil and Kisin have noticed that the extension $K_\infty$ of $K$, obtained by adding a compatible system of $p^n$th-roots of a fixed uniformizer, is well suited (more than the usual cyclomotic extension is some sense) to study semi-stable representations of the absolute Galois group of $K$. A natural question then arises: is there a good analogue of $(\phi,\Gamma)$-modules in this context? In this talk, I shall give a partial answer to this question together with a more complete, but still conjectural, one.

**John Coates** (Cambridge)

*On the conjecture of Birch and Swinnerton-Dyer *

The lecture will discuss recent progress on the conjecture of Birch and Swinnerton-Dyer, growing out of the work of Ye Tian on the classical Congruent Number Problem.

**Takako Fukaya** (Chicago)

*On some variants of Sharifi conjectures*

R. Sharifi has formulated mysterious conjectures on the relation between the arithmetic of cyclotomic fields and modular curves. I will discuss some variants of his conjectures. I will talk especially about the case of function fields. This is joint work with K. Kato and R.Sharifi.

**Henri Johnston** (Cambridge)

*$p$-adic group rings and the equivariant Tamagawa number conjecture*

We discuss applications of the understanding of the structure of group rings of finite groups over $p$-adic integers to the equivariant Tamagawa number conjecture. This is joint work with Andreas Nickel.

** Mahesh Kakde ** (KCL)

* Congruences in Iwasawa theory and applications*

An explicit description of $K_1$ groups of noncommutative Iwasawa algebras will be presented. This will be followed by applications including main conjecture for totally real fields and Kato's local epsilon conjecture.

** Jack Lamplugh ** (Cambridge)

*An analogue of the Washington-Sinnott theorem for elliptic curves with CM*

**David Loeffler** (Warwick)

* Euler systems for Rankin-Selberg convolutions of modular forms*

An Euler system is a certain compatible family of classes in the cohomology of a Galois representation, which plays a key role in relating arithmetical properties of the representation to values of the associated L-function. Only a few examples of such systems have been constructed to date, although they are conjectured to exist in quite general settings. I will describe a construction of an Euler system for the tensor product of the Galois representations of two modular forms, and an application to bounding Selmer groups. This is joint work with Antonio Lei and Sarah Zerbes.

** Jan Nekovář ** (Jussieu)

* Generalised Eichler-Shimura relations and Galois representations occurring in cohomology of certain Shimura varieties*

**Andreas Nickel** (Bielefeld)

*Equivariant Iwasawa theory and non-abelian Stark-type conjectures*

We discuss different formulations of the equivariant Iwasawa main conjecture focussing on a recent formulation due to Greither and Popescu. Using the results of Ritter and Weiss and of Kakde on the main conjecture, we give new evidence for non-abelian generalisations of Brumer's conjecture and the Coates-Sinnott conjecture.

**Sujatha Ramdorai** (Warwick)

* Residual representations and Iwasawa theoretic invariants*

Given two elliptic curves over $\mathbb{Q}$ that have good ordinary reduction at an odd prime $p$, and such that their residual Galois representations are irreducible and isomorphic at $p$, we discuss the behaviour of some Iwasawa theoretic invariants of these curves over the cyclotomic $\mathbb{Z}_p$-extension and the non-commutative False Tate extensions.

**Romyar Sharifi** (Arizona)

*Conjectures on cup products of units and modular symbols*

I will discuss conjectures relating cup products of cyclotomic units and modular symbols modulo an Eisenstein ideal. T. Fukaya and K. Kato have proven these conjectures, in part under mild hypotheses. I also hope to discuss joint work with Fukaya and Kato on variants, focusing on the case of elliptic units.