# Titles and Abstracts

**Tobias Berger** (Sheffield)

*A higher-rank analogue of Shintani's theta lift for imaginary quadratic fields*

**Andrew Booker** (Bristol)

*Converse theorems for $\mathrm{GL}(3)$*

The converse theorem of Jacquet, Piatetski-Shapiro and Shalika for GL(3) automorphic representations is well-known and has led to the resolution of some important cases of Langlands' functoriality conjectures. A less well-known variant of Cogdell and Piatetski-Shapiro uses analytic properties of the Rankin-Selberg twists by unramified GL(2) representations, but only over number fields with class number 1. I will report on work in progress with Muthu Krishnamurthy which removes the class number restriction when the input data comes from an Artin representation.

**Armand Brumer** (Fordham)

*On the Paramodular Conjecture*

**Kevin Buzzard** (Imperial)

*$C$-groups*

Given an automorphic representation (for example a Dirichlet character, modular eigenform, Maass form, grossencharacter...), Langlands' conjectures would predict the existence of some sort of associated "Galois representation" but sometimes the reality might be a lot vaguer -- one might have to deal with conjectural generalisations of Galois groups, for example the "global Langlands group", which still has no definition independent of conjectures. For cohomological automorphic representations (i.e. the ones for which one can hope to actually prove something) there is a conjecture where one can stick to Galois groups and everything is actually defined. However, perhaps surprisingly, one has to generalise Langlands' notion of an $L$-group slightly before one can state it. Toby Gee and I figured out the details of what the conjecture should be, and Christian Johansson has checked its compatibility with existing conjectures on the cohomology of Shimura varities (which are supposed to be the source of such representations). In the talk I will give an overview of the theory; I will explain what an $L$-group is, what a $C$-group is, and I will explain what the conjecture is through examples.

**Fred Diamond** (King's College London)

*Explicit Serre weights for two-dimensional Galois representations*

I will discuss joint work with Savitt on Serre weights for indecomposable two-dimensional mod p representations of Galois groups over ramified extensions of $\mathbb{Q}_p$. In particular, we prove results on the weight part of Serre's Conjecture in this context, indicating a structure of the set of weights and making it more explicit.

**Kai-Wen Lan** (Minnesota)

*Compactifications of PEL-type Shimura varieties and Kuga families with ordinary loci*

I will report on the construction of p-integral models of various algebraic compactifications of PEL-type Shimura varieties and Kuga families, allowing arbitrary ramification (including deep levels) at p, with good behaviors over the loci where certain (multiplicative) ordinary level structures are defined. I will explain why such a construction is useful for studying automorphic representations, and provide qualitative descriptions that might (it is hoped) make things easier.

**Judith Ludwig** (Imperial)

*P-adic Langlands functoriality for inner forms of unitary groups*

In this talk I will explain a notion of p-adic functoriality for inner forms of definite unitary groups. We will then study some properties of the classical Langlands functoriality and use them to construct a morphism between certain eigenvarieties attached to the unitary groups in question, which can be interpreted as an instance of *p*-adic functoriality.

**James Newton** (Cambridge)

*Serre weights and Shimura curves*

I will report on a joint project with Teruyoshi Yoshida, the goal of which is to study the mod $p$ étale cohomology of Shimura curves with full level at $p$ via the geometry of suitable semistable models for these curves. As a first step, we compute the de Rham cohomology of the mod $p$ special fibre of these models and relate it to the weight part of Serre's conjecture (as generalised by Buzzard, Diamond, Jarvis and Schein). This allows us to give a geometric interpretation of these conjectures.

**Roger Plymen** (Southampton)

*Local Langlands correspondence for inner forms of $\mathrm{SL}_n$*

A remarkable feature of the Langlands conjectures is that it is better to consider not just one reductive group at a time, but all inner forms simultaneously. The inner forms of $\mathrm{SL}_n$ come about as follows: let $F$ be a local nonarchimedean field and let $D$ be a division algebra, with centre $F$ and dimension $d^2$ over $F$. With $n = md$, the kernel of the reduced norm map $\mathrm{GL}_m(D) \to F^{\times}$ is an inner form of $\mathrm{SL}_n(F)$. The local Langlands correspondence for inner forms of $\mathrm{SL}_n(F)$ is due in characteristic 0 to Hiraga-Saito. In positive characteristic, this result is joint work with Aubert-Baum-Solleveld. I will illustrate this result with the example of $\mathrm{SL}_2(F)$ when $F$ is a local function field of characteristic 2. There are countably many supercuspidal $L$-packets for $\mathrm{SL}_2(F)$, and each packet has 4 elements. We compute the formal degrees of these supercuspidals – joint work with Sergio Mendes.

**David Savitt** (Arizona)

*Lattices in the cohomology of Shimura curves*

I will discuss joint work with Matthew Emerton and Toby Gee, in which we relate the geometry of tamely potentially Barsotti-Tate deformation rings for two-dimensional Galois representations to the integral structure of the cohomology of Shimura curves. As a consequence, we establish some conjectures of Breuil regarding this integral structure.

**Tony Scholl** (Cambridge)

*Remarks on monodromy and weights*

**Chris Skinner** (Princeton)

*Vanishing of $L$-functions and ranks of Selmer groups*

**Shaun Stevens** (UEA)

*On the local Langlands correspondence for classical groups*

**Jacques Tilouine** (Paris XIII)

*Big image of Galois and congruence ideals: The case of GSp(4)* (work in progress with H. Hida)

**Lynne Walling** (Bristol)

*Hecke operators on Siegel Eisenstein series*

**Teruyoshi Yoshida **(Cambridge)

*Integral models for $X(p)$ and Serre weights*

I will explain the semistable models of full level p modular and Shimura curves over a tamely ramified extension of the base field, and the representations of GL(2) over the residue field that appear in de Rham cohomology of their components, which are Deligne-Lusztig curves and Igusa curves. This is used in James Newton's talk on our joint work.