An irreducible representation of a group G is said to be distinguished with respect to a subgroup H if it carries a non-trivial H-invariant linear form. For a quadratic extension E/F of finite or p-adic fields and for a reductive algebraic group over F, Dipendra Prasad has formulated certain conjectures regarding distinction for the pair (G(E),G(F)). In this talk we will mainly focus on the results in the case G=SL(n) (which are joint with Dipendra Prasad). These results in turn require a finer understanding of the more well-known case of G=GL(n) and we will describe some of these new results (which are joint with Nadir Matringe).

I will recall the notion of overconvergent cohomology and its role in p-adic interpolation of Hecke eigenvalues associated to automorphic representations. I will then discuss a construction of the p-adic adjoint L-function associated to families of Hilbert modular forms. This is joint work with Matteo Longo.

I will present a map - The toric regulator, from motivic cohomology of algebraic varieties over p-adic fields with "totally degenerate reduction", e.g., p-adically uniformized varieties, to "toric intermediate Jacobians" which are quotients of algebraic tori by a discrete subgroup. The toric regulator recovers part of the l-adic etale regulator map for every prime l, and its logarithm recovers part of the log-syntomic regulator. Its valuation is related to a regulator constructed by Sreekantan. I expect the toric regulator to be the home of "refined" Beilinson conjectures and I will present some evidence for this claim. This is joint work with Wayne Raskind from Wayne State university.

I will discuss some recent developments in integral p-adic Hodge theory (starting with my joint work with Matthew Morrow and Peter Scholze) that were inspired by potential applications to the structure of algebraic K-theory.

I will survey some recent vanishing theorems for the mod p cohomology of Shimura varieties. I will mention some p-adic results and some l-adic results, where l is a prime different from p. Both settings rely on the geometry of the Hodge-Tate period morphism, but I will try to highlight the differently flavoured techniques that are needed. This is largely based on joint work with Daniel Gulotta and Christian Johansson, and on separate joint work with Peter Scholze.

The absolute cohomological purity for étale cohomology of Gabber-Thomason implies that an étale cohomology class on a regular scheme extends uniquely over a closed subscheme of large codimension. I will discuss the corresponding phenomenon for flat cohomology. The talk is based on joint work with Peter Scholze.

I will explain how to interpolate Kato's Euler system on the deformation space of a mod p, 2-dimensional, Galois representation. This is joint work with Wang Shanwen.

In this talk and the next one we will discuss explicit class field theory and Stark’s conjectures on special values of L-functions. In the first talk we motivate our work beginning with Stark’s conjecture and its relevance to Hilbert’s 12th problem. The Brumer-Stark conjecture postulates the existence of special elements, called Brumer-Stark units, that generate abelian extensions of totally real number fields. Our first result is a proof of this conjecture away from p=2. A conjecture of the speaker refining the Gross-Stark conjecture provides an explicit p-adic analytic formula for Brumer-Stark units. Our second result is a proof of this conjecture, up to an explicitly bounded and small root of unity. Hilbert’s 12th problem asks for the explicit construction of the abelian extensions of a number field by means of analytic data intrinsic to the number field. The Brumer-Stark units, together with other easily described elements generate the maximal abelian extension of a totally real ground field. The combination of our results can therefore be viewed as a p-adic analytic resolution of Hilbert’s 12th problem for totally real fields.

I will review some properties of the moduli stacks of (phi,Gamma)-modules which I recently constructed with Matthew Emerton.

Completed cohomology, as defined by Emerton, gives a natural candidate for general spaces of p-adic automorphic forms. I’ll give some motivated introduction to completed cohomology and its role in the p-adic Langlands program, and review a fundamental conjecture of Calegari-Emerton which predicts the qualitative properties of completed cohomology. In the latter part of the talk, I’ll explain a proof of the Calegari-Emerton conjecture in many new cases. The arguments involve a fun combination of perfectoid methods and more classical Shimura variety techniques, and I’ll try to highlight the main ingredients. This is joint work with Christian Johansson.

If G is any p-adic reductive group we prove that G admits an irreducible admissible supersingular (or equivalently supercuspidal) representation over any field of characteristic p. This is joint work with Karol Koziol and Marie-France Vigneras.

In this talk we will give details on our proof of the Brumer-Stark conjecture away from the prime p=2 and the conjecture on explicit formulae for Brumer-Stark units. In fact, we prove a strong form of the Brumer-Stark conjecture. The Brumer-Stark conjecture asks for an upper bound on the ideal class group. Our strategy is based on proving a lower bound on a modification of the class group by using group ring Hilbert modular forms and Ribet’s method. The most difficult part is the argument for constructing everywhere unramified classes in the “p-indistinguishable” case (i.e. when the Hecke eigenvalues at primes above p are not distinct modulo p). A key aspect of this argument is the construction of “extra congruences” using the trivial zeroes of the associated L-functions in this case. Next we use the class number formula to show that the inequality we obtained using Ribet’s method is actually an equality. This part rests on delicate induction arguments. We then deduce the Brumer-Stark conjecture (away from p=2) from the equality that we proved. Our proof of the conjecture on explicit formulae for Brumer-Stark units is based on a strong integral refinement of our earlier work on the Gross-Stark conjecture (jointly with Kevin Ventullo) and relies crucially on a group ring Hilbert modular form with constant term 1 that we construct.

I will talk about results of lifting mod l Galois representations to geometric l-adic representations, and their applications. This talk is mainly based on recent joint work with N. Fakhruddin and S. Patrikis.

Given a variation of Hodge structures V on a smooth complex quasi-projective variety S, its Hodge locus is the set of points s in S where the Hodge structure V_s admits exceptional Hodge tensors. A famous result of Cattani, Deligne and Kaplan shows that this Hodge locus is a countable union of irreducible algebraic subvarieties of S, called the special subvarieties of (S, V). In this talk I will describe the geometry of the Zariski closure of the union of the positive dimensional special subvarieties. Joint work with Ania Otwinowska.

For a rational elliptic curve with positive Mordell-Weil rank, we discuss some properties of the zeta elements constructed by Kato, and formulate a new conjecture. We give some results which support our conjecture, explaining relations of our conjecture with a conjecture by Perrin-Riou and a conjecture by Mazur and Tate on refined Birch Swinnerton-Dyer conjecture. This is joint work with D. Burns and T. Sano.

Given a locally symmetric space Y and a mod p system m of Hecke eigenvalues, the weight part of Serre's conjecture seeks to classify the coefficient systems on Y whose cohomology contains m. Through Taylor-Wiles patching, this problem is known to be closely linked to the Breuil-Mézard conjecture, which predicts a representation theoretic recipe for the special fibers of local Galois deformation rings (and more generally, the Emerton-Gee moduli stack of local Galois representations) out of atomic pieces corresponding to irreducible coefficient systems. We will describe recent progress on both questions in many cases, via a theory of local models, which relates (portions of) the Emerton-Gee stack to mixed characteristics versions of (deformed) affine Springer fibers. This is joint work with B. Levin and S. Morra.

Given a locally symmetric space Y and a mod p system m of Hecke eigenvalues, the weight part of Serre's conjecture seeks to classify the coefficient systems on Y whose cohomology contains m. Through Taylor-Wiles patching, this problem is known to be closely linked to the Breuil-Mézard conjecture, which predicts a representation theoretic recipe for the special fibers of local Galois deformation rings (and more generally, the Emerton-Gee moduli stack of local Galois representations) out of atomic pieces corresponding to irreducible coefficient systems. We will describe recent progress on both questions in many cases, via a theory of local models, which relates (portions of) the Emerton-Gee stack to mixed characteristics versions of (deformed) affine Springer fibers. This is joint work with B. Levin and S. Morra.

We explore the structure of the fibers of the Hitchin morphism for higher dimensional algebraic varieties.

I will explain an ongoing project with John Christian Ottem to establish several new classes of stably irrational complete intersections. Our results are based on degeneration techniques and a birational version of the nearby cycles functor that was developed in collaboration with Evgeny Shinder.

I will show how to compute the p-adic etale cohomology with compact support of period domains over local fields in the case of a basic isocrystal for quasi-split reductive groups. This follows the method used by Orlik in his computations of the l-adic etale cohomology using as a key new input the computation of Ext groups between mod p generalized Steinberg representations of p-adic groups. This is a joint work with Colmez, Dospinescu, and Hauseux.

We will explain vanishing theorems for the cohomology of the moduli space of abelian varieties using p-adic methods. This is joint work with G. Boxer.

Filtration by ramification groups of the Galois group of an extension of local fields with possibly imperfect residue fields is defined by Abbes and the speaker. The graded quotients are abelian groups and annihilated by the residue characteristic. We discuss the main ingredients of the proof and the construction of injections of the character groups of the graded quotients.

I will explain the explicit description of the Euler characteristic of automorphic l-adic local systems on the moduli stack A_g of principally polarised abelian varieties of dimension g (i.e. level one Siegel modular varieties) in terms of Galois representations taking values in GSpin groups. A crucial step is the explicit description of the cohomology of associated intersection complexes on the minimal compactification, i.e. the proof of (an unconditional formulation of) a particular case of a general conjecture of Kottwitz. The proof relies notably on work of Kottwitz and S. Morel, as well as automorphic and Galois-theoretic arguments.

In the Langlands program there are many interesting examples of "period integrals" that produce L-functions from automorphic forms. I will explain a duality that switches a period integral for a group G with a period integral for the dual group G^. Joint work with David Ben-Zvi and Yiannis Sakellaridis.

We bound the Bloch-Kato Selmer group of certain Rankin-Selberg L-functions, by constructing a Bertolini-Darmon-Kolyvagin system using the Gan-Gross-Prasad cycles on the product of unitary Shimura varieties. This allows us to relate the triviality/nontriviality of the Selmer group to the vanishing order of the L-function. We discuss some examples from elliptic curves. This is a joint work with Yifeng Liu, Yichao Tian, Liang Xiao and Wei Zhang.