The algebraic Sato-Tate group and Sato-Tate conjecture

Grzegorz Banaszak

Let $M$ be a motive over a number field $K$ in the Deligne's motivic category for absolute Hodge cycles. In an effort of proper setting of the Sato-Tate conjecture concerning the equidistribution of Frobenius elements in the representation of the Galois group $G_K$ on the $l$-adic realization of $M \otimes_{K} \overline{K}$, one of attempts is the introduction of the algebraic Sato-Tate group $AST_{K}(M)$. Maximal compact subgroups of $AST_{K}(M)(\mathbb{C})$ are expected to be the key tool for the statement of the Sato-Tate conjecture for $M$. An explicit construction of $AST_{K}(M)$ will be presented following an idea of J-P. Serre. The arithmetic properties of $AST_{K}(M)$ will be discussed along with applications for the Sato-Tate conjecture. This is joint work with Kiran Kedlaya.

Deformations of Saito-Kurokawa type I+II

Tobias Berger, Krzysztof Kłosin

We will report on joint work in progress on deformations of a 4-dimensional mod $p$ Galois representation $\overline{\sigma}$ of Saito-Kurokawa type, i.e. whose semisimplification equals $\chi^{k-2} \oplus \rho \oplus \chi^{k-1}$, where $\chi$ is the mod $p$ cyclotomic character and $\rho$ is an irreducible reduction of a $p$-adic Galois representation $\rho_f$ attached to an elliptic cusp form $f$ of weight $2k-2$. We show that under certain assumptions the universal crystalline (in Fontaine-Lafaille range) deformation ring is a dvr. When $k=2$ this allows for the verification of the Paramodular Conjecture of abelian surfaces over Q with a rational torsion point of order p when one can find a suitable Siegel modular form. When $k>2$, using a theorem due to Agarwal and Brown we obtain a modularity lifting theorem showing modularity of all such deformations of $\overline{\sigma}$.

Congruence primes for automorphic forms on unitary groups and applications to the arithmetic of Ikeda lifts.

Jim Brown

The problem of classifying congruences between automorphic forms has attracted a considerable amount of attention due not only to its inherent interest, but also because of the arithmetic applications of many of these congruences. For instance, congruences between automorphic forms have played a crucial role in recent progress on the Birch and Swinnerton-Dyer conjecture. In this talk we will discuss a sufficient condition for a prime to be a congruence prime for an automorphic form on $\textrm{U}(n,n)(\mathbb{A}_{F})$ where $F/\mathbb{Q}$ is a totally real field. This sufficient condition is given in terms of the divisibility of a certain special $L$-value of the automorphic form. We then apply this result to the case of the Hermitian Ikeda lift. This work is joint with Kris Klosin.

On the p-adic local Langlands correspondence

Pierre Colmez

Arithmetic of Picard modular surfaces modulo an inert prime

Ehud de Shalit

Picard modular surfaces are Shimura varieties attached to the group $U(2,1)$. They parametrize certain abelian 3-folds with endomorphisms by the ring of integers of a quadratic imaginary field $K$. Their geometry has been studied by many authors, including Larsen, Bellaiche, Bultel, Wedhorn, and Vollard. We concentrate on some properties of the reduction modulo a good prime $p$ which is inert in $K$, that have not been discussed before. Our main goal is to prepare the scene for the study of $p$-adic modular forms and $p$-adic L functions when $p$ is inert, but some of the results are interesting in their own right.
Joint work with Eyal Goren.

Geometric Serre weights for Hilbert modular forms

Fred Diamond

A (proven) version of Serre's Conjecture implies that an odd two-dimensional mod $p$ representation of a Galois group over $\mathbb{Q}$ is unramified at $p$ if and only if it arises from a mod $p$ modular form of weight one. I'll discuss some questions, conjectures and results pertaining to a generalization that predicts which mod $p$ representations of Galois groups over totally real fields arise from mod $p$ Hilbert modular forms of partial weight one.
(This is joint work with S. Sasaki.)

$p$-adic L-functions for unitary groups: a status report

Ellen Eischen

I will summarize a construction of $p$-adic L-functions associated to automorphic forms on unitary groups, joint with M. Harris, J.-S. Li, and C. Skinner. This project (which builds on work of several individuals, especially H. Hida and N. Katz) has grown and expanded since its initial conception in the early 2000s by Harris, Li, and Skinner. As part of the talk, I will address several important questions others frequently ask me about the project, including (but not limited to): What is its current status? What is now known, and what are the key ideas we have employed? Which challenges remain, and which do we expect to overcome? The $p$-adic Eisenstein measures from my own work will also feature prominently, since they are a key ingredient in the joint project.

Weil-étale cohomology and Zeta functions of arithmetic schemes

Matthias Flach

We report on joint work with Baptiste Morin in which we give a description of the vanishing order and leading Taylor coefficient of the Zeta function of a regular proper arithmetic scheme $\mathfrak{X}$ at any integer argument $n\in\mathbb{Z}$ in terms of Weil-étale cohomology complexes. This extends work of Lichtenbaum for the Dedekind Zeta function at $s=0$ and of Milne, Lichtenbaum and Geisser for varieties over finite fields. Our description is compatible with the Tamagawa number conjecture of Bloch, Kato, Fontaine, Perrin-Riou if $\mathfrak{X}$ is smooth over the ring of integers in a number field. We discuss in some detail the example of the Dedekind Zeta function at any $s=n\in\mathbb{Z}$ and elliptic curves over $\mathbb{Q}$ at $s=1$.

Reductions of Galois representations of small slopes

Eknath Ghate

We investigate the shape of the semisimplification of the reductions of certain crystalline representations of the Galois group of $\mathbb{Q}_p$. While the answer is now essentially completely known for all fractional slopes less than 2, thanks to work of Breuil, Buzzard-Gee, Bhattacharya-Ghate and others, in this talk we shall concentrate on describing the reduction in the very interesting missing case of integral slope 1.

The proof uses the compatibility between the mod $p$ and $p$-adic Local Langlands Correspondences with respect to the process of reduction, due to Berger, and reduces to a delicate harmonic analysis problem on the Bruhat-Tits tree, which we solve.

We also describe a subtler property of the reduction, namely how to distinguish between the peu and trés ramifiée reductions, in the relevant non-semisimple cases, using a more refined version of the above compatibility, due to Colmez.
This is joint work with Shalini Bhattacharya and Sandra Rozensztajn.

Cuspidal vs. Eisenstein cohomology

Günter Harder

To be announced

General Serre weight conjectures

Florian Herzig

We discuss Serre weight conjectures for $GL_n$ and some more general reductive groups, with a particular focus on explicit versions for Galois representations that are semisimple locally at $p$. This is joint work with T. Gee, and D. Savitt.

Higher derivatives of p-adic L-functions of Hilbert modular forms

Andrei Jorza

Greenberg and Stevens proved the Mazur-Tate-Teitelbaum conjecture on the exceptional zero of the $p$-adic L-function attached to an elliptic curve with split multiplicative reduction at $p$ using $p$-adic families of modular forms. A couple of years ago, in collaboration with Daniel Barrera and Mladen Dimitrov we applied the Greenberg-Stevens method to compute the first derivative of the $p$-adic L-function attached to a finite slope Hilbert modular form of not necessarily parallel weight. In recent work the three of us show that the $p$-adic L-function we construct satisfies a factorization formula which generalizes a result of Spiess in the ordinary parallel weight 2 case. Combining the factorization formula with our previous results yields a computations of higher derivatives of $p$-adic L-functions in terms of arithmetic L-invariants.

$F$-isocrystals with infinite monodromy

Joe Kramer-Miller

Let U be a smooth geometrically connected affine curve over $\mathbb{F}_p$ with compactification X. Following Dwork and Katz, a $p$-adic representation $\rho$ of $\pi_1(U)$ corresponds to an F-isocrystal. By work of Tsuzuki and Crew an F-isocrystal is overconvergent precisely when $\rho$ has finite monodromy at each $x \in X-U$. However, in practice most F-isocrystals arising geometrically are not overconvergent and have logarithmic growth at singularities (e.g. characters of the Igusa tower over a modular curve). We give a Galois-theoretic interpretation of these log growth F-isocrystals in terms of asymptotic properties of higher ramification groups.

Hida families and p-adic triple product L-functions.

Ming Lun Hsieh

We present a construction of the three-variable $p$-adic triple L-function attached to Hida families. This $p$-adic L-function is a three-variable power series with $p$-integral coefficients interpolating central L-values of triple product L-functions in the balanced case. We will give the explicit interpolation formula at all critical specialisations in the balanced range and discuss problems related to this $p$-adic L-function.

On the mod $p$ local Langlands program for Hilbert modular forms

Stefano Morra

Let $F=\mathbb{Q}_p$ be a fnite extension. The mod $p$-local Langlands program is expected to be realized in Hecke isotypical spaces in the cohomology of compact unitary groups with infnite level at $p$. In particular the local automorphic representations appearing in a local correspondence are expected to satisfy the constraints coming from Serre's conjectures (which is now a theorem by work of Gee and collaborators).

Breuil and Paskunas show that when $F\neq \mathbb{Q}_p$ is unramifed the Galois parameters are much fewer than $GL_2(F)$-representations, even taking into account the constraints on their $GL_2(\mathcal{O}_F)$-socle coming from Serre's conjectures. Nevertheless, they propose a purely local construction which attaches a family of $GL_2(F)$-representations to a single, suitably generic, Galois parameter.

In this talk we show that when a global Galois representation is tame above $p$ (and verifes standard technical conditions) the Hecke isotypical spaces in the mod $p$ cohomology of Shimura curves with $K_1$-level structure at $p$ satisfy a multiplicity one principle. In particular, the $K_1$-fxed vectors in the cohomology with infnite level at $p$ are those conjectured by Breuil-Paskunas, giving new constraints on the family of local representations attached by Breuil-Paskunas to a local Galois parameter.
This ongoing joint work with Dan Le and Benjamin Schraen.

Beyond the main conjecture in Iwasawa theory

Cristian Popescu

I will report on my recent proof (joint with Corey Stone) of a conjecture of Kurihara and its generalizations relating the higher Fitting ideals of various Iwasawa modules to equivariant $p$-adic L-functions.

Picard modular surfaces, Residual Albanese quotients and Rational Points

Dinakar Ramakrishnan

Picard modular surfaces $X$, which arise classically as (compactifications of) the quotients of the unit ball in $C^2$ by arithmetic lattices Gamma in $SU(2,1)$, have mirific properties, making them a crucial place to test various arithmetic and geometric conjectures. This talk will begin by describing the albanese variety and its residual quotients, and then Lang's conjecture over finitely generate fields. It will then move on to discuss an ongoing, partially completed, program with M. Dimitrov to establish the paucity of rational points on the open part.

Moduli stacks of potentially Barsotti-Tate Galois representations

David Savitt

I will discuss joint work with Ana Caraiani, Matthew Emerton, and Toby Gee in which we construct moduli stacks of two-dimensional tamely potentially Barsotti-Tate Galois representations, and relate their geometry to the weight part of Serre's conjecture for GL(2).

Explicit equations for modular curves associated to the normalizers of non-split Cartan subgroups.

Rene Schoof

The main obstacle in proving Serre's Uniformity Conjecture is the determination of the rational points of the modular curves associated to the normalizers of non-split Cartan subgroups. In this talk I will describe a method to compute explicit equations for these curves over $\mathbb{Q}$.

Main conjectures and special values of L-functions

Christopher Skinner

I will survey some of the recent progress on Iwasawa main conjectures for modular forms and some applications to special values of modular forms.

Galois representations and adelic point groups

Peter Stevenhagen

We show how the Galois representation of an elliptic curve over a number field can be used to determine the structure of the (topological) group of adelic points of that elliptic curve. As a consequence, we find that for almost all elliptic curves over a number field $K$, the adelic point group is a universal topological group depending only on the degree of $K$. Still, we can construct infinitely many pairwise non-isomorphic elliptic curves over $K$ that have an adelic point group not isomorphic to this universal group.

Congruence ideals and Selmer groups for higher symmetric powers

Jacques Tilouine

In a joint work with Hida we prove analogues of Iwasawa-Greenberg main conjectures relating congruence ideals and characteristic power of Selmer groups of certain higher symmetric powers of Hida families.

Euler systems and deformations of Galois representations

Eric Urban

I will explain some close relations between the existence of Euler systems and the properties of certain deformations of Galois representations. In particular, I will show that this approach allows to obtain Euler systems of rank $> 1$

On Galois representations of mod $p$ Hilbert eigenforms of weight one

Gabor Wiese

The talk will summarise the main ideas underlying the recent joint work with Mladen Dimitrov, proving that the existence of Hecke operators $T_\mathfrak{p}$,for $\mathfrak{p}$ dividing $p$, implies that the Galois representation attached to a mod $p$ Hilbert modular eigenform of parallel weight one and prime-to-$p$ level is unramified above $p$. This applies, in particular, to non-liftable mod $p$ eigenforms, and can be seen as a refinement of the weight aspect in generalisations of Serre's Modularity Conjecture to Hilbert modular forms.