## Schedule

Monday Tuesday Wednesday Thursday Friday
9:00 Registration
9:30 Andreatta Andreatta Andreatta Andreatta Andreatta
11:00 Caraiani Caraiani Caraiani Caraiani Caraiani
12:15 Hellmann Hellmann Allen Hellmann Hellmann

All talks will go for one hour. There will be coffee breaks from 10:30 to 11:00, and (on Monday, Tuesday, Thursday) from 16:00 to 16:30.

Useful local information: pdf

## Mini-courses

Fabrizio Andreatta (Milano) - $p$-adic variations of automorphic sheaves

I will present a technique for deforming coherent cohomology classes over modular curves. Applied to different sheaves one obtains in this way families of overconvergent modular forms and nearly overconvergent modular forms. I will provide applications to the construction of $p$-adic $L$-functions. This is based on several joint works with A. Iovita, V. Pilloni and G. Stevens.

Ana Caraiani (London) - The cohomology of locally symmetric spaces and applications to modularity

I will explain joint work in progress with Allen, Calegari, Gee, Helm, Le Hung, Newton, Scholze, Taylor, and Thorne on potential automorphy over CM fields. I will start by discussing locally symmetric spaces for $GL_n$ and giving an overview of the Calegari-Geraghty method and its prerequisites. I will then focus on one of the prerequisites, namely how to prove instances of local-global compatibility for the Galois representations associated to torsion classes occurring in the cohomology of locally symmetric spaces for $GL_n$. I will end with a more in depth discussion of automorphy lifting in this setting and explain how to combine Taylor's Ihara avoidance with an idea due to Khare-Thorne to make our argument go through.

Eugen Hellmann (Münster) - $p$-adic automorphic forms, eigenvarieties and criteria for classicality

In this course we will introduce and study $p$-adic modular forms and their higher dimensional generalizations, $p$-adic automorphic forms. We will start by introducing $p$-adic modular forms as $p$-adic limits of classical modular forms. These $p$-adic limits naturally vary in families, so called eigenvarieties. These are geometric objects over the $p$-adic numbers whose points parametrize $p$-adic modular forms. The problem how to decide whether a given $p$-adic modular form is in fact a classical form has been studies extensively over the last decades starting with Coleman’s criterion that small slope implies classical.

We will study such classicality criteria, ending with my recent joint work with C. Breuil and B. Schraen on classicality of $p$-adic automorphic forms on define unitary groups, which connects classicality criteria with geometric properties of eigenvarieties.

## Research talks

Patrick Allen (University of Illinois at Urbana-Champaign) - Automorphy of mod 3 representations over CM fields

Wiles's proof of the modularity of semistable elliptic curves over the rationals uses, as a starting point, the Langlands-Tunnell theorem, which implies that the mod 3 Galois representation attached to an elliptic curve over the rationals arises from a modular form of weight one. In order to feed this into modularity lifting theorems, one needs to use congruences between modular forms of weight one and modular forms of higher weight. Similar congruences are not known over CM fields, and Wiles's strategy runs into problems right from the start. We circumvent this congruence problem and show that mod 3 representations over CM field arise from the "correct" automorphic forms. Our argument relies on a 2-adic automorphy lifting theorem over CM fields together with a "2-3 switch" that gives a criterion for when a given mod 6 representation arises from an elliptic curve. This is joint work in progress with Chandrashekhar Khare and Jack Thorne.

Charlotte Chan (University of Michigan) - Period identities of CM forms on quaternion algebras

Waldspurger's formula gives an identity between the norm of a torus period and an $L$-function of the twist of an automorphic representation on $GL(2)$. For any two Hecke characters of a fixed quadratic extension, one can consider the two torus periods coming from integrating one character against the automorphic induction of the other. Because the corresponding $L$-functions agree, (the norms of) these periods---which occur on different quaternion algebras---are closely related. In this talk, we will discuss a direct proof of an explicit identity between the torus periods themselves and mention applications to p-adic automorphic forms.

Valentin Hernandez (Institut de Mathématiques de Jussieu/CNRS) - Families of Picard modular forms and application to the Bloch-Kato conjecture

About 15 years ago, Bellaïche and Chenevier developed a method to prove some cases of the Bloch-Kato conjecture using Eigenvarieties. More precisely, they used endoscopic results to construct certain non-tempered automorphic representations for the group $U(3)$, which is compact at infinity, and study the geometry of the associated Eigenvariety around the corresponding point to construct classes predicted by the Bloch-Kato conjecture. In this talk I will explain how to use geometric methods introduced by Andreatta, Iovita and Pilloni to construct an eigenvariety for the group $U(2,1)$ (for inert prime), and explain how we can adapt the method of Bellaïche and Chenevier to this situation to get new instances of the Bloch-Kato conjecture.

Cong Xue (University of Cambridge) - Cohomologies of stacks of shtukas

Let $G$ be a connected split reductive group over a function field. I will recall the definition of the classifying stacks of $G$-shtukas and their cohomologies. I will present the constant term morphisms of these cohomologies and the contractibility of deep enough strata. Then I will explain how we deduce that the cuspidal subspaces of the cohomologies are finite dimensional and that the whole cohomologies are Hecke modules of finite type. If time permits, I will talk about an application of the second result.

We will also have some questions and answers sessions for the courses. Further details will be announced during the school.