Special Program on $p$-adic $L$-functions and eigenvarieties

John Bergdall on Galois representations and eigenvarieties.

Mladen Dimitrov on $p$-adic families of $L$-functions and $\mathscr{L}$-invariants.

Nadir Matringe on Local representation theory.

A. Raghuram on Automorphic representations and $L$-functions.

Bergdall on Galois representations and eigenvarieties (Reading List)

1. Examples of Galois representations.
   The absolute Galois group of $\mathbb{Q}$. Examples of characters alone and in families. Tate modules of elliptic curves. Deformation spaces of Galois representations. Automorphic forms. Congruences between automorphic forms, in relation to deformation spaces.
2. Introduction to eigenvarieties.
   Hecke eigensystems and cohomology (review of consequences of Eichler-Shimura). A sample theorem from Hida's theory. $p$-adic weights and distributions, focused on the case of $\operatorname{GL}(2)$. Main theorems on distribution-valued cohomology and eigenvarieties. Brief description of $\operatorname{GL}(n)$ case.
3. Galois representations on eigenvarieties.
   Introduction to pseudo-representations. Main results on realizing and gluing pseudo-representations. Applications to eigenvarieties.
4. Applications and open questions.
   Discussion of the infinite fern for $\operatorname{GL}(2)$ and problems for $\operatorname{GL}(n)$. Further applications as the courses develop.

Dimitrov on $p$-adic families of $L$-functions and $\mathscr{L}$-invariants

1. Introduction to $p$-adic distributions.
   $p$-adic analysis, locally analytic functions, distributions, growth, $p$-adic Banach spaces, $p$-adic distributions on $\mathbb{Z}_p$. Admissible distributions and a theorem of Amice-Velu and Vishik. Amice and Mellin transforms.
2. $p$-adic $L$-functions
   The example of the Kubota-Leopoldt $p$-adic $L$-function. Locally analytic induction. Cohomological construction of distributions using automorphic symbols and overconvergent cohomology. Attaching $p$-adic $L$-functions to automorphic representations whose $p$-component is ordinary and spherical.
3. $p$-adic $L$-functions on eigenvarieties.
   $p$-adic $L$-functions live in families (and even on the eigenvariety). Coholological construction using bigger sheaves. Application: paucity of critical points. Example of Katz $p$-adic $L$.
4. $\mathscr{L}$-invariants
   The Gross-Stark Conjecture on the Kubota-Leopoldt $p$-adic $L$-function. $p$-adic version of the Birch and Swinnerton-Dyer Conjecture. Iwasawa theory and the quest of a $p$-adic version of the Bloch-Kato conjectures. Benois’ Trivial Zero Conjecture.

Nadir Matringe on Local representation theory

1. Smooth representations of non-Archimedean $\operatorname{GL}(n)$.
   We will define smooth and admissible representations of a locally profinite group $G$ and relate them with modules over the Hecke algebra $\mathcal{C}_c^\infty(G)$ of $G$. This will require introducing Haar measures on $G$. We will then define induction functors for smooth representations, and for $G=\operatorname{GL}_n(F)$ with $F$ a local non-archimdean local field we will define parabolic induction. We will study the spherical Hecke algebra of $\operatorname{GL}_n(F)$ and give a construction of spherical generic representations as induced from the Borel subgroup. We will relate this construction with Satake parameters. Time allowing we will state a rough version of the LLC, as a generalization of Artin reciprocity law, and explicate the case of generic spherical representations.
2. The theory of Whittaker models
   We will define Whittaker models for $\operatorname{GL}_n(F)$, and state their uniqueness (with a sketch of the proof). We will give a formula for spherical Whittaker functions. We will then define Rankin-Selberg integrals using such models, representationg local L-factors $L(s,\pi,\pi')$ attached to pairs of generic representations and state their functional equation. We will give a formula for these integrals in the spherical case.
3. Intertwining operators (and Hecke-Iwahori algebras)
   We will define standard intertwining operators between representations parabolically induced from associate parabolic subgroups. We will give the formula of Gindikin-Karpelevic which computes the image of the spherical vector of a spherical generic representation under such an operator, in terms of $L$-factors of pairs of the inducing data. If time allows we will say a few words on Hecke-Iwahori algebras and the subcategory of smooth representations with an Iwahori fixed vector.
4. Cuspidal representations and the Bernstein-Zelevinsky classification.
   We will define Jacquet functors for smooth representations of $\operatorname{GL}_n(F)$, in order to define cuspidal representations. Such representations are the most mysterious and contain substantial local arithmetic information that we will not discuss. We will then give the classification of discrete series representations in terms of cuspidal representations and that of irreducible representations in terms of discrete series (Langlands quotient theorem for $\operatorname{GL}_n(F)$). Finally we will relate this classification with the LLC, making it explicit up to the cuspidal case.

A. Raghuram on Automorphic representations and $L$-functions

1. Automorphic forms on $\operatorname{GL}(n)$.
   Dirichlet characters; Adeles/ideles and Hecke characters; automorphic representations; Whittaker models; the dictionary between modular forms and automorphic representations. (References: Gelbart's Automorphic forms on adele groups; Bump's Automorphic forms and representations; Raghuram's article: Notes on the arithmetic of Hecke characters.)
2. Cohomology of arithmetic groups.
   Adelic description of locally symmetric spaces; sheaves on such spaces; cohomology of arithmetic groups; cuspidal cohomology; the Eichler-Shimura isomorphism. (References: Chapter 8 of Shimura's book Introduction to the arithmetic theory of automorphic functions, Princeton University Press; Chapters 1-3 of Harder-Raghuram's book: Eisenstein cohomology and the special values of Rankin-Selberg $L$-functions, Annals of Math Studies, Vol 203.)
3. Special values of $L$-functions.
   The Mellin-transform of a modular form; Poincaré duality; cohomological interpretation of the Mellin-transform; theorems of Manin and Shimura on the special values of $L$-functions of a modular form; generalizations. (References: Shimura's Periods of modular forms, 1977 Math. Ann.; Raghuram-Tanabe: Notes on the arithmetic of Hilbert modular forms, 2011 JRMS.)
4. Langlands-Shahidi $L$-functions.
   Parabolically induced representations; Eisenstein series; standard intertwining operators; the (non-)constant-term of an Eisenstein series; Examples: Rankin-Selberg $L$-functions for $\operatorname{GL}(n) \times \operatorname{GL}(m)$; exterior-square and symmetric square $L$-functions for $\operatorname{GL}(n)$; Asai $L$-functions for $\operatorname{GL}(n)$. (References: Shahidi's Eisenstein series and $L$-functions.)

Hurley Entrance Photo 1

Hurley Entrance Photo 2