ICTS

The notion of admissibility of representations of p-adic groups goes back to Harish-Chandra. Bernstein, Jacquet and Vigneras have shown that smooth irreducible representations of connected reductive p-adic groups over algebraically closed fields of characteristic different from p are admissible. We use a Diamond diagram attached to a 2-dimensional reducible split mod p Galois representation of Gal_{Q_p^2} to construct a non admissible smooth irreducible mod p representation of GL_2(Q_p^2) following the approach of Daniel Le. This is joint work with Mihir Sheth

In this course we introduce the p-adically completed cohomology for Shimura curves associated to quaternion algebras over totally real fields. In particular, we study the properties of the Banach space representations obtained from them, their relation with the locally algebraic vectors, and their use in the Kisin--Taylor--Wiles patching.

Serre's Conjecture (over $\mathbf{Q}$) states that every odd irreducible representation $\rho:\mathrm{Gal}(K/\mathbf{Q}) \to \mathrm{GL}_2(\mathbf{F}_{p^r})$ arises from a modular form, with level determined by the local behavior of $\rho$ at primes other than $p$, and weight determined by its local behavior at $p$. Serre's Conjecture is now a theorem of Khare and Wintenberger, but the correctness of the prediction of the level and weight was proved first (through work of Mazur, Ribet, Gross, Coleman--Voloch, Gross, Edixhoven\ldots) and is key input for the argument of Khare and Wintenberger as well as for arithmetic applications of modularity results. This``refined part'' of Serre's Conjecture can also be viewed as a local-global compatibility result in the context of a mod $p$ Langlands Programme, driving the question of what shape it takes in conjectural relations between more general Galois and automorphic representations in characteristic $p$. From this point of view, a natural ``next case'' to consider is that of $\mathrm{GL}_2$ over a totally real field $F$, relating Hilbert modular forms and two-dimensional representations of Galois groups over $F$. Generalizing Serre's recipe for the weight to this setting turns out to reveal many new features. The lectures will start with a review of Serre's Conjecture over $\mathbf{Q}$, with particular emphasis on the recipe for the weight and a view to its generalizations. I'll then discuss two formulations of Serre's Conjecture in the context of $\mathrm{GL}_2$ over totally real fields and what is known about them. The first of these (formulated in successively greater generality by Buzzard--D--Jarvis, Schein and Gee) is in terms of Hecke eigenforms in the mod $p$ Betti (or \'etale) cohomology of Shimura varieties, and links more directly to the representation theory of $\mathrm{GL}_2$ of $p$-adic fields and to mod $p$ local Langlands correspondences. The second of these (formulated more recently with Sasaki) is in terms of sections of automorphic bundles in characteristic $p$, and relates more to the geometry of Shimura varieties and to non-cohomological automorphic representations.

In recent years, the formal deformation theory of Galois representations has been extended in various contexts to a theory of moduli stacks of Galois representations, in which the Galois representations vary in a genuinely algebraic (rather than merely formal) fashion.   I will describe some ideas related to moduli stacks of Galois representations, and their relationship to the theory of automorphic forms.   The talk will touch upon joint work of the speaker and Toby Gee, and also on work of Xinwen Zhu (some ongoing jointly with the speaker), as well as on the work and ideas of many other mathematicians.

There are two classical constructions of the space of p-adic modular forms — Serre’s construction via the p-adic completion of the q-expansions of classical modular forms, and Katz’s construction via functions on the Katz-Igusa cover of the ordinary locus. In this talk, we explain how to extend both constructions to give a space of p-adic *automorphic* forms — Serre’s construction is generalized by completing the Kirillov models of the automorphic representations generated by modular forms, and on the geometric side this amounts to replacing Katz’s cover with the big Igusa formal scheme of Caraiani-Scholze. This perspective yields new insight on Hida’s finiteness for ordinary p-adic modular forms, overconvergence, and the differential operator theta, and suggests a common generalization of the archimedean and p-adic theories of autumorphic forms via functions on (big) Igusa varieties.

In Kato's paper on Euler system, he constructed an explicit reciprocity law to compute Bloch-Kato's dual exponential map. In this talk, we give a variant of Kato's explicit reciprocity law and show that it is compatible with Bloch-Kato's dual exponential map. As a consequence, the new map is compatible with Kato's original one. This is a joint work with Pierre Colmez.

In this course we introduce the p-adically completed cohomology for Shimura curves associated to quaternion algebras over totally real fields. In particular, we study the properties of the Banach space representations obtained from them, their relation with the locally algebraic vectors, and their use in the Kisin--Taylor--Wiles patching.

(joint with Johannes Sprang) Let K be a CM field and L/K be an extension of degree n and χ be an algebraic critical Hecke character of L. Then we show that the L-value L(χ, 0) divided by carefully normalized Shimura-Katz periods is integral and construct a p-adic L-function for χ. This generalizes results by Damerell, Shimura and Katz for CM fields (L = K). Our method relies on a detailed analysis of new equivariant motivic Eisenstein classes and especially on the study of their de Rham real-izations and differs even in the CM case from the classical approach. It turns out that the de Rham realization of these Eisenstein classes can be explicitly described in terms of Eisenstein-Kronecker series and that working in equivariant cohomology allows to connect them with the L-function of χ. A further important feature of our work is an integral refinement of these classes in coherent cohomology relying on the completion of the Poincar ́e bundle. Here we were inspired by work of Bannai-Kobayshi in the imaginary quadratic case. With a technique pioneered in Sprang’s thesis, this approach leads directly to the construction of p-adic L-functions for χ.

These lectures are about mod p representations of the group GL2(K) for K a finite extension of Qp and their relation with an expected local Langlands correspondence. In a first part, I will recall what is the situation for the group GL2(Qp). I will then discuss the situation for GL2(K). I will conclude by a discussion of recent results concerning the Gelfand-Kirillov dimension and some application to deformation rings

Serre's Conjecture (over $\mathbf{Q}$) states that every odd irreducible representation $\rho:\mathrm{Gal}(K/\mathbf{Q}) \to \mathrm{GL}_2(\mathbf{F}_{p^r})$ arises from a modular form, with level determined by the local behavior of $\rho$ at primes other than $p$, and weight determined by its local behavior at $p$. Serre's Conjecture is now a theorem of Khare and Wintenberger, but the correctness of the prediction of the level and weight was proved first (through work of Mazur, Ribet, Gross, Coleman--Voloch, Gross, Edixhoven\ldots) and is key input for the argument of Khare and Wintenberger as well as for arithmetic applications of modularity results. This``refined part'' of Serre's Conjecture can also be viewed as a local-global compatibility result in the context of a mod $p$ Langlands Programme, driving the question of what shape it takes in conjectural relations between more general Galois and automorphic representations in characteristic $p$. From this point of view, a natural ``next case'' to consider is that of $\mathrm{GL}_2$ over a totally real field $F$, relating Hilbert modular forms and two-dimensional representations of Galois groups over $F$. Generalizing Serre's recipe for the weight to this setting turns out to reveal many new features. The lectures will start with a review of Serre's Conjecture over $\mathbf{Q}$, with particular emphasis on the recipe for the weight and a view to its generalizations. I'll then discuss two formulations of Serre's Conjecture in the context of $\mathrm{GL}_2$ over totally real fields and what is known about them. The first of these (formulated in successively greater generality by Buzzard--D--Jarvis, Schein and Gee) is in terms of Hecke eigenforms in the mod $p$ Betti (or \'etale) cohomology of Shimura varieties, and links more directly to the representation theory of $\mathrm{GL}_2$ of $p$-adic fields and to mod $p$ local Langlands correspondences. The second of these (formulated more recently with Sasaki) is in terms of sections of automorphic bundles in characteristic $p$, and relates more to the geometry of Shimura varieties and to non-cohomological automorphic representations.

In this talk, we prove a new result towards generalized Ogg's conjecture, which states as follows: For a positive integer N, the p-primary subgroup of the rational torsion subgroup of J_0(N) is equal to that of the rational cuspidal divisor class group of X_0(N) unless p^2 divides 12N.

Let K/F be a finite Galois extension of number fields and \sigma be an absolutely irreducible, self-dual representation of Gal(K/F). Given two elliptic curves E_1, E_2 with equivalent mod-p Galois representations, we study the variation of the parity of the multiplicities of \sigma in the representation space associated with the p∞-Selmer group of E_i over K. We also compare the root numbers for the twist of E_i/F by \sigma and show that the p-parity conjecture holds for the twist of E_1/F by \sigma if and only if it holds for the twist of E_2/F by \sigma. This is joint work with Somnath Jha and Sudhanshu Shekhar. 

In this course we introduce the p-adically completed cohomology for Shimura curves associated to quaternion algebras over totally real fields. In particular, we study the properties of the Banach space representations obtained from them, their relation with the locally algebraic vectors, and their use in the Kisin--Taylor--Wiles patching.

Serre's Conjecture (over $\mathbf{Q}$) states that every odd irreducible representation $\rho:\mathrm{Gal}(K/\mathbf{Q}) \to \mathrm{GL}_2(\mathbf{F}_{p^r})$ arises from a modular form, with level determined by the local behavior of $\rho$ at primes other than $p$, and weight determined by its local behavior at $p$. Serre's Conjecture is now a theorem of Khare and Wintenberger, but the correctness of the prediction of the level and weight was proved first (through work of Mazur, Ribet, Gross, Coleman--Voloch, Gross, Edixhoven\ldots) and is key input for the argument of Khare and Wintenberger as well as for arithmetic applications of modularity results. This``refined part'' of Serre's Conjecture can also be viewed as a local-global compatibility result in the context of a mod $p$ Langlands Programme, driving the question of what shape it takes in conjectural relations between more general Galois and automorphic representations in characteristic $p$. From this point of view, a natural ``next case'' to consider is that of $\mathrm{GL}_2$ over a totally real field $F$, relating Hilbert modular forms and two-dimensional representations of Galois groups over $F$. Generalizing Serre's recipe for the weight to this setting turns out to reveal many new features. The lectures will start with a review of Serre's Conjecture over $\mathbf{Q}$, with particular emphasis on the recipe for the weight and a view to its generalizations. I'll then discuss two formulations of Serre's Conjecture in the context of $\mathrm{GL}_2$ over totally real fields and what is known about them. The first of these (formulated in successively greater generality by Buzzard--D--Jarvis, Schein and Gee) is in terms of Hecke eigenforms in the mod $p$ Betti (or \'etale) cohomology of Shimura varieties, and links more directly to the representation theory of $\mathrm{GL}_2$ of $p$-adic fields and to mod $p$ local Langlands correspondences. The second of these (formulated more recently with Sasaki) is in terms of sections of automorphic bundles in characteristic $p$, and relates more to the geometry of Shimura varieties and to non-cohomological automorphic representations.

We will discuss different results on representations of reductive p-adic groups and of pro-p-Iwahori Hecke algebras, with coefficients in a field of characteristic p (or in an Artinian ring where p is nilpotent).

These lectures are about mod p representations of the group GL2(K) for K a finite extension of Qp and their relation with an expected local Langlands correspondence. In a first part, I will recall what is the situation for the group GL2(Qp). I will then discuss the situation for GL2(K). I will conclude by a discussion of recent results concerning the Gelfand-Kirillov dimension and some application to deformation rings.

The weight part of Serre's conjecture aims to classify the weights of congruent cohomological automorphic forms. We describe recent progress on the weight part of Serre's conjecture in some cases where the associated mod p Galois representation is generic at p. We also discuss how these results give an automorphic criterion for a mod p modular Galois representation to be tamely ramified at p. This is joint work with B.V. Le Hung, B. Levin, and S. Morra.

In the two talks (given by Haoran Wang and Yongquan Hu), we will report some recent results on the mod $p$ cohomology for $\mathrm{GL}_2$ from the point of view of the mod $p$ Langlands program.

Precisely, let $F$ be a totally real field unramified at all places above $p$ and $D$ be a quaternion algebra which splits at either none, or exactly one, of the infinite places. Let $\overline{r}:\mathrm{Gal}(\overline{F}/F)\rightarrow \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ be a continuous irreducible representation which, when restricted to a fixed place $v|p$, is non-semisimple and sufficiently generic. Under some mild assumptions, we prove that the admissible smooth representations of $\mathrm{GL}_2(F_v)$ occurring in the corresponding Hecke eigenspaces of the mod $p$ cohomology of Shimura varieties associated to $D$ have Gelfand-Kirillov dimension $[F_v:\mathbb{Q}_p]$. We also prove that any such representation can be generated as a $\mathrm{GL}_2(F_v)$-representation by its subspace of invariants under the first principal congruence subgroup. If moreover $[F_v:\mathbb{Q}_p]=2$, we prove that such representations have length $3$, providing strong evidence for a speculation of Breuil and Paˇsk ̄unas

In the two talks (given by Haoran Wang and Yongquan Hu), we will report some recent results on the mod $p$ cohomology for $\mathrm{GL}_2$ from the point of view of the mod $p$ Langlands program.

Precisely, let $F$ be a totally real field unramified at all places above $p$ and $D$ be a quaternion algebra which splits at either none, or exactly one, of the infinite places. Let $\overline{r}:\mathrm{Gal}(\overline{F}/F)\rightarrow \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ be a continuous irreducible representation which, when restricted to a fixed place $v|p$, is non-semisimple and sufficiently generic. Under some mild assumptions, we prove that the admissible smooth representations of $\mathrm{GL}_2(F_v)$ occurring in the corresponding Hecke eigenspaces of the mod $p$ cohomology of Shimura varieties associated to $D$ have Gelfand-Kirillov dimension $[F_v:\mathbb{Q}_p]$. We also prove that any such representation can be generated as a $\mathrm{GL}_2(F_v)$-representation by its subspace of invariants under the first principal congruence subgroup. If moreover $[F_v:\mathbb{Q}_p]=2$, we prove that such representations have length $3$, providing strong evidence for a speculation of Breuil and Paˇsk ̄unas

We give an ”etale” construction of Bellaiche’s θ-critical p-adic L-function. This is a joint work (in progress) with K. Buyukboduk.