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Abstract:
Let $p$ be a prime number and $G$ a connected split reductive
algebraic group over ${\mathbb Q}_p$ such that both $G$ and
its dual $\widehat G$ have connected center. Let $\rho:{\rm
Gal}(\overline {\mathbb Q}_p/{\mathbb Q}_p)\rightarrow \widehat
G(E)$ be a continuous group homomorphism where $E$ is a finite
extension of ${\mathbb Q}_p$. The aim of the $p$-adic Langlands
program is to associate to (the conjugacy class of) $\rho$
some hypothetical $p$-adic Banach space(s) $\Pi(\rho)$ over
$E$ endowed with a unitary continuous action of $G({\mathbb
Q}_p)$ such that $\Pi(\rho)$ occurs in some completed cohomology
groups when $\rho$ comes from some (pro)modular representation
of a global Galois group.
Assume that $\rho$ takes values in a Borel subgroup $\widehat
B(E)\subset \widehat G(E)$. I will explain how one can associate
to such a (sufficiently generic) $\rho$ a Banach space $\Pi(\rho)^{\rm
ord}$ endowed with a unitary continuous action of $G({\mathbb
Q}_p)$ and which is expected to be a closed subrepresentation
of $\Pi(\rho)$, namely its maximal closed subrepresentation
where all irreducible constituents are subquotients of unitary
continuous principal series. The representation $\Pi(\rho)^{\rm
ord}$ decomposes as $\Pi(\rho)^{\rm ord}=\oplus_{w\in W(\rho)}\Pi(\rho)^{\rm
ord}_w$ where $W(\rho)$ is a subset of the Weyl group $W$.
One important point is that its construction is directly inspired
by the study of the ``ordinary part'' of the tensor product
of the fundamental algebraic representations of $\widehat
G(E)$ (composed with $\rho$).
One can extend the construction of $\Pi(\rho)^{\rm ord}$ in
characteristic $p$ and associate to a (sufficiently generic)
$\overline\rho:{\rm Gal}(\overline {\mathbb Q}_p/{\mathbb
Q}_p)\rightarrow \widehat B(k_E) \subset \widehat G(k_E)$
a smooth representation:
$$\Pi(\overline\rho)^{\rm ord}=\oplus_{w\in W(\overline\rho)}\Pi(\overline\rho)^{\rm
ord}_w$$
of $G({\mathbb Q}_p)$ over $k_E$ where $k_E$ is a finite extension
of ${\mathbb F}_p$. When $G={\rm GL}_n$ and $\overline\rho$
comes from some modular Galois representation, I will explain
how one can use recent results of Gee and Geraghty on ordinary
Serre weights to prove that all ${\rm GL}_n({\mathbb Q}_p)$-representations
$\Pi(\overline\rho)^{\rm ord}_w$ really do occur in spaces
of automorphic forms modulo $p$ for definite unitary groups
which are outer forms of ${\rm GL}_n$.
The first lecture will be largely introductory, in particular
I will recall the situation for $G={\rm GL}_2$ and $\rho$
reducible as above. The second lecture will be devoted to
the construction of $\Pi(\rho)^{\rm ord}$ and I will stress
the parallel with the restriction to subgroups of $\widehat
B(E)$ of the tensor product of the fundamental algebraic representations
of $\widehat G(E)$. The last lecture will be devoted to the
local-global compatibility result in characteristic $p$ mentioned
above.
This is joint work with Florian Herzig.
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Christophe Breuil is one of the leading international
experts on p-adic Hodge theory and the p-adic
Langlands program, two of the central topics of the thematic
program.
In his thesis work, he introduced new p-adic Hodge
theoretic techniques for studying p-adic representations
of Galois groups of p-adic fields. The novelty of these
techniques was that they applied to integral representations
(i.e. for representations defined over Zp and
not just over Qp) and to ramified
p-adic fields. In 1999, Breuil joined with Brian Conrad,
Fred Diamond, and Richard Taylor in applying these techniques
to complete the proof of the modularity conjecture for elliptic
curves over Q.
The detailed computations involved in the proof of the modularity
conjecture led Breuil, together with Ariane Mezard, to formulate
a fundamental conjecture (now called the Breuil–Mezard
conjecture) which posits an intricate relationship between
p-adic Hodge theory, the representation theory of the
group GL2(Zp), and the
deformation theory of 2-dimensional p-adic representations
of Galois groups of p-adic Galois representations.
This conjecture is in some sense a quantitative local analogue
of the weight part of Serre’s celebrated conjecture on
modularity of mod p-representations.
Taking the ideas underlying the Breuil–Mezard conjecture
even further, Breuil then conjectured that there is a p-adic
local Langlands correspondence relating 2-dimensional p-adic
representations of the Galois group GQp
and the p-adic representation theory of the group GL
2(Qp). He
laid out the fundamental properties that such a correspondence
would have to satisfy, and in a series of papers gave compelling
evidence that this correspondence would exist.
Breuil’s ideas captured the imagination of number theorists
working on p-adic Hodge theory and the arithmetic of
automorphic forms, and over the course of the last decade
the p-adic Langlands correspondence has emerged as
one of the dominant themes in this area of number theory.
The p-adic Langlands correspondence for GL 2(Qp).has
been constructed in general by Pierre Colmez and Vytas Paskunas.
It has been used by Mark Kisin and by Matthew Emerton to provide
two different proofs of the Fontaine–Mazur conjecture
for odd 2-dimensional p-adic representations of GQ,
with Kisin argument simultaneously establishing the Breuil–Mezard
conjecture. All of this work serves to vindicate the deep
and original vision of Breuil.
Breuil himself continues to pursue the construction of a p-adic
Langlands correspondence, with the goal of moving beyond the
case of GL2(Qp)
to more general contexts. He has received several prizes and
honours in recognition of the importance of his contributions
to his field, including giving an invited talk at the 2010
ICM in Hyderabad.
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