Reduction modulo p of certain semi-stable representations (4 ninety-minute lectures)

Abstract

In this short lecture series, we will discuss Breuil&apos;s integral p-adic Hodge theory to compute mod-p reduction of semi-stable representations, and as an application we will construct deformation rings that parameterize certain semi-stable representations. We will focus on rather computational aspects than theoretical proof. Moreover, if time allows, we will talk about the conjecture, mod p local-global compatibility, that roughly states that a certain space of mod p algebraic automorphic forms on a unitary group determines a given Galois representation, which gives an evidence for mod p Langlands correspondence. In the rst lecture, we introduce the category of p-adic Galois representations of the absolute Galois group G_K for a fnite extension K of Q_p, and its subcategories whose objects are arising from geometry. These objects are called de Rham representations, semi-stable representations, and crystalline representations. We will give more attention to the semi-stable representations. By Colmez-Fontaine, it is known that the category of semi-stable representations is equivalent to the category of admissible filtered (phi,N)-modules. In the second lecture, we study how to compute mod p reductions of semi-stable representations. By Breuil and Liu, it is known that the category of Galois stable lattices in semi-stable representations with Hodge--Tate weights in [0, r] is equivalent to the category of strongly divisible modules, provided that r < p-1. To study mod p reduction, we study Breuil modules, which correspond to mod p reductions of strongly divisible modules. In the third lecture, we will quickly review Galois deformation theory, and then by making use of the parameterization of our families of strongly divisible modules discussed in the second lecture, we will be able to construct the irreducible components of those deformation rings. The last will be a colloquium style talk. For a given mod-p representation R of the absolute Galois group of a p-adic field K, one can dene a mod-p automoprhic representation P of GLn(K) by a certain space of mod-p algebraic automoprhic forms on a unitary group. We wish that P corresponds to R for a mod-p Langlands correspondence, but the structure of P is quite mysterious as a representation. It is natural to ask if P determines R, and we will answer this question in certain cases.