The weight in a Serre-type conjecture for tame n-dimensional Galois representations

We formulate a Serre-type conjecture for n-dimensional Galois representations that are tamely ramified at p. The weights are predicted using a representation-theoretic recipe. For n = 3 some of these weights were not predicted by the previous conjecture of Ash, Doud, Pollack, and Sinnott. Computational evidence for these extra weights is provided by calculations of Doud and Pollack. We obtain theoretical evidence for n = 4 using automorphic inductions of Hecke characters.Comment: 68 pages, revised (mainly added appendix that generalises Jantzen's theorem on the reduction modulo p of Deligne--Lusztig representations

Ordinary representations of G(Q_p) and fundamental algebraic representations

Let G be a split connected reductive algebraic group over Q_p such that both G and its dual group G-hat have connected centres. Motivated by a hypothetical p-adic Langlands correspondence for G(Q_p) we associate to an n-dimensional ordinary (i.e. Borel valued) representation rho : Gal(Q_p-bar/Q_p) to G-hat(E) a unitary Banach space representation Pi(rho)^ord of G(Q_p) over E that is built out of principal series representations. (Here, E is a finite extension of Q_p.) Our construction is inspired by the "ordinary part" of the tensor product of all fundamental algebraic representations of G. There is an analogous construction over a finite extension of F_p. In the latter case, when G=GL_n we show under suitable hypotheses that Pi(rho)^ord occurs in the rho-part of the cohomology of a compact unitary group. We also prove a weaker version of this result in the p-adic case.Comment: Revised (June 2014), 78 page

Adequate groups of low degree

The notion of adequate subgroups was introduced by Jack Thorne [42]. It is a weakening of the notion of big subgroups used in generalizations of the Taylor-Wiles method for proving the automorphy of certain Galois representations. Using this idea, Thorne was able to strengthen many automorphy lifting theorems. It was shown in [22] that if the dimension is small compared to the characteristic then all absolutely irreducible representations are adequate. Here we extend the result by showing that, in almost all cases, absolutely irreducible kG-modules in characteristic p, whose irreducible G+-summands have dimension less than p (where G+ denotes the subgroup of G generated by all p-elements of G), are adequate.Comment: 60 page

Potentially crystalline lifts of certain prescribed types

We prove several results concerning the existence of potentially crystalline lifts with prescribed Hodge-Tate weights and inertial types of a given n-dimensional mod p representation of the absolute Galois group of K, where K/Q_p is a finite extension. Some of these results are proved by purely local methods, and are expected to be useful in the application of automorphy lifting theorems. The proofs of the other results are global, making use of automorphy lifting theorems.Comment: 22 pages; final version, to appear in Document

On the irreducibility of pp-adic Banach principal series of pp-adic GL3\mathrm{GL}_3

We establish an optimal (topological) irreducibility criterion for $p$-adic Banach principal series of $\mathrm{GL}_{n}(F)$, where $F/\mathbb{Q}_p$ is finite and $n \le 3$. This is new for $n = 3$ as well as for $n = 2$, $F \ne \mathbb{Q}_p$ and establishes a refined version of Schneider's conjecture [Sch06, Conjecture 2.5] for these groups.Comment: 21 page