Invariant elements for $p$-modular representations of ${\mathbf {GL}}_{2}({\mathbf {Q}}_p)$

Abstract

Let p be an odd rational prime and F a p-adic field. We give a realization of the universal p-modular representations of GL 2 (F) in terms of an explicit Iwasawa module. We specialize our constructions to the case F = Q p , giving a detailed description of the invariants under principal congruence subgroups of irreducible admissible p-modular representations of GL 2 (Q p), generalizing previous work of Breuil and Paskunas. We apply these results to the local-global compatibility of Emerton, giving a generalization of the classical multiplicity one results for the Jacobians of modular curves with arbitrary level at p. Contents 1. Introduction 6625 1.1. Notation 6630 2. Reminders on the universal representations for GL 2 6633 2.1. Construction of the universal representation 6633 3. Structure theorems for universal representations 6637 3.1. Refinement of the Iwahori structure 6637 3.2. The case F = Q p 6647 4. Study of K t and I t invariants 6649 4.1. Invariants for the Iwasawa modules R − ∞,• 6649 4.2. Invariants for supersingular representations 6656 5. The case of principal and special series 6660 6. Global applications 6663 Acknowledgements 6665 References 6666

On crystabelline deformation rings of $$\mathrm {Gal}(\overline{\mathbb {Q}}_p/\mathbb {Q}_p)$$ Gal ( Q ¯ p / Q p ) (with an appendix by Jack Shotton)

“…We will use Lemma 7.3 to show that crystalline deformation rings in small HTweights are Cohen-Macaulay, which can be regarded as a complement of our main theorem. We need the following results of Breuil [5] and Morra [34,35]. We assume p ≥ 3 because this is imposed in [34,35].…”

“…We need the following results of Breuil [5] and Morra [34,35]. We assume p ≥ 3 because this is imposed in [34,35]. (i) Assume π is a supersingular representation.…”

“…(i) (1) is just[5, Cor.4.1.4] and (2) is a consequence of[34, Thm.1.1]. For (3), the first statement follows from[35, Thm.1.4] which determines π K1 ; note that the assumption θ ′ = θ ′s ensures that Ind K I θ ′ is indecomposable (see[8, §2]). The second statement follows from (1) by Frobenius reciprocity.…”

On Some Consequences of a Theorem of J. Ludwig

“…Yet another alternative is to use results of Morra [33], who actually computes the dimensions dim π In , where I n is a certain filtration of Iwahori subgroups by open normal subgroups.…”

Multiplicities of cohomological automorphic forms on $\mathrm{GL}_2$ and mod $p$ representations of $\mathrm{GL}_2(\mathbb{Q}_p)$

“…In fact, to do this we also need more refined structure theorems due to Morra [16,17]. Second, the theory of Paškūnas [20] allows us to pass to general admissible cotorsion representations.…”