Multivariable $$(\varphi ,\Gamma )$$ ( φ , Γ ) -modules and smooth o-torsion representations

Abstract

Let G be a Q p -split reductive group with connected centre and Borel subgroup B = T N . We construct a right exact functor D ∨ ∆ from the category of smooth modulo p n representations of B to the category of projective limits of finitely generated étale (ϕ, Γ)-modules over a multivariable (indexed by the set of simple roots) commutative Laurentseries ring. These correspond to representations of a direct power of Gal(Q p /Q p ) via an equivalence of categories. Parabolic induction from a subgroup P = L P N P gives rise to a basechange from a Laurent-series ring in those variables with corresponding simple roots contained in the Levi component L P . D ∨ ∆ is exact and yields finitely generated objects on the category SP A of finite length representations with subquotients of principal series as Jordan-Hölder factors. Lifting the functor D ∨ ∆ to all (noncommuting) variables indexed by the positive roots allows us to construct a G-equivariant sheaf Y π,∆ on G/B and a G-equivariant continuous map from the Pontryagin dual π ∨ of a smooth representation π of G to the global sections Y π,∆ (G/B). We deduce that D ∨ ∆ is fully faithful on the full subcategory of SP A with Jordan-Hölder factors isomorphic to irreducible principal series.

Drinfeld's lemma for perfectoid spaces and overconvergence of multivariate $(\varphi, \Gamma)$-modules

“…In particular, for each α ∈ ∆, there will be an element ̟ α arising from the factor of the product indexed by α; moreover, there will be a partial Frobenius lift ϕ α and a profinite group Γ K,α which act on ̟ α but not on the other variables. (The use of the symbols ∆ and α is meant to suggest roots of a Lie algebra; the relevance of this will not be apparent herein, but can be seen more directly in earlier work on the subject, especially [48]. )…”

“…The relevance of such representations may not be immediately clear from general considerations of arithmetic geometry; however, products of Galois groups occur naturally in the approach to geometric Langlands developed for GL 2 by Drinfeld [13] and extended to GL n by L. Lafforgue [31] and to other reductive groups by V. Lafforgue [32,33]. The relationship between multivariate (ϕ, Γ)-modules and Galois representations was previously explored by the third author [36,47] from a slightly different point of view: this line of inquiry emerged as part of a program to extend Colmez's construction of the p-adic local Langlands correspondence for GL 2 (Q p ) [11] by exhibiting analogues of (ϕ, Γ)-modules obtained from higherrank groups [38,48]. One motivation for consolidating the theory of multivariate (ϕ, Γ)-modules is to prepare for an eventual unification of this program with the work of V. Lafforgue described above; however, such a unification lies far beyond the scope of the present work.…”

Multivariable $(\varphi ,\Gamma )$-modules and representations of products of Galois groups: The case of the imperfect residue field

“…Berger's multivariable (ϕ, Γ)-modules is an attempt to generalize p-adic Langlands for GL 2 (F ), where F is a finite extension of Q p [6,7]. The third author of this current work also defines multivariable (ϕ, Γ)module over a m-variable Laurent series ring in an attempt to generalize p-adic Langlands for GL m (Q p ) [44,49,50]. One might also try to look at Zábrádi's multivariable (ϕ, Γ)-modules over Lubin-Tate extension to conjecturally understand p-adic Langlands for GL m (F ) [28].…”