Equidimensional adic eigenvarieties for groups with discrete series

Gulotta, Daniel R.

doi:10.2140/ant.2019.13.1907

Cited by 4 publications

(1 citation statement)

References 32 publications

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“…They construct families of automorphic forms extending over the boundary of weight space, to points in what can be viewed as an adic compactification of weight space, and show that the eigenvariety also extends to those points. (See also Gulotta [Gul19] for an analogous construction extending equidimensional eigenvarieties.) Consequently, we can compute the matrix coefficients of in an explicit basis for the space of forms over the ‘boundary weights’ given by monomials in the matrix coefficients of the dimension maximal lower unipotent subgroup of .…”

Section: Introductionmentioning

confidence: 99%

Slopes in eigenvarieties for definite unitary groups

2023

Compositio Math.

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We generalize bounds of Liu–Wan–Xiao for slopes in eigencurves for definite unitary groups of rank $2$ to slopes in eigenvarieties for definite unitary groups of any rank. We show that for a definite unitary group of rank $n$, the Newton polygon of the characteristic power series of the $U_p$ Hecke operator has exact growth rate $x^{1+2/{n(n-1)}}$, times a constant proportional to the distance of the weight from the boundary of weight space. The proof goes through the classification of forms associated to principal series representations. We also give a consequence for the geometry of these eigenvarieties over the boundary of weight space.

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Section: Introductionmentioning

confidence: 99%

Slopes in eigenvarieties for definite unitary groups

2023

Compositio Math.

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Cohomology of (, Γ)-Modules Over Pseudorigid Spaces

Bellovin

2023

International Mathematics Research Notices

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We study the cohomology of families of $(\varphi ,\Gamma )$-modules with coefficients in pseudoaffinoid algebras. We prove that they have finite cohomology, and we deduce an Euler characteristic formula and Tate local duality. We classify rank-$1$$(\varphi ,\Gamma )$-modules and deduce that triangulations of pseudorigid families of $(\varphi ,\Gamma )$-modules can be interpolated, extending a result of [29]. We then apply this to study extended eigenvarieties at the boundary of weight space, proving in particular that the eigencurve is proper at the boundary and that Galois representations attached to certain characteristic $p$ points are trianguline.

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Decompletion of cyclotomic perfectoid fields in positive characteristic

Berger¹,

Rozensztajn²

2022

Annales Henri Lebesgue

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Let E be a field of characteristic p. The group Z × p acts on E((X)) by a•f (X) = f ((1+X) a −1). This action extends to the X-adic completion E of ∪ n 0 E((X 1/p n )). We show how to recover E((X)) from the valued E-vector space E endowed with its action of Z × p . To do this, we introduce the notion of super-Hölder vector in certain E-linear representations of Z p . This is a characteristic p analogue of the notion of locally analytic vector in p-adic Banach representations of p-adic Lie groups.Résumé. -Soit E un corps de caractéristique p. Le groupeNous montrons comment récupérer E((X)) à partir du E-espace vectoriel valué E muni de son action de Z × p . Pour faire cela, nous introduisons la notion de vecteur super-Hölder dans certaines représentations E-linéaires de Z p . Ceci est un analogue en caractéristique p de la notion de vecteur localement analytique dans les représentations de groupes de Lie p-adiques sur des Banach p-adiques.

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Equidimensional adic eigenvarieties for groups with discrete series

Cited by 4 publications

References 32 publications

Slopes in eigenvarieties for definite unitary groups

Slopes in eigenvarieties for definite unitary groups

Cohomology of (, Γ)-Modules Over Pseudorigid Spaces

Decompletion of cyclotomic perfectoid fields in positive characteristic

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