Deformation spaces

1974 ◽  
pp. 48-70 ◽  
Author(s):  
Irwin Kra
Keyword(s):  
Deformation Spaces

scholarly journals Nearly ordinary Galois deformations over arbitrary number fields

2008 ◽  
Vol 8 (1) ◽  
pp. 99-177 ◽  
Author(s):  
Frank Calegari ◽  
Barry Mazur
Keyword(s):  
Arbitrary Number ◽  
Quadratic Field ◽  
Galois Representations ◽  
Galois Representation ◽  
Number Fields ◽  
Base Change ◽  
Deformation Spaces ◽  
Totally Real ◽  
Set Of Points ◽  
Dense Set

AbstractLet K be an arbitrary number field, and let ρ : Gal($\math{\bar{K}}$/K) → GL2(E) be a nearly ordinary irreducible geometric Galois representation. In this paper, we study the nearly ordinary deformations of ρ. When K is totally real and ρ is modular, results of Hida imply that the nearly ordinary deformation space associated to ρ contains a Zariski dense set of points corresponding to ‘automorphic’ Galois representations. We conjecture that if K is not totally real, then this is never the case, except in three exceptional cases, corresponding to: (1) ‘base change’, (2) ‘CM’ forms, and (3) ‘even’ representations. The latter case conjecturally can only occur if the image of ρ is finite. Our results come in two flavours. First, we prove a general result for Artin representations, conditional on a strengthening of the Leopoldt Conjecture. Second, when K is an imaginary quadratic field, we prove an unconditional result that implies the existence of ‘many’ positive-dimensional components (of certain deformation spaces) that do not contain infinitely many classical points. Also included are some speculative remarks about ‘p-adic functoriality’, as well as some remarks on how our methods should apply to n-dimensional representations of Gal($\math{\bar{\QQ}}$/ℚ) when n > 2.

scholarly journals Irreducible Components of Deformation Spaces: Wild 2-Adic Exercises

2014 ◽  
Vol 2015 (14) ◽  
pp. 5333-5356 ◽  
Author(s):  
Pierre Colmez ◽  
Gabriel Dospinescu ◽  
Vytautas Paškūnas
Keyword(s):  
Deformation Spaces ◽  
Irreducible Components

scholarly journals The deformation spaces of convex RP 2 -structures on 2-orbifolds

2005 ◽  
Vol 127 (5) ◽  
pp. 1019-1102 ◽  
Author(s):  
Suhyoung Choi ◽  
William Mark Goldman
Keyword(s):  
Deformation Spaces

scholarly journals Foliations in deformation spaces of local G-shtukas

2012 ◽  
Vol 229 (1) ◽  
pp. 54-78 ◽  
Author(s):  
Urs Hartl ◽  
Eva Viehmann
Keyword(s):  
Deformation Spaces

scholarly journals The smooth locus in infinite-level Rapoport–Zink spaces

2020 ◽  
Vol 156 (9) ◽  
pp. 1846-1872
Author(s):  
Alexander B. Ivanov ◽  
Jared Weinstein
Keyword(s):  
Elliptic Curves ◽  
Open Subset ◽  
Formal Group ◽  
Dense Open Subset ◽  
Additional Structure ◽  
Connected Component ◽  
Deformation Spaces ◽  
Modular Curve ◽  
Divisible Groups ◽  
Smooth Locus

Rapoport–Zink spaces are deformation spaces for $p$-divisible groups with additional structure. At infinite level, they become preperfectoid spaces. Let ${{\mathscr M}}_{\infty }$ be an infinite-level Rapoport–Zink space of EL type, and let ${{\mathscr M}}_{\infty }^{\circ }$ be one connected component of its geometric fiber. We show that ${{\mathscr M}}_{\infty }^{\circ }$ contains a dense open subset which is cohomologically smooth in the sense of Scholze. This is the locus of $p$-divisible groups which do not have any extra endomorphisms. As a corollary, we find that the cohomologically smooth locus in the infinite-level modular curve $X(p^{\infty })^{\circ }$ is exactly the locus of elliptic curves $E$ with supersingular reduction, such that the formal group of $E$ has no extra endomorphisms.

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