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Companion points and locally analytic socle conjecture for Steinberg case

Yiqin He

Abstract

In this paper, we will modify the Breuil-Hellmann-Schraen's (more generally, resp., Breuil-Ding's) local model for the trianguline variety (resp., Bernstein paraboline variety) to certain semistable (resp., potentially semistable) non-crystalline point with regular Hodge-Tate weights.Then we deduce several local-global compatibility results, including a classicality result, and the existence of expected companion points on the (definite) eigenvariety and locally analytic socle conjecture for such semistable non-crystalline Galois representations, under certain hypothesis on trianguline variety and the usual Taylor-Wiles assumptions. Moreover, we also discuss slightly the coherent sheaves obtained by patching argument and the coherent sheaves which are constructed from local models and Bezrukavnikov functor, under the route of the recently work of Hellmann-Hernandez-Schraen.

Companion points and locally analytic socle conjecture for Steinberg case

Abstract

In this paper, we will modify the Breuil-Hellmann-Schraen's (more generally, resp., Breuil-Ding's) local model for the trianguline variety (resp., Bernstein paraboline variety) to certain semistable (resp., potentially semistable) non-crystalline point with regular Hodge-Tate weights.Then we deduce several local-global compatibility results, including a classicality result, and the existence of expected companion points on the (definite) eigenvariety and locally analytic socle conjecture for such semistable non-crystalline Galois representations, under certain hypothesis on trianguline variety and the usual Taylor-Wiles assumptions. Moreover, we also discuss slightly the coherent sheaves obtained by patching argument and the coherent sheaves which are constructed from local models and Bezrukavnikov functor, under the route of the recently work of Hellmann-Hernandez-Schraen.
Paper Structure (34 sections, 60 theorems, 162 equations)

This paper contains 34 sections, 60 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General notation
  4. Lg
  5. Some preliminaries on potentially semistable deformation ring
  6. Local models on special $\Omega_{r}^{\oplus k}$-case
  7. Preliminaries
  8. Recollection of some groupoids
  9. Almost de Rham $(\varphi,\Gamma)$-modules
  10. Groupoids
  11. Variation of local models and its geometry
  12. Main constructions
  13. Representablity
  14. Description of local models through formal power series
  15. The case of $(\varphi,\Gamma)$-modules and Galois representations
  16. ...and 19 more sections

Key Result

Theorem 1.2

(Classicality, See Theorem Classicality) Assume Hypothesis TWhypo and Hypothesis introhypo-1. If the Galois representation $\rho:{\rm Gal}(\overline{F}/F)\rightarrow{\rm GL}_n(E)$ comes from a Steinberg type point $y=(\rho,\underline{\delta})\in Y(U^{{\mathfrak{p}}},\overline{\rho})$, then $\widehat

Theorems & Definitions (133)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Proposition 1.6
  • Remark 1.7
  • Theorem 1.8
  • Conjecture 1.9
  • Proposition 1.10
  • Remark 1.11
  • ...and 123 more