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Theta Operator Equals Fontaine Operator on Modular Curves

Yuanyang Jiang

Abstract

Inspired by [Pan22], we give a new proof that for an overconvergent modular eigenform $f$ of weight $1+k$ with $k\in\mathbb{Z}_{\ge1}$, assuming that its associated global Galois representation $ρ_{f}$ is irreducible, then $f$ is classical if and only if $ρ_{f}$ is de Rham at $p$. For the proof, we prove that theta operator $θ^{k}$ coincides with Fontaine operator in a suitable sense.

Theta Operator Equals Fontaine Operator on Modular Curves

Abstract

Inspired by [Pan22], we give a new proof that for an overconvergent modular eigenform of weight with , assuming that its associated global Galois representation is irreducible, then is classical if and only if is de Rham at . For the proof, we prove that theta operator coincides with Fontaine operator in a suitable sense.
Paper Structure (25 sections, 53 theorems, 228 equations)

This paper contains 25 sections, 53 theorems, 228 equations.

Table of Contents

  1. Introduction
  2. Organization
  3. Notations and conventions
  4. Acknowledgment
  5. Prerequisites
  6. Setup
  7. Locally analytic vectors
  8. Equivariant sheaves on ${\mathscr{F}\!l}$
  9. Automorphic sheaves on ${\mathscr{F}\!l}$
  10. Geometric Sen theory
  11. Arithmetic Sen operator and Fontaine operator
  12. Geometric Fontaine operator
  13. $\mathfrak{b}$-cohomology of $\mathcal{O}^{{\mathrm{la}}}$
  14. Geometric Fontaine operator
  15. Perversity
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $f\in M_{1+k}^{\dagger}(K^{p}):=\varinjlim_{K_{p}}M_{1+k}^{\dagger}(K^pK_p)$ be an overconvergent modular $\mathbb{T}^{S}$-eigenform of weight $1+k$ with $k\in\mathbb{Z}_{\ge 1}$. Assume that its associated Galois representation $\rho_{f}:\mathrm{Gal}_{\mathbb{Q}}\to \mathrm{GL}_{2}(\bar{\mathbb

Theorems & Definitions (144)

  • Theorem 1.1: Corollary \ref{['corClassicality']}
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4: Theorem \ref{['PanPilloni']}
  • Theorem 1.5: Theorem \ref{['mainthmFontaine=ThetaCohomologyVer']}
  • Definition 1.6
  • Theorem 1.7: Theorem \ref{['mainthmPanPilloni']}
  • Remark 1.8
  • Theorem 1.9: Theorem \ref{['thmFontaine=Theta']}
  • Remark 1.10: Comparison with Pan2209.06II
  • ...and 134 more