Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$p$-adic rigidity of eigenforms of infinite slope

Andrea Conti

Abstract

We give a notion of $p$-adic families of Hecke eigenforms that allows for the slope of the forms be infinite at $p$. We prove that, contrary to the case of finite slope when every eigenform lives in a Hida or Coleman family, the only families of infinite slope are either twists of Hida or Coleman families with Dirichlet characters of $p$-power conductor, or non-ordinary families with complex multiplication. Our proof goes via a local study of deformations of potentially trianguline Galois representations, relying on work of Berger and Chenevier, and a global input coming from an analogue of a result of Balasubramanyam, Ghate and Vatsal on a Greenberg-type conjecture for families of Hilbert modular forms.

$p$-adic rigidity of eigenforms of infinite slope

Abstract

We give a notion of -adic families of Hecke eigenforms that allows for the slope of the forms be infinite at . We prove that, contrary to the case of finite slope when every eigenform lives in a Hida or Coleman family, the only families of infinite slope are either twists of Hida or Coleman families with Dirichlet characters of -power conductor, or non-ordinary families with complex multiplication. Our proof goes via a local study of deformations of potentially trianguline Galois representations, relying on work of Berger and Chenevier, and a global input coming from an analogue of a result of Balasubramanyam, Ghate and Vatsal on a Greenberg-type conjecture for families of Hilbert modular forms.
Paper Structure (49 sections, 85 theorems, 55 equations)

This paper contains 49 sections, 85 theorems, 55 equations.

Table of Contents

  1. Topological lemmas
  2. Pseudorepresentations and their deformations
  3. Preliminaries on pseudorepresentations
  4. Residual pseudorepresentations
  5. The Hodge--Tate--Sen weights of a pseudorepresentation
  6. Hodge--Tate--Sen weights in families
  7. Universal pseudodeformations
  8. Pseudodeformations along rigid analytic spaces
  9. The determinant map
  10. Restriction of pseudorepresentations
  11. Induction of pseudorepresentations
  12. Convergence of pseudorepresentations
  13. Trianguline Galois representations
  14. $p$-adic Galois types
  15. Potentially trianguline representations
  16. ...and 34 more sections

Key Result

Theorem 2

Every $p$-supercuspidal (affinoid) family is a CM family.

Theorems & Definitions (228)

  • Definition 1: cf. Definitions \ref{['famdef']} and \ref{['afffamdef']}
  • Theorem 2: cf. Theorem \ref{['infslope']}
  • Theorem 3: cf. Theorem \ref{['bgvaff']}
  • Remark 1.1
  • Definition 1.2
  • Lemma 1.3
  • proof
  • Definition 1.4
  • Lemma 1.5: bellchen
  • Definition 1.6
  • ...and 218 more