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Slopes of modular forms and geometry of eigencurves

Ruochuan Liu, Nha Xuan Truong, Liang Xiao, Bin Zhao

Abstract

Under a stronger genericity condition, we prove the local analogue of ghost conjecture of Bergdall and Pollack. As applications, we deduce in this case (a) a folklore conjecture of Breuil--Buzzard--Emerton on the crystalline slopes of Kisin's crystabelian deformation spaces, (b) Gouvea's $\lfloor\frac{k-1}{p+1}\rfloor$-conjecture on slopes of modular forms, and (c) the finiteness of irreducible components of the eigencurve. In addition, applying combinatorial arguments by Bergdall and Pollack, and by Ren, we deduce as corollaries in the reducible and strongly generic case, (d) Gouvea--Mazur conjecture, (e) a variant of Gouvea's conjecture on slope distributions, and (f) a refined version of Coleman's spectral halo conjecture.

Slopes of modular forms and geometry of eigencurves

Abstract

Under a stronger genericity condition, we prove the local analogue of ghost conjecture of Bergdall and Pollack. As applications, we deduce in this case (a) a folklore conjecture of Breuil--Buzzard--Emerton on the crystalline slopes of Kisin's crystabelian deformation spaces, (b) Gouvea's -conjecture on slopes of modular forms, and (c) the finiteness of irreducible components of the eigencurve. In addition, applying combinatorial arguments by Bergdall and Pollack, and by Ren, we deduce as corollaries in the reducible and strongly generic case, (d) Gouvea--Mazur conjecture, (e) a variant of Gouvea's conjecture on slope distributions, and (f) a refined version of Coleman's spectral halo conjecture.
Paper Structure (46 sections, 74 theorems, 360 equations)

This paper contains 46 sections, 74 theorems, 360 equations.

Table of Contents

  1. Introduction
  2. Questions of slopes of modular forms
  3. Statement of main theorems
  4. Application A: Breuil--Buzzard--Emerton conjecture
  5. Application B: Gouvêa's $\lfloor\frac{k-1}{p+1}\rfloor$-conjecture
  6. Application C: Finiteness of irreducible components of eigencurves
  7. Application D: Gouvêa--Mazur conjecture
  8. Application E: Gouvêa's slope distribution conjecture
  9. Application F: refined Coleman--Mazur--Buzzard--Kilford spectral halo conjecture
  10. Overview of the proof of Theorem \ref{['T:local ghost theorem']}
  11. Notations
  12. Normalizations
  13. Recollection of the local ghost conjecture
  14. Space of abstract forms
  15. Power basis
  16. ...and 31 more sections

Key Result

Theorem 1.3

Assume $p\geq 11$ and that $\bar{r}: \mathop{\mathrm{Gal}}\nolimits_\mathbb{Q} \to \mathop{\mathrm{GL}}\nolimits_2(\mathbb{F})$ is an absolutely irreducible representation such that $\bar{r}_p$ is reducible and very generic (i.e. $2 \leq a \leq p-5$). Then for every $w_\star \in \mathfrak{m}_{\mathb

Theorems & Definitions (183)

  • Theorem 1.3: Ghost conjecture
  • Remark 1.4
  • Definition 1.5
  • Theorem 1.6: Local ghost theorem
  • Remark 1.7
  • Remark 1.8
  • Theorem 1.10: Breuil--Buzzard--Emerton conjecture
  • Remark 1.11
  • Theorem 1.13: Gouvêa's $\lfloor\frac{k-1}{p+1}\rfloor$-conjecture
  • Remark 1.14
  • ...and 173 more