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The eigencurve at crystalline points with scalar Frobenius and Gross-Stark regulators

Adel Betina, Alexandre Maksoud, Alice Pozzi

TL;DR

The paper provides a complete description of the local geometry of the $p$-adic eigencurve at $p$-irregular weight one points, showing that the local ring at the $p$-stabilized point $f_\alpha$ decomposes into four étale components over the weight space and that the ring is non-Gorenstein. It develops a framework connecting adjoint Selmer groups, $p$-adic regulators (including Greenberg–Stevens $\mathscr{L}$-invariants) and Gross–Stark regulators to the deformation theory of the residual Galois representation, with explicit polynomial criteria controlling residual slopes and regulator non-vanishing. By analyzing congruences between Hida families and residual slopes, the authors prove there are exactly four Hida families through $f_\alpha$, and show the four eigencurve components meet transversally at the irregular weight-one point, with each component étale over the weight space. They further relate these geometric phenomena to $p$-adic Hodge–theoretic structures via Ohta’s $\Lambda$-adic Eichler–Shimura theory and the KPX triangulation, highlighting obstructions to freeness in the ordinary $p$-adic cohomology of modular towers and linking these obstructions to regulators and transcendence conjectures. The results illuminate arithmetic consequences for Iwasawa theory and the behavior of adjoint $p$-adic $L$-functions in the irregular setting.

Abstract

A complete description of the local geometry of the $p$-adic eigencurve at $p$-irregular classical weight one cusp forms is given in the cases where the usual $R=T$ methods fall short. As an application, we show that the ordinary $p$-adic étale cohomology group attached to the tower of elliptic modular curves $X_1(Np^r)$ is not free over the Hecke algebra, when localized at a $p$-irregular weight one point.

The eigencurve at crystalline points with scalar Frobenius and Gross-Stark regulators

TL;DR

The paper provides a complete description of the local geometry of the -adic eigencurve at -irregular weight one points, showing that the local ring at the -stabilized point decomposes into four étale components over the weight space and that the ring is non-Gorenstein. It develops a framework connecting adjoint Selmer groups, -adic regulators (including Greenberg–Stevens -invariants) and Gross–Stark regulators to the deformation theory of the residual Galois representation, with explicit polynomial criteria controlling residual slopes and regulator non-vanishing. By analyzing congruences between Hida families and residual slopes, the authors prove there are exactly four Hida families through , and show the four eigencurve components meet transversally at the irregular weight-one point, with each component étale over the weight space. They further relate these geometric phenomena to -adic Hodge–theoretic structures via Ohta’s -adic Eichler–Shimura theory and the KPX triangulation, highlighting obstructions to freeness in the ordinary -adic cohomology of modular towers and linking these obstructions to regulators and transcendence conjectures. The results illuminate arithmetic consequences for Iwasawa theory and the behavior of adjoint -adic -functions in the irregular setting.

Abstract

A complete description of the local geometry of the -adic eigencurve at -irregular classical weight one cusp forms is given in the cases where the usual methods fall short. As an application, we show that the ordinary -adic étale cohomology group attached to the tower of elliptic modular curves is not free over the Hecke algebra, when localized at a -irregular weight one point.

Paper Structure

This paper contains 28 sections, 34 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Cohomology of Artin representations
  3. $p$-adic transcendence theory
  4. Cohomology of Artin representations
  5. Cohomology of $\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_p$
  6. Cohomology of $\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_p(1)$
  7. Cohomology of an Artin representation
  8. Families of adjoint Selmer groups and units
  9. Adjoint Selmer groups associated with an ordinary filtration
  10. Global units attached to $\mathop{\mathrm{ad}}\nolimits^0\rho$
  11. Slopes and cocycles of the first deformations of $\rho$
  12. A Greenberg-Stevens $\mathscr{L}$-invariant and other $p$-adic regulators
  13. Local description of a Selmer class
  14. Non-vanishing of regulators
  15. Regulators in dihedral cases
  16. ...and 13 more sections

Key Result

Theorem A

Let $\mathrm{x}$ be a point of the eigencurve $\mathcal{C}$ corresponding to the unique $p$--stabilization $f_\alpha$ of a $p$--irregular weight one newform $f$. Let $\mathcal{T}$ and $\varLambda \simeq \bar{\mathbb{Q}}_p\@undefined X \@undefined$ be the completed local ring of $\mathcal{C}$ at $\ma where the finite, flat map $\kappa^{\#}:\varLambda \to \mathcal{T}$ sends $X \mapsto \sum_{i=1}^4 X

Theorems & Definitions (78)

  • Theorem A
  • Theorem B: cf. Theorem \ref{['thm:freeness']} and Corollary \ref{['non-splittingOhta']}
  • Conjecture 2.1: Weak $p$-adic Schanuel conjecture
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Conjecture 2.4
  • Lemma 2.5
  • proof
  • ...and 68 more