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Eisenstein points on the Hilbert cuspidal eigenvariety

Adel Betina, Mladen Dimitrov, Sheng-Chi Shih

TL;DR

This work analyzes the geometry of Hilbert $p$-adic eigenvarieties at weight-$1$ intersections between cuspidal and Eisenstein loci, combining twisted Coleman families with Galois deformation theory and generalized matrix algebras to reveal precise local structures. The authors establish $R=T$ results in favorable cases, determine étaleness criteria for weight maps via $oldsymbol{ extL}$-invariants, and compute Eisenstein congruence ideals in full generality, tying intersection multiplicities to zeros of Deligne–Ribet $p$-adic $L$-functions. They also develop a robust framework for geometric $q$-expansions, residue maps, and the fundamental exact sequence, enabling geometric constructions of Eisenstein congruences and yielding new Iwasawa-theoretic consequences such as infinite unramified extensions and Gross–Stark-type results. The methods provide a versatile, deformation-theoretic approach that illuminates the arithmetic of Hilbert automorphic forms and offers potential extensions to unitary and symplectic Shimura varieties, with applications to $p$-adic $L$-functions and Euler systems.

Abstract

We present a comprehensive study of the geometry of Hilbert $p$-adic eigenvarieties at classical parallel weight one intersection points of their cuspidal and Eisenstein loci. For instance, we determine all such points at which the weight map is étale. The Galois theoretic approach presents genuine difficulties due to the lack of good deformation theory for pseudo-characters irregular at $p$ and reflects the richness of the local geometry. We believe that our geometric results lead to deeper insight into the arithmetic of Hilbert automorphic forms and we produce in support several applications in Iwasawa theory.

Eisenstein points on the Hilbert cuspidal eigenvariety

TL;DR

This work analyzes the geometry of Hilbert -adic eigenvarieties at weight- intersections between cuspidal and Eisenstein loci, combining twisted Coleman families with Galois deformation theory and generalized matrix algebras to reveal precise local structures. The authors establish results in favorable cases, determine étaleness criteria for weight maps via -invariants, and compute Eisenstein congruence ideals in full generality, tying intersection multiplicities to zeros of Deligne–Ribet -adic -functions. They also develop a robust framework for geometric -expansions, residue maps, and the fundamental exact sequence, enabling geometric constructions of Eisenstein congruences and yielding new Iwasawa-theoretic consequences such as infinite unramified extensions and Gross–Stark-type results. The methods provide a versatile, deformation-theoretic approach that illuminates the arithmetic of Hilbert automorphic forms and offers potential extensions to unitary and symplectic Shimura varieties, with applications to -adic -functions and Euler systems.

Abstract

We present a comprehensive study of the geometry of Hilbert -adic eigenvarieties at classical parallel weight one intersection points of their cuspidal and Eisenstein loci. For instance, we determine all such points at which the weight map is étale. The Galois theoretic approach presents genuine difficulties due to the lack of good deformation theory for pseudo-characters irregular at and reflects the richness of the local geometry. We believe that our geometric results lead to deeper insight into the arithmetic of Hilbert automorphic forms and we produce in support several applications in Iwasawa theory.
Paper Structure (27 sections, 38 theorems, 157 equations)

This paper contains 27 sections, 38 theorems, 157 equations.

Table of Contents

  1. Twisted Coleman families and Hilbert eigenvarieties
  2. Formal models for Hilbert modular schemes
  3. The Hodge--Tate map
  4. Andreatta--Iovita--Pilloni--Stevens' overconvergent modular sheaves
  5. Classicality for overconvergent modular forms
  6. Twisted $p$-adic families of arithmetic Hilbert modular forms
  7. Adic version of the formal function theorem
  8. Toroidal compactifications and descent to the minimal compactification
  9. The specialization map for overconvergent $p$-adic modular forms
  10. Geometric $q$-expansion, the residue map and the fundamental exact sequence
  11. The Eisenstein congruence ideal
  12. Eisenstein families
  13. Eigenvarieties
  14. Constant terms of Eisenstein families
  15. Hecke duality and Proof of Theorem \ref{['congruencemodule']}
  16. ...and 12 more sections

Key Result

Theorem A

Assume that $\Sigma_p^{\mathrm{irr}}=\{ \mathfrak{p} \}$ and that the Leopoldt defect $\delta_{F,p}$ vanishes. Set $d_{\mathfrak{p}}=[F_{\mathfrak{p}}:\mathbf{Q}_p]$.

Theorems & Definitions (77)

  • Theorem A
  • Theorem B
  • Proposition 1.1
  • proof
  • Proposition 1.2
  • proof
  • Remark 1.3
  • Corollary 1.4
  • proof
  • Theorem 1.5
  • ...and 67 more