Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Ramanujan and Sato-Tate Conjectures for Bianchi modular forms

George Boxer, Frank Calegari, Toby Gee, James Newton, Jack A. Thorne

TL;DR

This work proves the Ramanujan and Sato–Tate conjectures for regular algebraic, cuspidal automorphic representations on GL$_2$ over CM fields, including Bianchi modular forms, by establishing a potent automorphy framework for symmetric powers of 2–dimensional Galois representations. The authors develop a weight 0 automorphy lifting theorem using Ihara avoidance and the Emerton–Gee stacks, together with a generically reduced weight 0 crystalline deformation theory and a Dwork-family construction to produce ordinary points. Central technical innovations include a p–q–r switching via Harris tensor products to align local Hodge–Tate weights with automorphic inputs, and a detailed patching argument that handles arbitrary ramification at $p$. The results connect local crystalline lifting theory, deformation-ring geometry, and global automorphy to derive automorphy and then deduce Ramanujan-type bounds and Sato–Tate equidistribution for Bianchi and imaginary CM settings, with broader implications for potential automorphy in the Langlands program.

Abstract

We prove the Ramanujan and Sato-Tate conjectures for Bianchi modular forms of weight at least 2. More generally, we prove these conjectures for all regular algebraic cuspidal automorphic representations of $\mathrm{GL}_2(\mathbf{A}_F)$ of parallel weight, where $F$ is any CM field. We deduce these theorems from a new potential automorphy theorem for the symmetric powers of 2-dimensional compatible systems of Galois representations of parallel weight.

The Ramanujan and Sato-Tate Conjectures for Bianchi modular forms

TL;DR

This work proves the Ramanujan and Sato–Tate conjectures for regular algebraic, cuspidal automorphic representations on GL over CM fields, including Bianchi modular forms, by establishing a potent automorphy framework for symmetric powers of 2–dimensional Galois representations. The authors develop a weight 0 automorphy lifting theorem using Ihara avoidance and the Emerton–Gee stacks, together with a generically reduced weight 0 crystalline deformation theory and a Dwork-family construction to produce ordinary points. Central technical innovations include a p–q–r switching via Harris tensor products to align local Hodge–Tate weights with automorphic inputs, and a detailed patching argument that handles arbitrary ramification at . The results connect local crystalline lifting theory, deformation-ring geometry, and global automorphy to derive automorphy and then deduce Ramanujan-type bounds and Sato–Tate equidistribution for Bianchi and imaginary CM settings, with broader implications for potential automorphy in the Langlands program.

Abstract

We prove the Ramanujan and Sato-Tate conjectures for Bianchi modular forms of weight at least 2. More generally, we prove these conjectures for all regular algebraic cuspidal automorphic representations of of parallel weight, where is any CM field. We deduce these theorems from a new potential automorphy theorem for the symmetric powers of 2-dimensional compatible systems of Galois representations of parallel weight.
Paper Structure (34 sections, 54 theorems, 161 equations)

This paper contains 34 sections, 54 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. The new ideas in this paper
  3. Ihara avoidance and the Emerton--Gee stack
  4. Bianchi Modular Forms
  5. Recent work of Matsumoto
  6. Acknowledgements
  7. Notation
  8. The special fibres of weight $0$ crystalline lifting rings are generically reduced
  9. Breuil modules and strongly divisible modules
  10. Extensions of rank one Breuil modules
  11. Generic weight 0 crystalline representations
  12. Recollections on Emerton--Gee stacks
  13. Generic reducedness
  14. An automorphy lifting theorem in weight $0$
  15. Preliminaries
  16. ...and 19 more sections

Key Result

Theorem A

Let $F/{\mathbf Q}$ be an imaginary CM field. Let $\pi$ be a cuspidal algebraic automorphic representation for $\mathop{\mathrm{GL}}\nolimits(2)/F$ of parallel weight $k\ge 2$. Then $\pi_v$ is tempered for all finite places $v$; in particular, for places $v$ prime to the level of $\pi$, the Satake p

Theorems & Definitions (129)

  • Theorem A: Ramanujan Conjecture, Theorem \ref{['thm: Ramanujan thm main\n paper']}
  • Theorem B: Sato--Tate Conjecture, Theorem \ref{['thm_Sato_Tate_general_case']}
  • Theorem C: Potential automorphy of symmetric powers, Theorem \ref{['PO']}
  • Theorem D: Theorem \ref{['thm: special fibre weight 0 crystalline def ring generically\n reduced']}
  • Theorem E
  • Theorem F
  • Theorem G
  • proof
  • Definition 1.6.1: Parallel Weight
  • Remark 2.1.1
  • ...and 119 more