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Symmetric power functoriality for Hilbert modular forms

James Newton, Jack A. Thorne

TL;DR

The paper proves symmetric power functoriality for Hilbert modular forms by constructing all Sym^{n-1} lifts of cuspidal GL_2 representations over totally real fields. The authors develop an automorphy framework for untwisted tensor products and then a robust functoriality lifting theorem for tensor products, enabling propagation of automorphy through congruences. They extend these ideas to twisted tensor products using field-of-definition arguments and inertia-based inductions, culminating in an induction-on-n that yields Sym^{n-1} pi for all n. The approach combines Taylor–Wiles–Kisin patching, automorphy lifting in residually reducible settings, and level-raising techniques to achieve unconditional symmetric power functoriality; the results apply to Hilbert modular forms with regular weights and extend the scope beyond CM cases. The work advances the Langlands program by establishing broad symmetric power lifts in the Hilbert modular setting with rigorous automorphy-transfer machinery. All mathematical notation is kept within $...$ delimiters as required.

Abstract

Let $F$ be a totally real field. We prove the existence of all symmetric power liftings of those cuspidal automorphic representations of $\mathrm{GL}_2(\mathbf{A}_F)$ associated to Hilbert modular forms of regular weight.

Symmetric power functoriality for Hilbert modular forms

TL;DR

The paper proves symmetric power functoriality for Hilbert modular forms by constructing all Sym^{n-1} lifts of cuspidal GL_2 representations over totally real fields. The authors develop an automorphy framework for untwisted tensor products and then a robust functoriality lifting theorem for tensor products, enabling propagation of automorphy through congruences. They extend these ideas to twisted tensor products using field-of-definition arguments and inertia-based inductions, culminating in an induction-on-n that yields Sym^{n-1} pi for all n. The approach combines Taylor–Wiles–Kisin patching, automorphy lifting in residually reducible settings, and level-raising techniques to achieve unconditional symmetric power functoriality; the results apply to Hilbert modular forms with regular weights and extend the scope beyond CM cases. The work advances the Langlands program by establishing broad symmetric power lifts in the Hilbert modular setting with rigorous automorphy-transfer machinery. All mathematical notation is kept within delimiters as required.

Abstract

Let be a totally real field. We prove the existence of all symmetric power liftings of those cuspidal automorphic representations of associated to Hilbert modular forms of regular weight.
Paper Structure (11 sections, 42 theorems, 51 equations)

This paper contains 11 sections, 42 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Organization of this paper
  3. Notation
  4. Some comforting lemmas
  5. Automorphy of an untwisted tensor product
  6. A functoriality lifting theorem for tensor products
  7. Automorphy of a twisted tensor product
  8. Preparations for the proof
  9. Automorphy of a twisted tensor product -- first version
  10. Automorphy of a twisted tensor product -- second version
  11. Getting to the end

Key Result

Theorem 1

Let $F$ be a totally real field, and let $\pi$ be a cuspidal automorphic representation of $\mathrm{GL}_2(\mathbf{A}_F)$, without CM, such that $\pi_\infty$ is essentially square-integrable. Then for each $n \geq 2$, the $(n-1)^\text{th}$ symmetric power $\mathop{\mathrm{Sym}}\nolimits^{n-1} \pi$ ex

Theorems & Definitions (80)

  • Theorem 1
  • Conjecture 2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • ...and 70 more