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A $p$-adic approach to the existence of level-raising congruences

Jack A. Thorne

Abstract

We construct level-raising congruences between $p$-ordinary automorphic representations, and apply this to the problem of symmetric power functoriality for Hilbert modular forms. In particular, we prove the existence of the $n^\text{th}$ symmetric power lift of a Hilbert modular eigenform of regular weight for each odd integer $n = 1, 3, \dots, 25$.

A $p$-adic approach to the existence of level-raising congruences

Abstract

We construct level-raising congruences between -ordinary automorphic representations, and apply this to the problem of symmetric power functoriality for Hilbert modular forms. In particular, we prove the existence of the symmetric power lift of a Hilbert modular eigenform of regular weight for each odd integer .
Paper Structure (8 sections, 53 theorems, 150 equations)

This paper contains 8 sections, 53 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Homology of smooth representations
  3. The elliptic representations
  4. The definite unitary group
  5. Analysis of a derived tensor product
  6. Raising the level
  7. Application to symmetric power functoriality
  8. The mixed parity case

Key Result

Theorem 1

Fix a partition $n = n_1 + n_2$ and let $\pi_1, \pi_2$ be cuspidal, conjugate self-dual automorphic representations of $\mathrm{GL}_{n_1}(\mathbf{A}_F)$, $\mathrm{GL}_{n_2}(\mathbf{A}_F)$ such that $\pi = \pi_1 \boxplus \pi_2$ is regular algebraic and $\iota$-ordinary. Suppose that the following con Then there exists a regular algebraic, conjugate self-dual, cuspidal, $\iota$-ordinary automorphic

Theorems & Definitions (97)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • ...and 87 more