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A GL(3) converse theorem via a "beyond endoscopy'' approach

Valentin Blomer, Wing Hong Leung

Abstract

We give a new proof of the converse theorem for Maass forms on ${\rm GL}(3)$ using a technique that is inspired by Langlands' philosophy of "beyond endoscopy", thereby implementing these ideas for the first time in a higher rank setting.

A GL(3) converse theorem via a "beyond endoscopy'' approach

Abstract

We give a new proof of the converse theorem for Maass forms on using a technique that is inspired by Langlands' philosophy of "beyond endoscopy", thereby implementing these ideas for the first time in a higher rank setting.
Paper Structure (31 sections, 22 theorems, 410 equations)

This paper contains 31 sections, 22 theorems, 410 equations.

Table of Contents

  1. Introduction
  2. Converse theorems
  3. The method
  4. Automorphic forms on GL(3)
  5. Hecke relations
  6. Voronoi summation
  7. Kloosterman sums
  8. Integral kernels
  9. Choice of the test function and properties of its integral transform
  10. The Kuznetsov formula
  11. Beginning of the proof of Theorem \ref{['thm2']} -- the easy cases
  12. The Eisenstein contribution
  13. Minimal Eisenstein series
  14. Maximal Eisenstein series
  15. Sum6 contribution
  16. ...and 16 more sections

Key Result

Theorem 1.1

Suppose that a sequence $B(n, m)$ of complex numbers satisfies the usual Hecke relations, the Ramanujan conjecture and the Voronoi summation formula. Then this sequence is the set of Fourier coefficients of an automorphic form on ${\rm GL}(3)$ whose archimedean Langlands parameter is determined by t

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2: MS
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 30 more