Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Ramanujan-Petersson and Selberg conjectures for Maass forms

Andr'e Unterberger

TL;DR

The paper develops automorphic distributions on the plane $\mathbb{R}^2$ and uses Weyl/pseudodifferential analysis to prove the Ramanujan–Petersson conjecture for Maass forms on $\mathrm{SL}(2,\mathbb{Z})$, with the Selberg eigenvalue conjecture for $\Gamma_0(M)$ as a corollary. Central to the approach is the transform $\Theta$ that connects plane distributions to non-holomorphic modular forms, enabling a transfer of spectral questions to a plane-based setting where the Euler operator $2i\pi\mathcal{E}$ (and its square) plays a key role. The paper constructs a generating object $\mathfrak B^j$ from modular distributions and analyzes Hecke operators in the plane via $T_p^{\mathrm{dist}}$, providing uniform bounds on their powers and a spectral localization framework. Through a combination of automorphic-distribution theory, Weyl calculus, and careful decomposition into Eisenstein and cuspidal components, the authors obtain sharp Ramanujan–Petersson bounds and Selberg-type spectral results, illustrating a novel analytic route bridging PDE techniques and automorphic form theory.

Abstract

We prove the Ramanujan-Petersson conjecture for Maass forms of the group $SL(2,Z)$, with the help of automorphic distribution theory and pseudodifferential analysis. The first notion is an alternative to classical automorphic function theory, in which the plane takes the place usually ascribed to the hyperbolic half-plane. The Selberg conjecture for Hecke's group $Γ_0(M)$ follows as well.

The Ramanujan-Petersson and Selberg conjectures for Maass forms

TL;DR

The paper develops automorphic distributions on the plane and uses Weyl/pseudodifferential analysis to prove the Ramanujan–Petersson conjecture for Maass forms on , with the Selberg eigenvalue conjecture for as a corollary. Central to the approach is the transform that connects plane distributions to non-holomorphic modular forms, enabling a transfer of spectral questions to a plane-based setting where the Euler operator (and its square) plays a key role. The paper constructs a generating object from modular distributions and analyzes Hecke operators in the plane via , providing uniform bounds on their powers and a spectral localization framework. Through a combination of automorphic-distribution theory, Weyl calculus, and careful decomposition into Eisenstein and cuspidal components, the authors obtain sharp Ramanujan–Petersson bounds and Selberg-type spectral results, illustrating a novel analytic route bridging PDE techniques and automorphic form theory.

Abstract

We prove the Ramanujan-Petersson conjecture for Maass forms of the group , with the help of automorphic distribution theory and pseudodifferential analysis. The first notion is an alternative to classical automorphic function theory, in which the plane takes the place usually ascribed to the hyperbolic half-plane. The Selberg conjecture for Hecke's group follows as well.

Paper Structure

This paper contains 9 sections, 19 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Homogeneous and bihomogeneous functions in the plane
  3. From the plane to the half-plane
  4. Modular distributions and modular forms
  5. Hecke operators and their powers
  6. A generating object for modular distributions
  7. The operator with $(2i\pi{\mathcal{E}})^2{\mathfrak B}$ for symbol
  8. The Ramanujan-Petersson estimate for Maass forms
  9. The Selberg conjecture

Key Result

Proposition 4.1

Given a non-trivial character $\chi$ on $\mathbb Q^{\times}$, such that $|\chi(\frac{m}{n})|\leq |mn|^C$ for some $C>0$, and $\lambda\in \mathbb R$, the even distribution ${\mathfrak N}_{\chi,i\lambda}\in {\mathcal{S}}'(\mathbb R^2)$ defined by the equation satisfies the identity $\,\langle{\mathfrak N}_{\chi,i\lambda}\,,\,h\,\circ\,\left(\right)\rangle\,=\,\langle{\mathfrak N}_{\chi,i\lambda}\,,

Theorems & Definitions (35)

  • Proposition 4.1
  • Proposition 4.2
  • Proposition 4.3
  • proof
  • Proposition 4.4
  • Theorem 5.1
  • proof
  • Lemma 5.2
  • proof
  • Proposition 5.3
  • ...and 25 more