Multiplicativity of Fourier coefficients of Maass forms for $\operatorname{SL}(n,\mathbb{Z})$

Dorian Goldfeld; Eric Stade; Michael Woodbury

doi:10.4171/rlm/1081

JournalsrlmVol. 36, No. 3pp. 501–512

Multiplicativity of Fourier coefficients of Maass forms for $SL (n, Z)$

Dorian Goldfeld
Columbia University, New York, USA
Eric Stade
University of Colorado Boulder, USA
Michael Woodbury
Rutgers, The State University of New Jersey, Piscataway, USA

$Multiplicativity of Fourier coefficients of Maass forms for $\operatorname{SL}(n,\mathbb{Z})$ cover$

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Abstract

The Fourier coefficients of a Maass form $ϕ$ for $SL (n, Z)$ are complex numbers $A_{ϕ} (M)$ , where $M = (m_{1}, m_{2}, \dots, m_{n - 1})$ and $m_{1}, m_{2}, \dots, m_{n - 1}$ are non-zero integers. It is well known that coefficients of the form $A_{ϕ} (m_{1}, 1, \dots, 1)$ are eigenvalues of the Hecke algebra and are multiplicative. We prove that the more general Fourier coefficients $A_{ϕ} (m_{1}, \dots, m_{n - 1})$ are also eigenvalues of the Hecke algebra and satisfy the multiplicativity relations

A_{ϕ} (m_{1} m_{1}^{'}, m_{2} m_{2}^{'}, \dots, m_{n - 1} m_{n - 1}^{'}) = A_{ϕ} (m_{1}, m_{2}, \dots, m_{n - 1}) \cdot A_{ϕ} (m_{1}^{'}, m_{2}^{'}, \dots, m_{n - 1}^{'})

provided the products $\prod_{i = 1}^{n - 1} m_{i}$ and $\prod_{i = 1}^{n - 1} m_{i}^{'}$ are relatively prime to each other.

Cite this article

Dorian Goldfeld, Eric Stade, Michael Woodbury, Multiplicativity of Fourier coefficients of Maass forms for $SL (n, Z)$ . Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 36 (2025), no. 3, pp. 501–512

DOI 10.4171/RLM/1081