Multiplicativity of Fourier coefficients of Maass forms for $\operatorname{SL}(n,\mathbb{Z})$
Abstract
The Fourier coefficients of a Maass form \phi for \operatorname{SL}(n,\mathbb{Z}) are complex numbers A_{\phi}(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_{\phi}(m_{1},1,\dots,1) are eigenvalues of the Hecke algebra and are multiplicative. We prove that the more general Fourier coefficients A_{\phi}(m_{1},\dots,m_{n-1}) are also eigenvalues of the Hecke algebra and satisfy the multiplicativity relations A_{\phi}(m_{1}m_{1}',m_{2}m_{2}',\dots,m_{n-1}m_{n-1}')=A_{\phi}(m_{1},m_{2},\dots,m_{n-1})\cdot A_{\phi}(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.
Fields of Study & Subjects
Advanced Algebra and Geometry
Analytic Number Theory Research
Spectral Theory in Mathematical Physics
Publication details
- Journal / source
- Rendiconti Lincei Matematica e Applicazioni
- Volume
- 36
- Issue
- 3
- Pages
- 501–512
- Publication date
- 4 March 2026
- Language
- EN