Multiplicativity of Fourier Coefficients of Maass Forms for SL($n,\mathbb Z$)
Dorian Goldfeld, Eric Stade, Michael Woodbury
TL;DR
The paper addresses the multiplicativity of Fourier coefficients $A_\phi(m_1,\dots,m_{n-1})$ for Maass forms on SL$(n,\mathbb Z)$ by proving that all such coefficients are eigenvalues of the Hecke algebra and satisfy the multiplicativity relation under the coprimality condition $\gcd(\prod_i m_i,\prod_i m_i')=1$. It develops the $A$-formula and related lemmas to show each $A_\phi(p^{K_1},\dots,p^{K_r},1,\dots,1)$ is a Hecke eigenvalue, and deduces the general multiplicativity from this eigenstructure. The paper provides explicit combinatorial expressions for coefficients with a single $p$ in a fixed position and for coefficients with a single $p$-power in one coordinate, via recurrences and tailored Hecke operator combinations. These results justify the Euler product description of the Godement–Jacquet $L$-function in terms of general Fourier coefficients and fill a gap in the literature by providing a complete proof of multiplicativity for all $A_\phi(m_1,\dots,m_{n-1})$.
Abstract
The Fourier coefficients of a Maass form $φ$ for SL$(n,\mathbb Z)$ are complex numbers $A_φ(M)$, where $M=(m_1,m_2,\ldots,m_{n-1})$ and $m_1,m_2,\ldots ,m_{n-1}$ are nonzero integers. It is well known that coefficients of the form $A_φ(m_1,1,\ldots,1)$ are eigenvalues of the Hecke algebra and are multiplicative. We prove that the more general Fourier coefficients $A_φ(m_1,\ldots,m_{n-1})$ are also eigenvalues of the Hecke algebra and satisfy the multiplicativity relations $$A_φ\big(m_1m_1',\;m_2m_2', \;\ldots\; m_{n-1}m_{n-1}'\big) = A_φ\big(m_1,m_2,\ldots,m_{n-1})\cdot A_φ(m_1',m_2',\ldots,m_{n-1}'\big)$$ provided the products $\prod\limits_{i=1}^{n-1} m_i$ and $\prod\limits_{i=1}^{n-1} m_i'$ are relatively prime to each other.