Multiplicativity of Fourier Coefficients of Maass Forms for SL(n,\mathbb Z)

2025/02/05 by Goldfeld, Dorian, Stade, Eric, Woodbury, Michael
#11F55 #11F72 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2502.03562

Abstract

The Fourier coefficients of a Maass form ϕ for SL(n,\mathbb Z) are complex numbers A_ϕ(M), where M=(m₁,m₂,…,m_n-1) and m₁,m₂,… ,m_n-1 are nonzero integers. It is well known that coefficients of the form A_ϕ(m₁,1,…,1) are eigenvalues of the Hecke algebra and are multiplicative. We prove that the more general Fourier coefficients A_ϕ(m₁,…,m_n-1) are also eigenvalues of the Hecke algebra and satisfy the multiplicativity relations A_ϕ(m₁m₁', m₂m₂', … m_n-1m_n-1') = A_ϕ(m₁,m₂,…,m_n-1)⋅ A_ϕ(m₁',m₂',…,m_n-1') provided the products ∏_i=1^n-1 m_i and ∏_i=1^n-1 m_i' are relatively prime to each other.

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