Multiplicativity of Fourier Coefficients of Maass Forms for SL($n,\mathbb Z$)
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909479827668992 |
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| author | Goldfeld, Dorian Stade, Eric Woodbury, Michael |
| author_facet | Goldfeld, Dorian Stade, Eric Woodbury, Michael |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_03562 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiplicativity of Fourier Coefficients of Maass Forms for SL($n,\mathbb Z$) Goldfeld, Dorian Stade, Eric Woodbury, Michael Number Theory 11F55, 11F72 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. |
| title | Multiplicativity of Fourier Coefficients of Maass Forms for SL($n,\mathbb Z$) |
| topic | Number Theory 11F55, 11F72 |
| url | https://arxiv.org/abs/2502.03562 |