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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.18016 |
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| _version_ | 1866910263084580864 |
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| author | Hattori, Shin |
| author_facet | Hattori, Shin |
| contents | Let $p$ be a rational prime, let $q>1$ be a $p$-power integer, let $\mathbb{F}_q$ be the field of $q$ elements and let $A=\mathbb{F}_q[t]$ be the polynomial ring over $\mathbb{F}_q$. Let $\mathfrak{n}\in A$ be a nonzero element and let $\wp\in A$ be a monic irreducible polynomial of positive degree. Let $k\geq 2$ and $r\geq 1$ be integers. Let $S_k(Γ_1(\mathfrak{n}\wp^r))$ be the space of Drinfeld cuspforms of level $Γ_1(\mathfrak{n}\wp^r)$ and weight $k$. In this paper, we prove that the multiplicity of a Hecke eigensystem of finite $\wp$-slope in $S_k(Γ_1(\mathfrak{n}\wp^r))$ is equal to $q^{(r-1)\mathrm{deg}(\wp)}$ times that in $S_k(Γ_1(\mathfrak{n}\wp))$. In particular, this shows that a Hecke eigensystem of finite $\wp$-slope appears in $S_k(Γ_1(\mathfrak{n}\wp^r))$ if and only if it appears in $S_k(Γ_1(\mathfrak{n}\wp))$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_18016 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On some constancy of Hecke eigensystems for Drinfeld cuspforms of finite slope Hattori, Shin Number Theory Let $p$ be a rational prime, let $q>1$ be a $p$-power integer, let $\mathbb{F}_q$ be the field of $q$ elements and let $A=\mathbb{F}_q[t]$ be the polynomial ring over $\mathbb{F}_q$. Let $\mathfrak{n}\in A$ be a nonzero element and let $\wp\in A$ be a monic irreducible polynomial of positive degree. Let $k\geq 2$ and $r\geq 1$ be integers. Let $S_k(Γ_1(\mathfrak{n}\wp^r))$ be the space of Drinfeld cuspforms of level $Γ_1(\mathfrak{n}\wp^r)$ and weight $k$. In this paper, we prove that the multiplicity of a Hecke eigensystem of finite $\wp$-slope in $S_k(Γ_1(\mathfrak{n}\wp^r))$ is equal to $q^{(r-1)\mathrm{deg}(\wp)}$ times that in $S_k(Γ_1(\mathfrak{n}\wp))$. In particular, this shows that a Hecke eigensystem of finite $\wp$-slope appears in $S_k(Γ_1(\mathfrak{n}\wp^r))$ if and only if it appears in $S_k(Γ_1(\mathfrak{n}\wp))$. |
| title | On some constancy of Hecke eigensystems for Drinfeld cuspforms of finite slope |
| topic | Number Theory |
| url | https://arxiv.org/abs/2605.18016 |