Proportion of Atkin-Lehner sign patterns and Hecke Eigenvalue Equidistribution
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915624014315520 |
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| author | Ross, Erick van Lidth, Alexandre Wolf, Martha Rose Xue, Hui |
| author_facet | Ross, Erick van Lidth, Alexandre Wolf, Martha Rose Xue, Hui |
| contents | Let $N \ge 1$, $k \ge 2$ even, and $σ$ denote a sign pattern for $N$. In this paper, we first determine the exact proportion of forms in $S_k(N)$ and $S_k^\mathrm{new}(N)$ with a given Atkin-Lehner sign pattern $σ$. Then we study the asymptotic behavior of the Hecke operators $T_p$ over the subspaces of $S_k(N)$ and $S_k^{\mathrm{new}}(N)$ with Atkin-Lehner sign pattern $σ$. In particular, for the $p$-adic Plancherel measure $μ_p$, we show that the Hecke eigenvalues for $T_p$ over these subspaces are $μ_p$-equidistributed as $N+k \to \infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_13969 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Proportion of Atkin-Lehner sign patterns and Hecke Eigenvalue Equidistribution Ross, Erick van Lidth, Alexandre Wolf, Martha Rose Xue, Hui Number Theory Primary 11F25, Secondary 11F72 and 11F11 Let $N \ge 1$, $k \ge 2$ even, and $σ$ denote a sign pattern for $N$. In this paper, we first determine the exact proportion of forms in $S_k(N)$ and $S_k^\mathrm{new}(N)$ with a given Atkin-Lehner sign pattern $σ$. Then we study the asymptotic behavior of the Hecke operators $T_p$ over the subspaces of $S_k(N)$ and $S_k^{\mathrm{new}}(N)$ with Atkin-Lehner sign pattern $σ$. In particular, for the $p$-adic Plancherel measure $μ_p$, we show that the Hecke eigenvalues for $T_p$ over these subspaces are $μ_p$-equidistributed as $N+k \to \infty$. |
| title | Proportion of Atkin-Lehner sign patterns and Hecke Eigenvalue Equidistribution |
| topic | Number Theory Primary 11F25, Secondary 11F72 and 11F11 |
| url | https://arxiv.org/abs/2511.13969 |