Signs of the Second Coefficients of Hecke Polynomials
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929421467779072 |
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| author | Ross, Erick Xue, Hui |
| author_facet | Ross, Erick Xue, Hui |
| contents | Let $T_m(N, k, χ)$ be the $m$-th Hecke operator of level $N$, weight $k \ge 2$, and nebentypus $χ$, where $N$ is coprime to $m$. We first show that for any given $m \ge 1$, the second coefficient of the characteristic polynomial of $T_m(N, k, χ)$ is nonvanishing for all but finitely many triples $(N,k,χ)$. Furthermore, for $χ$ trivial and any fixed $m$, we determine the sign of the second coefficient for all but finitely many pairs $(N,k)$. Finally, for $χ$ trivial and $m=3,4$, we compute the sign of the second coefficient for all pairs $(N,k)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_10951 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Signs of the Second Coefficients of Hecke Polynomials Ross, Erick Xue, Hui Number Theory Let $T_m(N, k, χ)$ be the $m$-th Hecke operator of level $N$, weight $k \ge 2$, and nebentypus $χ$, where $N$ is coprime to $m$. We first show that for any given $m \ge 1$, the second coefficient of the characteristic polynomial of $T_m(N, k, χ)$ is nonvanishing for all but finitely many triples $(N,k,χ)$. Furthermore, for $χ$ trivial and any fixed $m$, we determine the sign of the second coefficient for all but finitely many pairs $(N,k)$. Finally, for $χ$ trivial and $m=3,4$, we compute the sign of the second coefficient for all pairs $(N,k)$. |
| title | Signs of the Second Coefficients of Hecke Polynomials |
| topic | Number Theory |
| url | https://arxiv.org/abs/2407.10951 |