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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2411.12754 |
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| _version_ | 1866917841494605824 |
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| author | Nicolas, Jean-Louis |
| author_facet | Nicolas, Jean-Louis |
| contents | Let $Δ= \sum_{m=0}^\infty q^{(2m+1)^2} \in \mathbb{F}_2[[q]]$ be the reduction mod 2 of the $Δ$ series. A modular form $f$ modulo $2$ of level 1 is a polynomial in $Δ$. If $p$ is an odd prime, then the Hecke operator $T_p$ transforms $f$ in a modular form $T_p(f)$ which is a polynomial in $Δ$ whose degree is smaller than the degree of $f$, so that $T_p$ is nilpotent. The order of nilpotence of $f$ is defined as the smallest integer $g=g(f)$ such that, for every family of $g$ odd primes $p_1,p_2,\ldots,p_g$, the relation $T_{p_1}T_{p_2}\ldots T_{p_g}(f)=0$ holds. We show how one can compute explicitly $g(f)$; if $f$ is a polynomial of degree $d\geqslant 1$ in $Δ$, one finds that $g(f) < \frac 32 \sqrt d$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_12754 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Formes modulaires modulo $2$ : L'ordre de nilpotence des opérateurs de Hecke (version développée) Nicolas, Jean-Louis Number Theory 14J15 G.2 Let $Δ= \sum_{m=0}^\infty q^{(2m+1)^2} \in \mathbb{F}_2[[q]]$ be the reduction mod 2 of the $Δ$ series. A modular form $f$ modulo $2$ of level 1 is a polynomial in $Δ$. If $p$ is an odd prime, then the Hecke operator $T_p$ transforms $f$ in a modular form $T_p(f)$ which is a polynomial in $Δ$ whose degree is smaller than the degree of $f$, so that $T_p$ is nilpotent. The order of nilpotence of $f$ is defined as the smallest integer $g=g(f)$ such that, for every family of $g$ odd primes $p_1,p_2,\ldots,p_g$, the relation $T_{p_1}T_{p_2}\ldots T_{p_g}(f)=0$ holds. We show how one can compute explicitly $g(f)$; if $f$ is a polynomial of degree $d\geqslant 1$ in $Δ$, one finds that $g(f) < \frac 32 \sqrt d$. |
| title | Formes modulaires modulo $2$ : L'ordre de nilpotence des opérateurs de Hecke (version développée) |
| topic | Number Theory 14J15 G.2 |
| url | https://arxiv.org/abs/2411.12754 |