Congruence properties of the coefficients of the classical modular polynomials
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917683695452160 |
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| author | Wang, Haiyang |
| author_facet | Wang, Haiyang |
| contents | The classical modular polynomials $Φ_\ell(X,Y)$ give plane curve models for the modular curves $X_0(\ell)/\mathbb{Q}$ and have been extensively studied. In this article, we provide closed formulas for $\ell$ nontrivial coefficients of the classical modular polynomials $Φ_\ell(X,Y)$ in terms of the Fourier coefficients of the modular invariant function $j(z)$ for a prime $\ell$. Our interest in the formulas were motivated by our conjectures on congruences modulo powers of the primes $2,3$ and $5$ satisfied by the coefficients of these polynomials. We deduce congruences from these formulas supporting the conjectures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_01986 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Congruence properties of the coefficients of the classical modular polynomials Wang, Haiyang Number Theory The classical modular polynomials $Φ_\ell(X,Y)$ give plane curve models for the modular curves $X_0(\ell)/\mathbb{Q}$ and have been extensively studied. In this article, we provide closed formulas for $\ell$ nontrivial coefficients of the classical modular polynomials $Φ_\ell(X,Y)$ in terms of the Fourier coefficients of the modular invariant function $j(z)$ for a prime $\ell$. Our interest in the formulas were motivated by our conjectures on congruences modulo powers of the primes $2,3$ and $5$ satisfied by the coefficients of these polynomials. We deduce congruences from these formulas supporting the conjectures. |
| title | Congruence properties of the coefficients of the classical modular polynomials |
| topic | Number Theory |
| url | https://arxiv.org/abs/2406.01986 |