Divisibility of the coefficients of modular polynomials
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914095775612928 |
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| author | Breuer, Florian |
| author_facet | Breuer, Florian |
| contents | Let $N>1$ and let $Φ_N(X,Y)\in\mathbb{Z}[X,Y]$ be the modular polynomial which vanishes precisely at pairs of $j$-invariants of elliptic curves linked by a cyclic isogeny of degree $N$. In this note we study the divisibility of the coefficients of $Φ_N(X+J, Y+J)$ for certain algebraic numbers $J$, in particular $J=0$ and other singular moduli. It turns out that these coefficients are highly divisible by small primes at which $J$ is supersingular. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06423 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Divisibility of the coefficients of modular polynomials Breuer, Florian Number Theory 11G07, 11G15 Let $N>1$ and let $Φ_N(X,Y)\in\mathbb{Z}[X,Y]$ be the modular polynomial which vanishes precisely at pairs of $j$-invariants of elliptic curves linked by a cyclic isogeny of degree $N$. In this note we study the divisibility of the coefficients of $Φ_N(X+J, Y+J)$ for certain algebraic numbers $J$, in particular $J=0$ and other singular moduli. It turns out that these coefficients are highly divisible by small primes at which $J$ is supersingular. |
| title | Divisibility of the coefficients of modular polynomials |
| topic | Number Theory 11G07, 11G15 |
| url | https://arxiv.org/abs/2509.06423 |