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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | https://arxiv.org/abs/2603.29789 |
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| _version_ | 1866915902951260160 |
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| author | Colò, Leonardo |
| author_facet | Colò, Leonardo |
| contents | We propose a bridge between oriented supersingular elliptic curves and the arithmetic of modular curves. To an $\mathcal{O}$-oriented supersingular curve, we attach a class in the relative homology group $H(X_0(N),C,\mathbb{Z})$, i.e. modular symbols, compatible with the Hecke action. We then compute vectors of $\ell$-adic periods by pairing with weight $2$ cusp forms via Coleman integration. This yields an explicit, computable map from short combinatorial homology representatives to truncated vectors in $(\mathbb{Z}/\ell^m\mathbb{Z})^d$. Motivated by this encoding, we formulate the Modular Symbol Inversion (MSI) problem -- recovering a short homology representative from its truncated $\ell$-adic period data -- and discuss its arithmetic structure, its relation to path problems on isogeny graphs and Bruhat-Tits trees, and potential applications to cryptographic constructions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_29789 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | From Orientations to $\ell$-adic Period Vectors Colò, Leonardo Number Theory We propose a bridge between oriented supersingular elliptic curves and the arithmetic of modular curves. To an $\mathcal{O}$-oriented supersingular curve, we attach a class in the relative homology group $H(X_0(N),C,\mathbb{Z})$, i.e. modular symbols, compatible with the Hecke action. We then compute vectors of $\ell$-adic periods by pairing with weight $2$ cusp forms via Coleman integration. This yields an explicit, computable map from short combinatorial homology representatives to truncated vectors in $(\mathbb{Z}/\ell^m\mathbb{Z})^d$. Motivated by this encoding, we formulate the Modular Symbol Inversion (MSI) problem -- recovering a short homology representative from its truncated $\ell$-adic period data -- and discuss its arithmetic structure, its relation to path problems on isogeny graphs and Bruhat-Tits trees, and potential applications to cryptographic constructions. |
| title | From Orientations to $\ell$-adic Period Vectors |
| topic | Number Theory |
| url | https://arxiv.org/abs/2603.29789 |