Algebraic relations over finite fields that preserve the endomorphism rings of CM $j$-invariants
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866910305538277376 |
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| author | Campagna, Francesco Dill, Gabriel Andreas |
| author_facet | Campagna, Francesco Dill, Gabriel Andreas |
| contents | We characterise the integral affine plane curves over a finite field $k$ with the property that all but finitely many of their $\overline{k}$-points have coordinates that are $j$-invariants of elliptic curves with isomorphic endomorphism rings. This settles a finite field variant of the André-Oort conjecture for $Y(1)^2_\mathbb{C}$, which is a theorem of André. We use our result to solve the modular support problem for function fields of positive characteristic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_10976 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Algebraic relations over finite fields that preserve the endomorphism rings of CM $j$-invariants Campagna, Francesco Dill, Gabriel Andreas Number Theory Algebraic Geometry 11G15, 11G18 We characterise the integral affine plane curves over a finite field $k$ with the property that all but finitely many of their $\overline{k}$-points have coordinates that are $j$-invariants of elliptic curves with isomorphic endomorphism rings. This settles a finite field variant of the André-Oort conjecture for $Y(1)^2_\mathbb{C}$, which is a theorem of André. We use our result to solve the modular support problem for function fields of positive characteristic. |
| title | Algebraic relations over finite fields that preserve the endomorphism rings of CM $j$-invariants |
| topic | Number Theory Algebraic Geometry 11G15, 11G18 |
| url | https://arxiv.org/abs/2308.10976 |