Compactified moduli spaces and Hecke correspondences for elliptic curves with a prescribed $N$-torsion scheme
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| Format: | Preprint |
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2025
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| _version_ | 1866917981489987584 |
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| author | Studnia, Elie |
| author_facet | Studnia, Elie |
| contents | Given an integer $N \geq 3$, we prove that for any ring $R$ and any finite locally free $R$-group scheme $G$ which is fppf-locally (over $R$) isomorphic the $N$-torsion subscheme of some elliptic curve $E/R$, there is a smooth affine curve $Y_G(N)$ parametrizing elliptic curves over $R$-schemes whose $N$-torsion subscheme is isomorphic to $G$. We also describe compactifications $X_G(N)$ of these curves when $R$ is a regular excellent Noetherian ring in which $N$ is invertible, as well as construct the Hecke correspondences they are endowed with. As an application, we show that the equations for $X_G(N)$ found over base fields for $N=7,8,9,11,13$ (by Halberstadt--Kraus, Poonen--Schaefer--Stoll, Chen and Fisher) are in fact valid over regular excellent Noetherian bases that are $\mathbb{Q}$-algebras. Finally, we describe in detail the equivalence of this construction with the point of view of Galois twists that these authors use. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_06855 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Compactified moduli spaces and Hecke correspondences for elliptic curves with a prescribed $N$-torsion scheme Studnia, Elie Number Theory Algebraic Geometry 11F32, 14D22, 14G35, 11G18, 14H52 Given an integer $N \geq 3$, we prove that for any ring $R$ and any finite locally free $R$-group scheme $G$ which is fppf-locally (over $R$) isomorphic the $N$-torsion subscheme of some elliptic curve $E/R$, there is a smooth affine curve $Y_G(N)$ parametrizing elliptic curves over $R$-schemes whose $N$-torsion subscheme is isomorphic to $G$. We also describe compactifications $X_G(N)$ of these curves when $R$ is a regular excellent Noetherian ring in which $N$ is invertible, as well as construct the Hecke correspondences they are endowed with. As an application, we show that the equations for $X_G(N)$ found over base fields for $N=7,8,9,11,13$ (by Halberstadt--Kraus, Poonen--Schaefer--Stoll, Chen and Fisher) are in fact valid over regular excellent Noetherian bases that are $\mathbb{Q}$-algebras. Finally, we describe in detail the equivalence of this construction with the point of view of Galois twists that these authors use. |
| title | Compactified moduli spaces and Hecke correspondences for elliptic curves with a prescribed $N$-torsion scheme |
| topic | Number Theory Algebraic Geometry 11F32, 14D22, 14G35, 11G18, 14H52 |
| url | https://arxiv.org/abs/2504.06855 |