The stacky Batyrev-Manin conjecture and modular curves
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913125245124608 |
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| author | Darda, Ratko Han, Changho |
| author_facet | Darda, Ratko Han, Changho |
| contents | Let $\mathcal{X}_0(N)$ be the Deligne--Rapoport modular stack of elliptic curves endowed with a cyclic rational $N$-isogeny over a number field $F$. Let $N\in\{1,2,3,4,5,6,7,8,9,10,12,13,16,18,25\},$ which are precisely the values for which the coarse moduli space of $\mathcal{X}_0(N)$ is isomorphic to $\mathbb{P}^1$. We show that the stacky Batyrev--Manin conjecture [DY24] holds for the naive height on $\mathcal{X}_0(N)$ when $F=\mathbb{Q}$. In the process, we give a concrete description of $\mathcal{X}_0(N)$ as a square root stack over a stacky curve. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_19771 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The stacky Batyrev-Manin conjecture and modular curves Darda, Ratko Han, Changho Number Theory Algebraic Geometry Primary 11G50, 14G05, Secondary 14A20, 11G18 Let $\mathcal{X}_0(N)$ be the Deligne--Rapoport modular stack of elliptic curves endowed with a cyclic rational $N$-isogeny over a number field $F$. Let $N\in\{1,2,3,4,5,6,7,8,9,10,12,13,16,18,25\},$ which are precisely the values for which the coarse moduli space of $\mathcal{X}_0(N)$ is isomorphic to $\mathbb{P}^1$. We show that the stacky Batyrev--Manin conjecture [DY24] holds for the naive height on $\mathcal{X}_0(N)$ when $F=\mathbb{Q}$. In the process, we give a concrete description of $\mathcal{X}_0(N)$ as a square root stack over a stacky curve. |
| title | The stacky Batyrev-Manin conjecture and modular curves |
| topic | Number Theory Algebraic Geometry Primary 11G50, 14G05, Secondary 14A20, 11G18 |
| url | https://arxiv.org/abs/2602.19771 |