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arXiv · 2602.19771

The stacky Batyrev-Manin conjecture and modular curves

Ratko Darda·Changho Han
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Abstract

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.

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Ratko Darda, Changho Han. 2026-02-23. The stacky Batyrev-Manin conjecture and modular curves. https://arxiv.org/abs/2602.19771

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