The stacky Batyrev-Manin conjecture and modular curves

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.

34 pages. v3: improved exposition (especially in the introduction)