Stacky Batyrev-Manin conjecture and modular curves

Abstract: Let $\mathscr{X}_0(N)$ 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 $\mathscr{X}_0(N)$ is isomorphic to $\mathbb{P}<sup>1$. We show that the stacky Batyrev--Manin conjecture [DY24] holds for the naive height on $\mathscr{X}_0(N)$ when $F=\mathbb{Q}$. In the process, we give a concrete description of $\mathscr{X}_0(N)$ as a square root stack over a stacky curve.