The stacky Batyrev-Manin conjecture and modular curves

TL;DR

This paper verifies the stacky Batyrev--Manin conjecture for certain modular stacks over the rationals, providing explicit descriptions of these stacks as square root stacks over stacky curves.

Contribution

It demonstrates the conjecture for specific modular stacks and offers a concrete description of these stacks as square root stacks over stacky curves.

Findings

The conjecture holds for the naive height on these stacks over .

Provides explicit descriptions of _0(N) as square root stacks.

Identifies the cases where the coarse moduli space is ^1.

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.