Arithmetic geometry of the moduli stack of Weierstrass fibrations over $\mathbb{P}^1$

TL;DR

This paper constructs and analyzes the moduli stack of Weierstrass fibrations over the projective line, extending classical GIT methods to stacks, and computes its motive and point counts over finite fields.

Contribution

It provides an explicit stack-theoretic construction of the moduli of Weierstrass fibrations, proving smoothness, separation, and calculating motives and point counts.

Findings

The moduli stack $ ext{W}_n$ is smooth and algebraic.

The substack of minimal Weierstrass fibrations is a separated Deligne-Mumford stack.

The motive of the stable Weierstrass fibrations is $ ext{L}^{10n - 2}$, with point count $q^{10n - 2}$ over finite fields.

Abstract

Coarse moduli spaces of Weierstrass fibrations over the (unparameterized) projective line were constructed by the classical work of [Miranda] using Geometric Invariant Theory. In our paper, we extend this treatment by using results of [Romagny] regarding group actions on stacks to give an explicit construction of the moduli stack $\mathcal{W}_n$ of Weierstrass fibrations over an unparameterized $\mathbb{P}^{1}$ with discriminant degree $12n$ and a section. We show that it is a smooth algebraic stack and prove that for $n \geq 2$, the open substack $\mathcal{W}_{\mathrm{min},n}$ of minimal Weierstrass fibrations is a separated Deligne-Mumford stack over any base field $K$ with $\mathrm{char}(K) \neq 2,3$ and not dividing $n$. Arithmetically, for the moduli stack $\mathcal{W}_{\mathrm{sf},n}$ of stable Weierstrass fibrations, we determine its motive in the Grothendieck ring of stacks to be $\{\mathcal{W}_{\mathrm{sf},n}\} = \mathbb{L}^{10n - 2}$ in the case that $n$ is odd, which results in its weighted point count to be $\#_q(\mathcal{W}_{\mathrm{sf},n}) = q^{10n - 2}$ over $\mathbb{F}_q$. In the appendix, we show how our methods can be applied similarly to the classical work of [Silverman] on coarse moduli spaces of self-maps of the projective line, allowing us to construct the natural moduli stack and to compute its motive.