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

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.