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 $n\ge 2$, the open substack ${\mathcal{W}}_{\min ,n}$ of minimal Weierstrass fibrations is a separated Deligne-Mumford stack over any base field $K$ with $\mathrm{char}(K)\ne 2,3$ 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.

38 pages; comments very welcome!

References in corpus (4)

• The Grothendieck group of algebraic stacks
• Infinitesimal automorphisms of algebraic varieties and vector fields on elliptic surfaces
• Enumerating odd-degree hyperelliptic curves and abelian surfaces over ${\mathbb{P}}^1$
• Moduli of elliptic $K3$ surfaces: monodromy and Shimada root lattice strata

Cited by in corpus (1)

• The Chow rings of moduli spaces of elliptic surfaces over ${\mathbb{P}}^1$