Arithmetic of the moduli of semistable elliptic surfaces

Abstract

We prove a new sharp asymptotic with the lower order term of zeroth order on ${\mathcal{Z}}_{{\mathbb{F}}_q(t)}(\mathcal{B})$ for counting the semistable elliptic curves over ${\mathbb{F}}_q(t)$ by the bounded height of discriminant $\Delta (X)$. The precise count is acquired by considering the moduli of nonsingular semistable elliptic fibrations over ${\mathbb{P}}^1$, also known as semistable elliptic surfaces, with $12n$ nodal singular fibers and a distinguished section. We establish a bijection of $K$-points between the moduli functor of semistable elliptic surfaces and the stack of morphisms ${\mathcal{L}}_{1,12n}\cong {\mathrm{Hom}}_n({\mathbb{P}}^1,{\overline{\mathcal{M}}}_{1,1})$ where ${\overline{\mathcal{M}}}_{1,1}$ is the Deligne-Mumford stack of stable elliptic curves and $K$ is any field of characteristic $\ne 2,3$. For $\mathrm{char}(K)=0$, we show that the class of ${\mathrm{Hom}}_n({\mathbb{P}}^1,\mathcal{P}(a,b))$ in the Grothendieck ring of $K$-stacks, where $\mathcal{P}(a,b)$ is a 1-dimensional $(a,b)$ weighted projective stack, is equal to ${\mathbb{L}}^{(a+b)n+1}-{\mathbb{L}}^{(a+b)n-1}$. Consequently, we find that the motive of the moduli ${\mathcal{L}}_{1,12n}$ is ${\mathbb{L}}^{10n+1}-{\mathbb{L}}^{10n-1}$ and the cardinality of the set of weighted ${\mathbb{F}}_q$-points to be ${\#}_q({\mathcal{L}}_{1,12n})=q^{10n+1}-q^{10n-1}$. In the end, we formulate an analogous heuristic on ${\mathcal{Z}}_{\mathbb{Q}}(\mathcal{B})$ for counting the semistable elliptic curves over $\mathbb{Q}$ by the bounded height of discriminant $\Delta$ through the global fields analogy.

17 pages, To Appear at Mathematische Annalen (2019)