Arithmetic of the moduli of semistable elliptic surfaces

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 $Δ(X)$. 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 $\neq 2,3$. 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 $Δ$ through the global fields analogy.

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