$\ell$-adic \'etale cohomology of the moduli of stable elliptic fibrations

TL;DR

This paper computes the $\, ext{ell}$-adic étale cohomology and Frobenius eigenvalues for the moduli stack of stable elliptic fibrations over the projective line with specified singular fibers, over finite fields.

Contribution

It explicitly determines the $\, ext{ell}$-adic étale cohomology and Frobenius eigenvalues for a class of moduli stacks of elliptic fibrations, extending understanding of their arithmetic properties.

Findings

Computed the $\, ext{ell}$-adic étale cohomology groups.

Determined eigenvalues of the geometric Frobenius.

Analyzed the structure over finite fields with characteristic not 2 or 3.

Abstract

We determine the $\ell$-adic \'etale cohomology and the eigenvalues of the geometric Frobenius for the moduli stack $\mathcal{L}_{1,12n} := \mathrm{Hom}_{n}(\mathbb{P}^1, \overline{\mathcal{M}}_{1,1})$ of stable elliptic fibrations over $\mathbb{P}^{1}$ with $12n$ nodal singular fibers and a marked Weierstrass section over $\overline{\mathbb{F}}_q$ with $\mathrm{char}(\overline{\mathbb{F}}_q) \neq 2,3$.