$\ell$-adic \'etale cohomology & Galois representations of the moduli stack of stable elliptic surfaces.

We determine the compactly supported $\ell$-adic \'etale cohomology and the eigenvalues of the geometric Frobenius map acting on this cohomology as isomorphisms of Galois representations for the moduli stack $\mathcal{L}_{1,12n} := \mathrm{Hom}_{n}(\mathbb{P}^1, \overline{\mathcal{M}}_{1,1})$ of stable elliptic fibrations over $\mathbb{P}^{1}$, also known as stable elliptic surfaces, with $12n$ nodal singular fibers and a marked Weierstrass section over $\overline{\mathbb{F}}_q$ with $\mathrm{char}(\overline{\mathbb{F}}_q) \neq 2,3$. In the end, we consider the Hasse-Weil zeta function for $\mathcal{L}_{1,12n}$ and show that it is equal to the rational function $\mathrm{Z}(\mathcal{L}_{1,12n},t)=\frac{\left( 1 - q^{10n-1} \cdot t \right)}{\left( 1 - q^{10n+1} \cdot t \right)}$