Yun's zeta function and the overorder zeta function for Gorenstein cubic orders

Abstract

For every Gorenstein cubic $\mathbb Z$-order, we prove an identity equating Yun's zeta function, defined by counting finite-index submodules of the trace dual, with the explicit overorder zeta function introduced in our previous work on Beyond Endoscopy for $\mathrm{GL}_3$. This is posed as Conjecture A in an early draft of Deng-Espinosa and Lee subsequently proved the functional equation of the overorder zeta function, making the Deng--Espinosa isolation of the trivial representation fully unconditional. His argument computes the local factors explicitly. Our proof is independent of Lee's and does not evaluate the individual cubic overorder factors: it matches natural decompositions of the two sides and concludes by induction. As an application, we give a short, uniform evaluation of the local $\mathrm{GL}_3 $ Kloosterman Dirichlet series in the Poisson-summation argument of Deng--Espinosa. The direct local analysis of the Kloosterman series occupies nearly ninety pages in Deng--Espinosa while our treatment here replaces its case-by-case enumeration with a short uniform proof.