Beyond Endoscopy for $\mathrm{GL}_3(\mathbb{Q})$: Functional equation for the $L$-function of a cubic order

Abstract

The Beyond Endoscopy strategy, proposed by Langlands, aims to establish the principle of functoriality by analyzing the trace formula. Recently, Deng and Espinosa advanced this program for $\mathrm{GL}_3(\mathbb{Q})$ by isolating the contribution of the trivial representation from the elliptic regular part. Their work relies on a conjectural factorization formula for the $L$-function associated with a cubic order, which yields the functional equation for the completed $L$-function. In this paper, we provide an unconditional proof of this functional equation for every Gorenstein order in a cubic number field. As a consequence, their isolation of the trivial representation for $\mathrm{GL}_3(\mathbb{Q})$ becomes fully unconditional.