Archimedean Non-vanishing, Cohomological Test Vectors, and Standard $L$-functions of $\mathrm{GL}_{2n}$: Real Case

Abstract

The standard $L$-functions of $\mathrm{GL}_{2n}$ expressed in terms of the Friedberg-Jacquet global zeta integrals have better structure for arithmetic applications, due to the relation of the linear periods with the modular symbols. The most technical obstacles towards such arithmetic applications are (1) non-vanishing of modular symbols at infinity and (2) the existance or construction of uniform cohomological test vectors. Problem (1) is also called the non-vanishing hypothesis at infinity, which was proved by Binyong Sun, by establishing the existence of certain cohomological test vectors. In this paper, we explicitly construct an archimedean local integral that produces a new type of a twisted linear functional $\Lambda_{s,\chi}$, which, when evaluated with our explicitly constructed cohomological vector, is equal to the local twisted standard $L$-function $L(s,\pi\otimes\chi)$ as a meromorphic function of $s\in \mathbb{C}$. With the relations between linear models and Shalika models, we establish (1) with an explicitly constructed cohomological vector, and hence recovers a non-vanishing result of Binyong Sun via a completely different method. Our main result indicates a complete solution to (2), which will be presented in a paper of Dihua Jiang, Binyong Sun and Fangyang Tian with full details and with applications to the global period relations for the twisted standard $L$-functions at critical places.