ARCHIMEDEAN NON-VANISHING AND COHOMOLOGICAL TEST VECTOR (Analytic, geometric and $p$-adic aspects of automorphic forms and $L$-functions)

Abstract

The standard L-functions of GL2n 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. In this paper, we just give an overview for our recent work on the archimedean local integrals of Friedberg-Jacquet([CJLT19], [LT20]). We will focus on the complex case, explicitly construct a uniform cohomological test vector v for a new twisted linear functional ∧s, χ and establish the non-vanishing property for the archimedean local Friedberg-Jacquet integral when evaluating at v.