Archimedean non-vanishing, cohomological test vectors, and standard L-functions of GL2n: real case

• Published in: Mathematische Zeitschrift Vol. 296; no. 1-2; pp. 479 - 509
• Main Authors: Chen, Cheng, Jiang, Dihua, Lin, Bingchen, Tian, Fangyang
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 2020; Springer Nature B.V
• Subjects: Arithmetic; Existence theorems; Infinity; Integrals; Mathematics; Mathematics and Statistics; Modular construction; Symbols; Primary 22E45; Friedberg–Jacquet integral; Archimedean non-vanishing; Cohomological test vector; Standard; Shalika model; Secondary 11F67; Linear model; functions for general linear groups
• ISSN: 0025-5874; 1432-1823
• DOI: 10.1007/s00209-019-02453-z

The standard L -functions of GL 2 n 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 existence or construction of uniform cohomological test vectors . Problem (1) is also called the non-vanishing hypothesis at infinity , which was proved by Sun [Duke Math J 168(1):85–126, (2019), Theorem 5.1], 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 Λ s , χ , which, when evaluated with our explicitly constructed cohomological vector, is equal to the local twisted standard L -function L ( s , π ⊗ χ ) for all complex values s . With the relations between linear models and Shalika models, we establish (1) with an explicitly constructed cohomological vector using classical invariant theory, and hence proves the non-vanishing results of Sun [ 24 , Theorem 5.1] via a completely different method.