Archimedean non-vanishing, cohomological test vectors, and standard L -functions of $${\mathrm {GL}}_{2n}$$ GL 2 n :
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Mathematische Zeitschrift
Archimedean non-vanishing, cohomological test vectors, and standard L-functions of GL2n : real case Cheng Chen1 · Dihua Jiang1 · Bingchen Lin2 · Fangyang Tian3 Received: 27 March 2019 / Accepted: 21 October 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
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. 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 nonvanishing 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 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. Keywords Linear model · Shalika model · Friedberg–Jacquet integral · Archimedean non-vanishing · Cohomological test vector · Standard L-functions for general linear groups Mathematics Subject Classification Primary 22E45; Secondary 11F67
The research of Jiang is supported in part by the NSF Grants DMS–1600685 and DMS–1901802; that of Lin is supported in part by the China Scholarship Council No.201706245006; and that of Tian is is supported in part by AcRF Tier 1 grant R-146-000-277-114 of National University of Singapore.
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Fangyang Tian [email protected] Cheng Chen [email protected] Dihua Jiang [email protected] Bingchen Lin [email protected]
1
School of Mathematics, University of Minnesota, Minneapolis, USA
2
School of Mathematics, Sichuan University, Chengdu, China
3
School of Mathematics, National University of Singapore, Singapore, Singapore
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C. Chen et al.
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Cohomological representations of GL2n (R) . . . . . . 1.2 Shalika model and linear model . . . . . . . . . . . . . 1.3 Cohomological test vectors and non-vanishing property 2 Cohomological representations and Shalika models . . . . . 3 Cohomological representations and linear models . . . . . . 3.1 The GL2 (R) case . . . . . . . . . . . . . . . . . . . . 3.2 A new construction of linear model . . . . . . . . . . . 4 Cohomological vectors in the induced representation . . . . 4.1 Some reductions . . . . . . . . . . . . . . . . . . . . . 4.2 On certain minimal K -type functions . . . . . . . . . . 4.
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