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Archimedean non-vanishing, cohomological test vectors, and standard L-functions of \({\mathrm {GL}}_{2n}\): real case

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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 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 \(\Lambda _{s,\chi }\), which, when evaluated with our explicitly constructed cohomological vector, is equal to the local twisted standard L-function \(L(s,\pi \otimes \chi )\) 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.

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Authors and Affiliations

  1. School of Mathematics, University of Minnesota, Minneapolis, USA

    Cheng Chen & Dihua Jiang

  2. School of Mathematics, Sichuan University, Chengdu, China

    Bingchen Lin

  3. School of Mathematics, National University of Singapore, Singapore, Singapore

    Fangyang Tian

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  1. Cheng Chen
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  2. Dihua Jiang
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  3. Bingchen Lin
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  4. Fangyang Tian
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Correspondence to Fangyang Tian.

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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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Chen, C., Jiang, D., Lin, B. et al. Archimedean non-vanishing, cohomological test vectors, and standard L-functions of \({\mathrm {GL}}_{2n}\): real case. Math. Z. 296, 479–509 (2020). https://doi.org/10.1007/s00209-019-02453-z

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