Archimedean non-vanishing, cohomological test vectors, and standard L-functions of GL 2n: real case

Cheng Chen, Dihua Jiang, Bingchen Lin, Fangyang Tian

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The standard L-functions of 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 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.

Original languageEnglish (US)
Pages (from-to)479-509
Number of pages31
JournalMathematische Zeitschrift
Issue number1-2
StatePublished - Oct 1 2020

Bibliographical note

Publisher Copyright:
© 2019, Springer-Verlag GmbH Germany, part of Springer Nature.


  • Archimedean non-vanishing
  • Cohomological test vector
  • Friedberg–Jacquet integral
  • Linear model
  • Shalika model
  • Standard L-functions for general linear groups


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