On zeroes of the twisted tensor \(L\)-function
type: Item
instance of: scholarly article
title: On zeroes of the twisted tensor \(L\)-function
zbMATH Open document ID: 0786.11030
DOI: 10.1007/BF01459497
author: Yuval Z. Flicker
published in: Mathematische Annalen
publication date: 1994-05-09T00:00:00Z
full work available at URL: https://eudml.org/doc/165127
review text: This paper concerns the twisted tensor \(L\)-function \(L(s,r(\pi)\otimes \omega)\) -- which has been introduced and studied in the reviewer's (*) ``Twisted tensors and Euler products'', Bull. Soc. Math. Fr. 116, 295-313 (1988; Zbl 0674.10026) -- associated with a cuspidal representation \(\pi\) of \(GL(n,\mathbb{A}_ E)\) and a character \(\omega\) of \(\mathbb{A}_ F^ \times/ F^ \times\). Here \(E/F\) is a quadratic extension of global fields. When \(\pi\) is distinguished, and has a supercuspidal component, the following is shown. Theorem. Let \(s_ 0\) be a complex number such that for every separable field extension \(L(T)\) of \(F\) of degree \(n\), the \(L\)-function \(L(s,\omega_ T)\) (associated to the character \(\omega_ T\) of the idele class group of \(L(T)\) obtained on restricting \(\omega(\text{det})\) to the multiplicative group of \(L(T)\), viewed as a torus in \(GL(n,F)\)) vanishes to the order \(m\) at \(s=s_ 0\). Then \(L(s,r(\pi)\otimes \omega)\) vanishes at \(s=s_ 0\) to the order \(m\). A cuspidal \(\pi\) is called distinguished if its space contains a form whose integral (``period'') over the ``cycle'' \(PGL(n,\mathbb{A}_ F)/PGL(n,F)\) is non-zero. They emerge in (*) as the only representations for which the twisted tensor \(L\)-function may have poles (at \(s=1,0\)). A precise conjecture parametrizing the distinguished \(\pi\) is made (and proven for \(GL(2)\)) in the reviewer's ``On distinguished representations'', J. Reine Angew. Math. 418, 139-172 (1991; Zbl 0725.11026). The proof in the reviewed paper is made relatively simple on using ideas from the proof of Deligne and Kazhdan of their simple trace formula. Similar ideas were used in the split case \(E=F\oplus F\) and the adjoint representation \(L\)-function \(L(s,\omega\otimes \pi\times \check\pi)/ L(s,\omega)\), in the reviewer's ``The adjoint representation \(L\)-function for \(GL(n)\)'', Pac. J. Math. 154, 231-244 (1992; Zbl 0766.11028). This in turn was motivated by the work of \tex…
reviewed by: Yuval Z. Flicker
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11S40
Mathematics Subject Classification ID: 11R39
Mathematics Subject Classification ID: 11F66
Mathematics Subject Classification ID: 22E55
zbMATH DE Number: 537344
zbMATH Keywords: zeros
zbMATH Keywords: twisted tensor \(L\)-function
zbMATH Keywords: cuspidal representation
zbMATH Keywords: distinguished representations
zbMATH Keywords: trace formula
zbMATH Keywords: local factor
zbMATH Keywords: global \(L\)- function
zbMATH Keywords: analytic continuation
zbMATH Keywords: functional equation
zbMATH Keywords: poles
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 537344