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On zeroes of the twisted tensor $$L$$-function. (English) Zbl 0786.11030
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 H. Jacquet and D. Zagier, “Eisenstein series and the Selberg trace formula, II”, Trans. Am. Math. Soc. 300, 1-48 (1987; Zbl 0625.10024), which dealt with the split case and $$n=2$$ for $$\pi$$ which is not assumed to have a supercuspidal component.
The Appendix – “On the local twisted tensor $$L$$-function” – to this paper, defines – and studies the main properties of – the local factor in the possibly ramified places, of the global $$L$$-function under consideration. This extends the work of (*) to show that the complete (rather than partial) $$L$$-function $$L(s,r(\pi)\otimes\omega)$$ has analytic continuation, functional equation, and poles as specified in (*).

##### MSC:
 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11S40 Zeta functions and $$L$$-functions 11R39 Langlands-Weil conjectures, nonabelian class field theory 11F66 Langlands $$L$$-functions; one variable Dirichlet series and functional equations 22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
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