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On poles of twisted tensor \(L\)-functions. (English) Zbl 0983.11023

Let \(E\) be a quadratic separable field extension of a global field \(F\). Denote by \(\mathbb{A}_E\), \(\mathbb{A}_F\) the corresponding rings of adeles. Put \(\text{G}_n\) for \(\text{GL}_n\) and \(Z_n\) for its center. Then \(Z_n(\mathbb{A}_E)\) is the group \(\mathbb{A}_E^\times\) of ideles of \(\mathbb{A}_E\). Fix a cuspidal representation \(\pi\) of the adele group \(\text{G}_n(\mathbb{A}_E)\). Without lost of generality, we may assume that the central character of \(\pi\) is trivial on the split component of \(\mathbb{A}_E^\times\). This is the multiplicative group \(\mathbb{R}^\times\) of the field of real numbers embedded in \(\mathbb{A}_E^\times\) via \(x\mapsto (x,\dots, x,1,\dots)\) (\(x\) in the archimedean, 1 in the finite components). Let \(S\) be a finite set of places of \(F\) (depending on \(\pi\)), including the places where \(E/F\) ramify, and the archimedean places, such that for each place \(v'\) of \(E\) above a place \(v\) outside \(S\), the component \(\pi_{v'}\) of \(\pi\) is unramified. Following [Y. Flicker, Bull Soc. Math. Fr. 116, 295-313 (1988; Zbl 0674.10026)], let \(r\) be the twisted tensor representation of \(\widehat{\text{G}}= [\text{GL}(n,\mathbb{C})\times \text{GL}(n,\mathbb{C})]\times \text{Gal}(E/F)\) on \(\mathbb{C}^n\otimes \mathbb{C}^n\). It acts by \(r((a,b))(x\otimes y)= ax\otimes by\) and \(r(\sigma)(x\otimes y)= y\otimes x\) \((\sigma\in \text{Gal}(E/F)\), \(\sigma\neq 1)\). Let \(q_v\) be the cardinality of the residue field \(R_v/\pi_vR_v\) of the ring \(R_v\) of integers in \(F_v\). We define the twisted tensor \(L\)-function to be the Euler product \[ L(s,r(\pi),S)= \prod_{v\not\in S}\text{det} [1-q_v^{-s} r(t_v)]^{-1}. \] The authors show that the only possible pole of the twisted tensor \(L\)-functions in \(\text{Re}(s)\geq 1\) is located as \(s=1\) for all quadratic extensions of global fields.

MSC:

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations

Citations:

Zbl 0674.10026
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References:

[1] Y. Flicker: Twisted tensors and Euler products. Bull. Soc. Math. France, 116, 295-313 (1988). · Zbl 0674.10026
[2] Y. Flicker: On the local twisted tensor L-function. Appendix to : On zeroes of the twisted tensor L-function. Math. Ann., 297, 199-219 (1993). · Zbl 0786.11030 · doi:10.1007/BF01459497
[3] H. Jacquet and J. Shalika: Exterior square L-functions. Automorphic Forms, Shimura Varieties, and L-functions (ed. L. Clozel and S. Milne), pp. 143-226 (1990). · Zbl 0695.10025
[4] H. Jacquet and J. Shalika : On Euler products and the classification of automorphic representations. I. II. Amer. J. Math., 103, 449-558; 777-815 (1981). JSTOR: · Zbl 0473.12008 · doi:10.2307/2374103
[5] T. Kon-no: The residual spectrum of SU{2,2) (preprint).}
[6] F. Shahidi: On certain L-functions. Amer. J. Math., 103, 297-355 (1981). JSTOR: · Zbl 0467.12013 · doi:10.2307/2374219
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