Piatetski-Shapiro, Ilya I.; Rallis, Stephen Rankin triple \(L\)-functions. (English) Zbl 0637.10023 Compos. Math. 64, 31-115 (1987). Let \(k\) be an algebraic number field and \(K\) be either a cubic extension of \(k\) or the product of \(k\) and a quadratic extension of \(k\). Let \(G=\text{PGL}(2,K)\). The \(L\)-group of \(G\) has an obvious 8-dimensional representation \(r\). Corresponding to an irreducible cuspidal representation \(\pi\) of \(G(\mathbb A)\) there is Langland’s \(L\)-function \(L(\pi,r)\). In this paper the authors give an integral representation of \(L(\pi,r)\) involving an Eisenstein series on \(\text{PSp}(6)\) (observe that \(G\) has a finite covering which can be imbedded into \(\text{PSp}(6)\) ). From a detailed study of this integral and the properties of the Eisenstein series they deduce the analytic continuation of \(L(\pi,r)\) with exact location of the poles. Reviewer: J. G. M. Mars (Utrecht) Cited in 6 ReviewsCited in 79 Documents 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 11R39 Langlands-Weil conjectures, nonabelian class field theory 22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings Keywords:orbit structure; intertwining operators; Rankin triple L functions. Langland’s L-function; integral representation; Eisenstein series; analytic continuation; poles PDFBibTeX XMLCite \textit{I. I. Piatetski-Shapiro} and \textit{S. Rallis}, Compos. Math. 64, 31--115 (1987; Zbl 0637.10023) Full Text: Numdam EuDML References: [1] W. Casselman : Introduction to the theory of admissible representations of p-adic groups , preprint. · Zbl 0472.22004 [2] P. Garrett : Decomposition of Eisenstein Series: Rankin Triple Products , preprint. · Zbl 0625.10020 · doi:10.2307/1971310 [3] R. Gustafson : The Degenerate Principal Series for Sp(2n) , Ph.D. Thesis Yale (1979). · Zbl 0482.22013 [4] H. Jacquet : On the residual spectrum of GL(n) , Springer Lecture Notes 1041 (1984) pp. 185-280. · Zbl 0539.22016 [5] H. Jacquet , I. Piatetski-Shapiro and J. Shalika : Rankin-Selberg Convolutions , Amer. Jour. of Math. 105 (1979) pp. 367-464. · Zbl 0525.22018 · doi:10.2307/2374264 [6] M. Kashiwara and M. Vergne : On the Segal-Shale-Weil representations and harmonic polynomials , Inv. Math. 44 (1978) 1-47. · Zbl 0375.22009 · doi:10.1007/BF01389900 [7] R. Langlands : Euler Products, Yale Mathematicae Monographs 1 , Yale University Press (1971). · Zbl 0231.20016 [8] I. Piatetski-Shapiro : Euler subgroups . In: Lie Groups and Their Representations , Budapest (1971) pp. 597-620. · Zbl 0329.20028 [9] I. Piateski-Shapiro and S. Rallis : L-functions of automorphic forms on simple classical groups . In: R. Rankin (ed.) Modular Forms , Ellis Horwood (1983) pp. 251-263. · Zbl 0566.10022 [10] I. Piatetski-Shapiro and S. Rallis : L-functions in Classical Groups (notes at Institute for Advanced Study in 1983-1984) , pp. 1-82. [11] I. Piatetski-Shapiro and S. Ralls: \epsilon -factors of representations of classical groups , Proc. Nat. Acad. Sciences 83 (1986) pp. 4589-4953. · Zbl 0599.12012 · doi:10.1073/pnas.83.13.4589 [12] F. Shahidi : On certain L-functions , Amer. Jour. of Math. 103 (1981) 297-356. · Zbl 0467.12013 · doi:10.2307/2374219 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.