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Rankin triple \(L\)-functions. (English) Zbl 0637.10023
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.

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
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