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.


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
Full Text: Numdam EuDML


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