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Abstract

In Part I [Automorphic forms, representation theory and arithmetic, Pap. Colloq., Bombay 1979, 303-395 (1981; Zbl 0484.10020)] the second author gave his interesting investigations on the integral I(s), against an Eisenstein series, of the kernel of the Selberg trace formula. This paper presents them in the setting of F-adeles for a global field F; included essentially is an adelic generalization of the second author's well-known paper on modular forms with Fourier coefficients involving zeta-functions of quadratic fields. \par A principal application is the divisibility of I(s) by the zeta-function for F and the consequent holomorphy of the symmetric square of the L- series attached to a cusp form (a fact due to Shimura in the classical case and proved in an adelic set-up by the first author). Another application in view is to deduce the usual trace formula itself (by determining the residue of I(s) at s=1); complicated explicit calculations of the terms constituting I(s) are necessary. \par The attraction of this paper, however, consists in obtaining a generalization of the trace formula with the various terms expressed as products of local integrals; this renders the method amenable to generalization for higher dimensions.