Abstract. We consider various generalizations of linear homogeneous distributions on adeles and construct a number of algebras of non-linear generalized functions on adeles and totally disconnected groups such as the discrete adeles.1. Introduction. The algebra of adeles over the field of rational numbers Q was introduced by A. Weil. Invertible adeles, now known as ideles, were introduced by C. Chevalley [4]. Adeles became a powerful analytic tool in number theory [20]; they have applications in representation theory [9] and in modern mathematical physics [6].There is no interesting topology on the field of rational numbers. However it is possible to embed this field into the canonically associated algebra of adeles having a non-trivial topology and a Haar measure. Thus numerous tools of functional analysis can be applied. Here we summarize some developments in linear and non-linear theory of generalized functions on adeles. See [5] for basics of the non-linear theory on the real axis. An abstract approach to the non-linear theory is presented in [2,15].
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