Modular shadows and the Levy-Mellin ∞-adic transform

Abstract

This paper continues the study of the structures induced on the "invisible boundary" of the modular tower and extends some results of [MaMar1]. We start with a systematic formalism of pseudo–measures generalizing the well–known theory of modular symbols for SL(2). These pseudo–measures, and the related integral formula which we call the Lévy–Mellin transform, can be considered as an "∞–adic" version of Mazur's p–adic measures that have been introduced in the seventies in the theory of p–adic interpolation of the Mellin transforms of cusp forms, cf. [Ma2]. A formalism of iterated Lévy–Mellin transform in the style of [Ma3] is sketched. Finally, we discuss the invisible boundary from the perspective of non–commutative geometry.

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