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Modular shadows and the Levy-Mellin infinity-adic transform

Yuri Manin·Matilde Marcolli
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Abstract

This paper continues the study of the structures induced on the ``invisible boundary'' of the modular tower and extends some results of math.NT/0102006. 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 Levy-Mellin transform, can be considered as an ``infinity-adic'' version of Mazur's p-adic measures introduced in the seventies in the theory of p-adic interpolation of Mellin transforms of cusp forms. A formalism of iterated Levy-Mellin transform in the style of math.NT/0502576 is sketched. Finally, we discuss the invisible boundary from the perspective of non-commutative geometry.

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Yuri Manin, Matilde Marcolli. 2007-07-13. Modular shadows and the Levy-Mellin infinity-adic transform. https://arxiv.org/abs/math/0703718

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