Pseudodifferential Analysis and Modular Forms

Abstract

This chapter is meant as another motivation (cf. introduction) for the construction of the ascending pseudodifferential analysis, rather than as an introduction to a new point of view in modular form theory: we hope to come back to possible developments in this direction at some later occasion.We here wish to show that the parallel treatments of anaplectic analysis and associated alternative pseudodifferential analysis on one hand, of usual analysis and pseudodifferential analysis on the other hand, extend, up to some point, as parallel sources of holomorphic modular forms on one hand, nonholomorphic modular forms on the other hand. In particular, let us direct the interested reader to Remark 5.1.1 at the end of this section, which puts on an absolutely equal footing the Rankin–Cohen brackets of holomorphic modular forms, well known to modular form theorists, and their analogues in nonholomorphic modular form theory, which do not seem to have kept their attention. As explained in the introduction, making this possible was one of the initial aims of the present work.