Pierre Bieliavsky

Title: Modular forms, Rankin-Cohen brackets and (strict) deformation quantization [pdf]
Abstract: In the late 90', Ehzoler and later Zagier defined a formal deformation of the algebra of (even weighted) modular forms as a formal superposition of Rankin-Cohen brackets. Since then, these deformations have been studied and used in various directions. In particular, Connes and Moscovici gave a beautiful application in their study of the transverse geometry of codimension one foliations. However, these considerations remain at the formal level. In the present work, we will present a re-summation of the Rankin-Cohen formal product series under the form of an oscillatory integral. We will then discuss the possibilities that yields this non-formal formula in the framework of operator algebras. In particular, the non-formal product closes on the space of square integrable functions on the group ax+b as a Hilbert algebra. This result strongly supports the possibility of a new example of a locally compact quantum group based on arithmetic considerations.