Rankin–Cohen brackets and identities among eigenforms

Arvind Kumar and Jaban Meher

Vol. 297 (2018), No. 2, 381–403

DOI: 10.2140/pjm.2018.297.381

Abstract

We investigate the cases for which the Rankin–Cohen brackets of two quasimodular eigenforms give rise to eigenforms. More precisely, we characterise all the cases in a subspace of the space of quasimodular forms for which Rankin–Cohen brackets of two quasimodular eigenforms are again eigenforms. In the process, we obtain some new polynomial identities among quasimodular eigenforms. To prove the results on quasimodular forms, we prove several results in the theory of nearly holomorphic modular forms. These new results in the theory of nearly holomorphic modular forms are of independent interest.

Keywords

eigenforms, nearly holomorphic modular forms, quasimodular forms, Rankin–Cohen brackets.

Mathematical Subject Classification 2010

Primary: 11F25, 11F37

Secondary: 11F30, 11F67

Milestones

Received: 17 September 2017

Revised: 25 February 2018

Accepted: 28 April 2018

Published: 20 December 2018