Rankin-Cohen brackets on quasimodular forms

François Martin (1) , Emmanuel Royer (1)

1 LMBP - Laboratoire de Mathématiques Blaise Pascal

We give the algebra of quasimodular forms a collection of Rankin-Cohen operators. These operators extend those defined by Cohen on modular forms and, as for modular forms, the first of them provide a Lie structure on quasimodular forms. They also satisfy a ``Leibniz rule'' for the usual derivation. Rankin-Cohen operators are useful for proving arithmetic identities. In particular we give an interpretation of the Chazy equation and explain why such an equation has to exist.

Mots clés: Rankin-Cohen operators; quasimodular forms; Leibniz rule; Chazy; Ramanujan; differential equation

Domaines: Théorie des nombres [math.NT]

François Martin, Emmanuel Royer. Rankin-Cohen brackets on quasimodular forms. Journal of the Ramanujan Mathematical Society, 2009, 24 (3), pp.213-233. ⟨hal-00009145v2⟩