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Seminar

Date/Time: 
Friday, July 14, 2017 - 11:30 to 12:30
Venue: 
SMS Seminal Hall
Speaker: 
Arvind Kumar
Affiliation: 
HRI, Allahabad
Title: 
Rankin-Cohen brackets and identities among eigenforms
Abstract :- Identities connecting modular forms (in particular, eigenforms) have attracted the attention of many mathematicians since they imply nice identities among their Fourier coefficients. It is quite natural to ask whether the property of being a Hecke eigenform is preserved under multiplication or more generally, under the Rankin-Cohen bilinear operators. In this talk, we will give a brief survey of the existing results in this direction. We then consider a subclass of the space of quasimodular forms and nearly holomorphic modular forms for our purpose. In the main result, we classify all the cases when the Rankin-Cohen bracket of two eigenforms results in an eigenform. In the process, we obtain some new polynomial identities among quasimodular eigenforms. We extensively use the Rankin's method and the non-vanishing properties of modular $L$-functions in appropriate half-planes. We also establish some interesting properties of the space of nearly holomorphic modular forms in the course of the proof.

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