Products of Nearly Holomorphic Eigenforms

TL;DR

This paper proves that the product of two nearly holomorphic Hecke eigenforms is a Hecke eigenform only in finitely many cases, highlighting a rare multiplicative property in this class of forms.

Contribution

It establishes a finiteness result for the products of nearly holomorphic Hecke eigenforms being eigenforms, a new insight in the structure of these forms.

Findings

Finitely many pairs produce eigenforms upon multiplication.

The product of two nearly holomorphic eigenforms is rarely an eigenform.

Provides a classification of such products.

Abstract

We prove that the product of two nearly holomorphic Hecke eigenforms is again a Hecke eigenform for only finitely many choices of factors.