On the Petersson scalar product of arbitrary modular forms

TL;DR

This paper extends the Petersson scalar product to all modular forms of integral weight for certain groups, analyzing its properties and the behavior of Hecke operators, including cases of degeneracy and nondegeneracy.

Contribution

It introduces a natural extension of the Petersson scalar product to all modular forms and investigates its properties, including Hecke operator adjoints and degeneracy conditions.

Findings

Hecke operators have the same adjoints as for cusp forms under the extended product.

The Petersson product is nondegenerate for (N) and weight > 2.

Examples show degeneracy can occur at weight 2.

Abstract

We consider a natural extension of the Petersson scalar product to the entire space of modular forms of integral weight $k\ge 2$ for a finite index subgroup of the modular group. We show that Hecke operators have the same adjoints with respect to this inner product as for cusp forms, and we show that the Petersson product is nondegenerate for $\Gamma_1(N)$ and $k>2$. For $k=2$ we give examples when it is degenerate, and when it is nondegenerate.