Quadratic forms of holomorphic cusp forms and the decay of their $\ell^p$-norms for $0 < p < 2$

TL;DR

This paper investigates the structure and decay properties of quadratic forms of holomorphic cusp forms, showing they are generally not expressible as linear combinations of basis forms for large weights, and their lp-norms tend to zero for certain p-values.

Contribution

It provides new insights into the expressibility of quadratic forms of cusp forms and their lp-norm decay, highlighting finiteness and asymptotic behaviors.

Findings

Quadratic forms of cusp forms are not generally expressible as linear combinations for large weights.

The lp-norms of such quadratic forms tend to zero when 0 < p < 2.

Finiteness of solutions to certain modular form equations is demonstrated.

Abstract

In this paper, we demonstrate that, given an orthonormal basis of holomorphic Hecke cusp forms, conditionally, quadratic forms composed of cusp forms -- each expressed as a bounded linear combination of holomorphic Hecke cusp forms -- are generally not themselves expressible as bounded linear combinations of holomorphic Hecke cusp forms when the sum of the weights exceeds some absolute constant, provided that the coefficients of the quadratic form satisfy appropriate nonvanishing and boundedness conditions. This illustrates the finiteness of the number of solutions to the linear equation of modular forms equated to a quadratic form of large weight. We also show that, conditionally, for $0 < p < 2$, the $\ell^p$-norm of such quadratic forms in holomorphic Hecke cusp forms tends to zero asymptotically with respect to expansion in this orthonormal basis of Hecke eigenforms.