The effect of the trace operator on the duality of modular grids in genus zero levels

TL;DR

This paper investigates how the trace operator influences the duality of canonical bases in genus zero modular forms, identifying conditions under which duality persists after applying the trace operator.

Contribution

It precisely characterizes the conditions under which the duality of canonical bases is preserved when the trace operator is applied to modular forms for genus zero levels.

Findings

Duality holds under specific conditions after the trace operator

The trace operator can preserve or break coefficient duality

Conditions depend on the level and properties of the modular forms

Abstract

Griffin, the second author, and Molnar studied coefficient duality for canonical bases for a broad range of spaces of weakly holomorphic modular forms, showing that the Fourier coefficients of canonical basis elements appear as negatives of Fourier coefficients for elements of a canonical basis of a related space of forms. We investigate the effect of the trace operator on this duality for modular forms for $\Gamma_0(N)$ of genus zero and show exactly when duality still holds after applying the trace operator.