Siegel modular forms of degree two and level five

TL;DR

This paper constructs a specific ring of meromorphic Siegel modular forms of degree 2 and level 5, identifying generators and their properties, and proves a minimal generating set for the holomorphic forms of this level.

Contribution

It introduces a new construction of a ring of meromorphic Siegel modular forms and determines a minimal generating set for holomorphic forms at level 5.

Findings

Ring generated by four singular theta lifts and their Jacobian.

Minimal generating set of 18 forms for holomorphic Siegel modular forms.

Explicit weights of generators for the modular forms.

Abstract

We construct a ring of meromorphic Siegel modular forms of degree 2 and level 5, with singularities supported on an arrangement of Humbert surfaces, which is generated by four singular theta lifts of weights 1, 1, 2, 2 and their Jacobian. We use this to prove that the ring of holomorphic Siegel modular forms of degree 2 and level $\Gamma_0(5)$ is minimally generated by eighteen modular forms of weights 2, 4, 4, 4, 4, 4, 6, 6, 6, 6, 10, 11, 11, 11, 13, 13, 13, 15.