Explicit computations of Siegel modular forms of degree two

TL;DR

This paper introduces a unified computational approach for Siegel modular forms of degree two, implementing algorithms in Sage to compute various rings and types of these forms, which previously lacked general methods.

Contribution

It presents a novel unified framework and algorithms for computing Siegel modular forms of degree two, implemented in Sage, covering a wide range of forms and introducing the concept of formal Siegel modular forms.

Findings

Implemented algorithms for computing rings of Siegel modular forms

Unified modeling of various Siegel modular forms using formal Siegel modular forms

Demonstrated the effectiveness of the Sage implementation

Abstract

Unlike classical modular forms, there is currently no general way to implement the computation of Siegel modular forms of arbitrary weight, level and character, even in degree two. There is however, a way to do it in a unified way. After providing a survey of known computations we describe the implementation of a class modeling Siegel modular forms of degree two in Sage. In particular, we describe algorithms to compute a variety of rings of Siegel modular forms, many of which are implemented in our class. A wide variety of Siegel modular forms (e.g., both vector- and scalar-valued) can be modeled via this class and we unify these via a construct we call a formal Siegel modular form. We define this notion and discuss it in detail.