Evaluating Igusa functions

TL;DR

This paper introduces a novel method to evaluate Igusa functions by leveraging Siegel Eisenstein series and classical modular forms, providing an explicit algorithm for precise computation.

Contribution

It presents a new approach to compute Igusa functions using Siegel Eisenstein series and Fourier coefficients of modular forms, offering an explicit evaluation algorithm.

Findings

Developed an explicit algorithm for Igusa function evaluation.

Connected Fourier coefficients of Siegel modular forms to classical modular forms.

Demonstrated the effectiveness of the method with computational examples.

Abstract

The moduli space of principally polarized abelian surfaces is parametrized by three Igusa functions. In this article we investigate a new way to evaluate these functions by using Siegel Eisenstein series. We explain how to compute the Fourier coefficients of certain Siegel modular forms using classical modular forms of half-integral weight. One of the results in this paper is an explicit algorithm to evaluate the Igusa functions to a prescribed precision.