j-kieffer / hdme

C library for the evaluation of modular equations in dimension 2

Description

This repository contains a C library for evaluating higher-dimensional modular equations by analytic methods. As it stands now, the following types of modular equations in genus g=2 are supported, either over the rationals or prime finite fields:

• Modular equations of Siegel type;
• Modular equations of Hilbert type, using parametrizations of Humbert surfaces for all fundamental discriminants less than 100, and Hilbert surfaces for all fundamental discriminants less than 20.

The choice of coordinates is dynamic, so that all corner cases (extra endomorphisms, vanishing of I4...) can be covered.

Along the way, the library provides functionality for the evaluation of genus 2 theta constants in quasi-linear time, and for the action of Hecke operators on Siegel modular forms. The algorithms are described in the associated paper [K21].

Prerequisites

An installation of Arb version 2.21.0 or later, and Flint version 2.8.0 or later.

Installation

Run

./configure

to create the Makefile. To compile the library, run

make

This will produce a shared library in the source directory. The Makefile also responds to the standard commands

make tests
make check
make clean

Documentation

A detailed documentation for each function existed in previous versions, but is now outdated. For examples of usage, look at the test files in the test subfolder of each module.

Credits

• The code for fast evaluation of theta constants using Arb is inspired by the cmh library by Andreas Enge and Emmanuel Thomé.
• The code written by Enea Milio has been useful in the implementation of Mestre's algorithm and to implement the inversion of the Hilbert embedding.
• The design and semantics of many functions are inspired from existing functions in the libraries Flint and Arb, maintained by William Hart and Fredrik Johansson.