Computational Arithmetic of Modular Forms (Course notes)

TL;DR

This paper provides an accessible, detailed explanation of algorithms for computing modular forms and their arithmetic properties, emphasizing implementation and the use of group cohomology.

Contribution

It introduces an explicit, elementary presentation of the modular symbols algorithm suitable for implementation over various rings, with foundational tools from homological algebra.

Findings

Provides a clear derivation of the modular symbols algorithm

Enables implementation over arbitrary rings with sufficient linear algebra support

Includes foundational homological algebra tools for computation

Abstract

These course notes are about computing modular forms and some of their arithmetic properties. Their aim is to explain and prove the modular symbols algorithm in as elementary and as explicit terms as possible, and to enable the devoted student to implement it over any ring (such that a sufficient linear algebra theory is available in the chosen computer algebra system). The chosen approach is based on group cohomology and along the way the needed tools from homological algebra are provided.