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Computing Classical Modular Forms for Arbitrary Congruence Subgroups

Eran Assaf

TL;DR

It is proved the existence of an efficient algorithm for the computation of modular forms of weight $k$ and level $Gamma$ where $\Gamma \subseteq SL_{2}({\mathbb{Z}})$ is an arbitrary congruence subgroup.

Abstract

In this paper, we prove the existence of an efficient algorithm for the computation of $q$-expansions of modular forms of weight $k$ and level $Γ$, where $Γ\subseteq SL_{2}({\mathbb{Z}})$ is an arbitrary congruence subgroup. We also discuss some practical aspects and provide the necessary theoretical background.

Computing Classical Modular Forms for Arbitrary Congruence Subgroups

TL;DR

It is proved the existence of an efficient algorithm for the computation of modular forms of weight and level where is an arbitrary congruence subgroup.

Abstract

In this paper, we prove the existence of an efficient algorithm for the computation of -expansions of modular forms of weight and level , where is an arbitrary congruence subgroup. We also discuss some practical aspects and provide the necessary theoretical background.

Paper Structure

This paper contains 41 sections, 62 theorems, 165 equations, 1 table.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main Results
  4. Recovering known results
  5. New results
  6. Related Literature
  7. Organization
  8. Acknowledgements
  9. Setup and Notation
  10. Congruence Subgroups
  11. Modular Forms
  12. Explicit Computation of
  13. Modular Symbols
  14. Manin Symbols
  15. Modular Symbols with Character
  16. ...and 26 more sections

Key Result

Theorem 1.2.5

There exists an algorithm that given a group of real type $G \subseteq GL_{2}(\mathbb{Z}/N\mathbb{Z})$, a prime $p \nmid N$ such that $p \bmod N \in \det(G)$ and an integer $k \ge 2$, computes the Hecke operator $T_p$ on the space of modular forms $S_k(\Gamma_G)$ using $O(C d \cdot k \log k \cdot

Theorems & Definitions (157)

  • Conjecture 1.1.1: Serre's uniformity problem over $\mathbb{Q}$, serre1972proprietes
  • Definition 1.2.1
  • Remark 1.2.2
  • Remark 1.2.3
  • Remark 1.2.4
  • Theorem 1.2.5: Corollary \ref{['cor:Complexity Hecke Operator']}
  • Example 1.2.6
  • Theorem 1.2.7: Corollary \ref{['cor:complexity-double-coset-operator']}
  • Example 1.2.8
  • Remark 1.2.9
  • ...and 147 more