Computing with adèles and idèles

A SageMath package for computing with adèles and idèles.

To use this package, you need to import it:

sage: from adeles.all import *

After this, all functionality of this package is available:

sage: Adeles(QQ)
Adèle Ring of Rational Field

This package is based on and part of the following master’s thesis:

[Her2021] Mathé Hertogh, Computing with adèles and idèles, master’s thesis, Leiden University, 2021.

Contents of the package

The package can be seen to consist out of four parts.

Part 1 corresponds to Chapters 3–6 of [Her2021] and provides the functionality to compute with adèles and idèles over number fields. It consists out of these files:

• Profinite Integers

• Profinite Numbers

• Infinite Completions of Number Fields

• Adèles

• Multiplicative p-adics

• Idèles

• Ray Class Groups

Part 2 corresponds to Chapter 7 of [Her2021] and implements profinite graphs, which visualize graphs of functions from and to the ring of rational profinite integers. In particular, the profinite Fibonacci function is implemented. Part 2 consists of out two files:

• Profinite Functions

• Profinite Graphs

Part 3 corresponds to Chapter 8 of [Her2021] and implements the adèlic matrix factorization algorithms discussed there. This resides in the file:

• Adèlic Matrix Factorization

Part 4 corresponds to Chapter 9 of [Her2021] and implements the computation of Hilbert class fields of imaginary quadratic number fields using Shimura’s reciprocity law. It consists of the files:

• Modular Functions and their Action

• Shimura Connecting Homomorphism

• Hilbert Class Field Computations

Table of Contents

• Profinite Integers
  • Profinite Integers over \(\QQ\)
  • The general case
• Profinite Numbers
  • Profinite Numbers over \(\QQ\)
  • Profinite Integers over \(K\)
• Infinite Completions of Number Fields
• Adèles
  • Examples with \(\QQ\)-adèles
  • Adèles over Number Fields
• Multiplicative \(p\)-adics
• Idèles
• Ray Class Groups
• Profinite Functions
• Profinite Graphs
  • Factorial digits
  • The visualization function
  • Graphing a profinite function
• Adèlic Matrix Factorization
• Modular Functions and their Action
• Shimura Connecting Homomorphism
• Hilbert Class Field Computations
  • Example 1
    • Finding a class invariant
    • Computing the minimal polynomial
  • Example 2
  • Example 3

Indices and Tables

• Index

• Module Index