Number Theory Informed by Computation

Jennifer S. Balakrishnan, Bjorn Poonen, Akshay Venkatesh

• Publisher: American Mathematical Society, IAS/Park City Mathematics Institute
• Year: 2025
• Topic: Mathematics
• eBook ISBN:
  9781470484514
• Subtopic: Mathematics General
• Index: Mathematics

This book is a collection of lecture notes from the Graduate Summer School "Number Theory Informed by Computation" held at the IAS/Park City Mathematics Institute 2022. The lectures address topics at the interface between number theory and computation, both terms interpreted broadly. Included are lectures on polynomial-time algorithms for problems in algebraic number theory, algorithms for counting points on mod $p$ reduction of a variety working with many values of $p$ simultaneously, arithmetic statistics, the theory of lattices, Brauer–Manin obstruction, the theory of rigid cocycles, and the inverse Galois problem. The volume starts with the notes of Hendrik Lenstra's public lecture, aimed at a general audience, where the speaker explains how modern abstract algebra can be used to explain concrete properties of integers. This volume is aimed at students with a background in graduate level number theory. For some expositions, undergraduate abstract algebra may be sufficient, while others might require basic algebraic geometry.

Table of contents

• Cover
• Title page
• Contents
• Introduction
• Algebra in the Real World
• Polynomial-time Algorithms in Algebraic Number Theory: Based on lectures by Hendrik Lenstra
• Counting Points on Hyperelliptic Curves over Finite Fields
• Number Field Counting, Class Group Heuristics, and Computation
• An Introduction to Lattices, Lattice Reduction, and Lattice-Based Cryptography
• Rational Points on Varieties and the Brauer–Manin Obstruction
• Rigid Cocycles and Singular Moduli for Real Quadratic Fields
• Around the Inverse Galois Problem