Lecture on Classical Theory and Arithmetic Statistics

Overview

• Four courses in one week: Lecture in the morning, exercises in the afternoon.
• Focus on getting a general feel of the topics, not on mastering all content.
• Classical Theory will be covered, followed by practical computations.
• Use of computational tools like Parry or Magma for numerical data.
• Afternoon sessions will focus on exercises involving these theories.

Key Themes

Importance of Hands-On Experience

• Importance of understanding by doing, not just memorizing theorems.
• Exercises can be done by hand or using computational tools.

Arithmetic Statistics

• Understanding average behavior of mathematical objects like class groups or fields.
• Theorem boundaries define randomness in arithmetic statistics.
• Cohen-Lenstra heuristic: What does a random group look like?

Algebraic Number Theory (ANT)

Motivation

• Solving equations in integers/rational numbers often requires larger number fields.
• Example: Fermat's equation (x^n + y^n = z^n).

Key Concepts

• Number Fields and Rings: Fields larger than rational numbers, e.g., cyclotomic fields.
• Integral Closure: OK, the ring of integers in a number field K, is integral closure of Z in K.
• Prime Factorization: Unique for ideals in OK, leading to concepts like class group.
• Units and Class Groups: Structures determining factorization, unique elements up to units.

Computation in ANT

• Discriminant: Important invariant, relates to size and splitting behavior of primes.
• Class Group Computation: Techniques involve factorizing primes and computing relations.
• Unit Groups: Structurally easier to handle than arbitrary number fields.

Cyclotomic Fields

• Generated by roots of unity, with accessible group actions by Z/mZ*.
• These fields are Galva and fully understood via their group actions.

Key Theorems

• Minkowski Bound: Determining finiteness of class groups and units.
• Tchebotarev Density Theorem: Describes distribution of Frobenius elements in Galva groups.

Applications and Practical Use

• Ray Class Field: Generalization for any number field K, replacing simple cyclotomic fields.

Important Results in Cyclotomic Fields

• Every abelian extension of Q is cyclotomic (Kronecker-Weber theorem).
• Extends understanding to any number field K.

Practical Examples

• Examination of primes that can be written as x^2 + ny^2, with explicit examples and analyses (e.g., n = 1, 2, 5).

• Relation to splitting in quadratic fields and concepts of class groups.

• Illustrations on explicit computations for small example cases.

• Emphasis on understanding transformations and structures involving cyclotomic and ray class fields.

Conclusion

• Hands-on usage of class field theory is pivotal for solving practical problems in arithmetic statistics.
• Encouragement to actively engage and solve problems during exercises and discussion sessions.