Song Wang & Yichao Tian

Course Description

The distribution of primes and Diophantine equations are main topics in Number Theory. There are some examples: (i) There exist infinitely many primes. (ii) There does not exist right triangle with integral side lengths whose area is a square.

The main goal of this course is to prove the following two results:

• Dirichlet’s theorem: Given a, N ≥ 1 coprime integers, there exist infinitely many primes congruent to a modulo N.
• Mordell’s theorem: For a projective smooth curve \[ E: y^2z=x^3+axz^2+bz^3, \quad a,b \in \mathbb{Q},\] the set of rational points on E has a natural finitely generated abelian group structure.

In this course, we will cover some basics on group theory: abelian groups, characters, and structure of finitely generated abelian group. Prerequisite for this course is Calculus and Linear Algebra.

Homework will be assigned regularly, and some project problems will be proposed.