Fall 2025 Introduction to number theory

The goal of this course is to introduce diophantine problems on quadratic and cubic equations.

We will emphasize on how ideas/tools from algebra, geometry and analysis can be used to attack

these problems. We begin with the topic on integers represented by quadratic forms.

After explaining a proof of the Hasse-Minkowski's theorem, we will move on to cubic equations:

elliptic curves, the associated Hasse-Weil L-functions and the famous Birch and Swinnerton-Dyer conjecture.

Time: Thursday 2,3,4 (9:10-12:10)

The first class is online on September 4th. Link

Textbook:

A course in arithmetic GTM7 / by P. Serre.

Number theory / by Z.I. Borevich and I.R. Shafarevich

Rational points on elliptic curves / by J. Silverman and J. Tate

Office hour: TBA Astro-Math 412.

Prerequisites: Linear algebra, basic knowledge on finite fields and complex analysis

Syllabus:

Pell's equation and continued fractions (09/04 Online).

Finite fields, Gauss sum and Quadratic reciprocity laws. 

The number of solution of Fermat equations over finite fields.  

Topological and algebraic structures of p-adic integers and fields. Hensel's lemma. 

Hilbert symbol 

Quadratic forms and Hasse-Minkowski's theorem. 

Dirichlet theorem on primes in arithmetic progressions.

Elliptic curves: definitions and the group law. 

Elliptic curves over complex numbers. 

Find rational torsion points in elliptic curves over Q---Nagell-Lutz theorem. 

Mordell-Weil theorem (I) :  Heights. 

Mordell-Weil theorem (II) : The method of 2-descent.  

評量方式:

作業 60%+期末考40%