Franz Lemmermeyer

# Algebraic Number Theory

### Schedule

```  Mo 13:40 - 15:30, SAZ 02
We 15:40 - 17:30, SAZ 02 ```

### Motivation

A standard course in algebraic number theory discusses the proofs of the main results on integral bases, discriminants, Dedekind rings, class groups, Dirichlet's unit theorem, etc. In this semester, I will instead concentrate on quadratic extensions of the rationals and of the rational function fields and introduce elliptic curves. This will allow us to do a lot of explicit calculations that cannot be done (by hand) for extensions of higher degree.

### Topics

• I. Quadratic Number Fields. Here we will discuss failure of unique factorization, ideal arithmetic, class number computation, and units. Applications: Bachet-Mordell equations y2 = x3+k, some simple 3-descents on elliptic curves like x3+y3 = Az3, possibly the cubic reciprocity law.
• II. Quadratic Function Fields. These are quadratic extensions of the rational function fields Fp[X]; their arithmetic has a lot in common with quadratic number fields.
• III. Arithmetic of Elliptic and Hyperelliptic Curves. These can essentially be identified with quadratic extensions of function fields.

### Homework

Homework is always due one week after hand-out except when stated otherwise. Solutions will be posted after all students have turned their homework in.

### Schedule

• Mo 11.09.06: Euler and the failure of unique factorization
• We 13.09.06: Euclidean rings; Principal Ideal Domains
• Mo 18.09.06: Unique Factorization Domains; Ideals
• We 20.09.06: Ideals
• Mo 25.09.06: Algebraic Integers, modules
• We 27.09.06, 4:40 - 5:30 pm Ideals.
• Mo 02.10.06: Unique Factorization into prime ideals
• We 04.10.06 3:40 - 4:30 examples
• Mo 09.10.06: Units
• We 11.10.06 3:40 - 5:30: Units
• Mo 16.10.06 Class groups
• We 18.10.06: NO CLASS, moved to 11.10.06
• Mo 23.10.06 no class
• We 25.10.06 no class
• Mo 30.10.06 Class group calculations continued
• We 02.11.06 Diophantine equations
• Mo 06.11.06 Genus theory and quadratic reciprocity
• We 09.11.06 3:40 - 5:30: quadratic forms
• Mo 13.11.06 quadratic forms
• We 15.11.06 no class; moved to 09.11.
• Sa 18.11.06 12:00 - 14:00, SAZ04: Midterm 1 covering everything up to class group calculations. Proctor: Adem Ersin Ureyen. Here are the solutions
• Mo 20.11.06 reduction of quadratic forms
• We 22.11.06 quadratic forms
• Mo 27.11.06 Bhargava composition of quadratic forms
• We 29.11.06 No Class.
• Mo 04.12.06 Class groups in orders
• We 06.12.06 Bijection between forms and modules.
• Mo 11.12.06 Quadratic Forms in Function Fields
• We 13.12.06 Elliptic and Hyperelliptic Curves. Here are the updated lecture notes
• Sa 16.12.06 15:00, SAZ 04, Midterm 2. Here are the solutions. Average 76/100
• Mo 18.12.06 Elliptic and Hyperelliptic Curves
• We 20.12.06 No class (16.12.)
• Jan 04, 2007 NO FINAL (postponed)
• Jan 06, 2007; 14:00 - 16:00, SAZ03 final.

### Books

Here are a few books I recommend.
• H. Cohn, Advanced Number Theory. One of the few books with a readable account of quadratic forms.
• D. Flath, Introduction to number theory. Probably the best presentation of the theory of quadratic forms.
• Cox, Primes of the form x2 + ny2: Chapter 1 contains the theory of binary quadratic forms; the other two chapters deal with class field theory and complex multiplication.
• Marcus: Number Fields. My favorite for the theory of general number fields; it has lots of exercises.
• Ireland and Rosen: A classical introduction to modern number theory. An all-time classic.
• Murty and Esmonde: another problem collection.
• Swinnerton-Dyer: A Brief Guide to Algebraic Number Theory: concise, not expensive.
There are also various lecture notes to be found here.