GRAduate Number Theory Seminar:
Algebraic Number Theory and Elliptic Curves

If you are looking for the web page of last Spring's Kolyvagin seminar, you want here instead.

Practical Information

Organizers: Alex Ghitza (aghitza@math.mit.edu) and Brian Osserman (osserman@math.mit.edu)
When: Fall 2000, 2 hours/week, MoFr 11-12
Where: MIT, Rm 24-110


Seminar Description

Format: a semester-long seminar giving a rapid introduction to algebraic number theory and elliptic curves. Hopefully, the material will end up including exactly what is needed for an elegant proof of the class number 1 problem for imaginary quadratic extensions, which we will then be able to present at the end of the semester. All participants will be expected to give lectures, and to prepare TeX lecture handouts.

Topics: Dedekind domains, rings of integers, scheme-theoretic curves, finite morphisms thereof, splitting and ramification, the Tchebotarov density theorem and class field theory, selected introductory topics from elliptic curve theory, complex multiplication, modular curves, and the solution to Gauss' class number 1 problem.

Prerequisites: A semester of graduate algebraic geometry, and familiarity with the commutative algebra required therein.


Lecture Notes

(Some lecture notes were not originally typed, and are not yet available online) Algebraic Number Theory Elliptic Curve Theory Further Topics


Lecture Schedule

Introduction - 1 lecture (9/11)
Ramblings on how elementary number theory motivates algebraic number theory, discussion of the class number 1 problem.

Dedekind Domains and Rings of Integers - 1 lecture (9/15)
The groups of fractional ideals and unique factorization in Dedekind domains, every ring of integers is a Dedekind domain.

Ideal Class Groups - 1 lecture (9/18)
Some further basic properties of number fields, the ideal class group of a Dedekind domain, and its finiteness for a ring of integers.

Splitting and Ramification of Prime Ideals - 1 lecture (9/22)
The splitting of prime ideals in a finite extension of Dedekind domains, and ramification at only finitely many places.

Local Completions - 1 lecture (9/29)
Completions of local rings and a survey of their properties.

Frobenius Elements - 1 lecture (10/2)
Decomposition and inertial groups, Frobenius elements.

Tchebotarov density and global class field theory - 2 lectures (10/6, 10/13)
Tchebotarov's density theorem and applications. Main theorems of class field theory and applications (proofs omitted).

Fundamentals of Elliptic Curve Theory - 4 lectures (10/16, 10/20, 10/23, 10/27)
As much of chapter III of Silverman as possible, including the group law, isogenies, torsion points, and the j-invariant.

Elliptic Curves over C - 1 lecture (10/30)
The structure of elliptic curves over C as C modulo a lattice, and corollaries.

Torsion Points and Integrality Properties - 1 lecture (11/3)
The Nagell-Lutz theorem and the theorem on reduction of torsion points mod p.

The Mordell-Weil theorem - 1 lecture (11/6)
The proof that elliptic curves over number fields have finite rank as abelian groups.

Elliptic Curves Over Finite Fields - 1 lecture (11/13)
Basic methodology of handling elliptic curves over finite fields.

Orders and their Ideal Class Groups - 1 lecture (11/17)
Orders of rings of integers and their ideal class groups. The relationship to the ideal class group of the rings of integers.

Complex Multiplication - 1 lecture (11/20)
The theory of complex multiplication of elliptic curves, and generation of ring class fields.

Modular Curves - 2 lectures (11/27, 12/1)
An introduction to the theory of modular curves, and a development of X(N) over Q.

The Class Number 1 Problem - 1 lecture (12/4)
The proof of the class number 1 problem for imaginary quadratic extensions, using complex multiplication theory and the theory of modular curves.


References

Here are the relevant references for material for the seminar, starting with background:

Some random supplemental notes I've written up:

Norms of elements of number fields (or DVI)

Integral monic minimal polynomials and Gauss' lemma (or DVI)

Line bundles and divisors on curves (or DVI)

If you have any other suggestions for references, please email me.