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
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.
Dedekind Domains and Rings of Integers - 1 lecture (9/15)
Ideal Class Groups - 1 lecture (9/18)
Splitting and Ramification of Prime Ideals - 1 lecture (9/22)
Local Completions - 1 lecture (9/29)
Frobenius Elements - 1 lecture (10/2)
Tchebotarov density and global class field theory - 2 lectures (10/6, 10/13)
Fundamentals of Elliptic Curve Theory - 4 lectures (10/16, 10/20, 10/23,
10/27)
Elliptic Curves over C - 1 lecture (10/30)
Torsion Points and Integrality Properties - 1 lecture (11/3)
The Mordell-Weil theorem - 1 lecture (11/6)
Elliptic Curves Over Finite Fields - 1 lecture (11/13)
Orders and their Ideal Class Groups - 1 lecture (11/17)
Complex Multiplication - 1 lecture (11/20)
Modular Curves - 2 lectures (11/27, 12/1)
The Class Number 1 Problem - 1 lecture (12/4)
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.
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.
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.
The groups of fractional ideals and unique factorization in Dedekind
domains, every ring of integers is a Dedekind domain.
Some further basic properties of number fields, the ideal class group
of a Dedekind domain, and its finiteness for a ring of integers.
The splitting of prime ideals in a finite extension of Dedekind domains,
and ramification at only finitely many places.
Completions of local rings and a survey of their properties.
Decomposition and inertial groups, Frobenius elements.
Tchebotarov's density theorem and applications. Main theorems of class
field theory and applications (proofs omitted).
As much of chapter III of Silverman as possible, including the group
law, isogenies, torsion points, and the j-invariant.
The structure of elliptic curves over C as C modulo a lattice, and
corollaries.
The Nagell-Lutz theorem and the theorem on reduction of torsion points
mod p.
The proof that elliptic curves over number fields have finite rank as
abelian groups.
Basic methodology of handling elliptic curves over finite fields.
Orders of rings of integers and their ideal class groups. The relationship
to the ideal class group of the rings of integers.
The theory of complex multiplication of elliptic curves, and generation of
ring class fields.
An introduction to the theory of modular curves, and a development of
X(N) over Q.
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: