Algebraic Number Theory and Elliptic Curves

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 (PS) or DVI
- Ideal Class Groups (PS) or DVI
- Splitting and Ramification of Primes (PS) or DVI
- Local Fields (PS) or DVI
- Tchebotarev Density and Global Class Field Theory (PS) or DVI

- Invariant Differentials and Dual Isogenies (PS) or DVI
- The Tate Module and Endomorphisms (PS) or DVI
- Torsion Points (PS) or DVI
- Elliptic Curves Over Finite Fields (PS) or DVI

- Orders and Their Class Groups (PS) or DVI
- Complex Multiplication (PS) or DVI
- Modular Curves (PS) or DVI
- The Class Number One Problem (PS) or DVI

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.

- Commutative Algebra: Atiyah and MacDonald,
*Introduction to Commutative Algebra* - Commutative Algebra: David Eisenbud,
*Commutative Algebra* - Algebraic Geometry: Robin Hartshorne,
*Algebraic Geometry* - Dedekind Domains: Jean-Pierre Serre,
*Local Fields* - Algebraic Number Theory: Serge Lang,
*Algebraic Number Theory* - Algebraic Number Theory: David Cox,
*Primes of the Form x^2+ny^2* - Elliptic Curve Theory: Joseph Silverman,
*Arithmetic of Elliptic Curves*

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.