# Tutorial on complex multiplication of elliptic curves

## Fall 2014

Mondays and Thursdays 4-5:30 pm in Science Center 530, Harvard

**Taught by **

Rong Zhou

Yihang Zhu

**Office hour:** 3:30 pm, Fridays, common room of math department.

**Syllabus**, including some suggested final project topics and a reading list: Click here. (Updated 9/8/2014)

## Lectures

Lecture 1, 9/8/14: Review of Galois theory, separability, finite fields. Sum of two squares. Notes.

Lecture 2, 9/11/14: Sum of two squares. Number fields and rings of integers. Dedekind domains. For proofs see the first three sections of Neukirch's book. Notes.

Lecture 3, 9/15/14: The class group. Prime factorization. Decomposition groups in a Galois extension. The main reference is the first chapter of Neukirch's book. Notes.

Lecture 4, 9/18/14: Inertia subgroups. Frobenius. Unramified class field theory. Notes.

Lecture 5, 9/22/14: Riemann surfaces. Complex tori. The modular curve. Notes. (Modified 9/30/2014)

Lecture 6, 9/25/14: Modular functions. Notes.

Lecture 7, 9/29/14: Elliptic functions. Notes.

Lecture 8, 10/2/14: Affine algebraic varieties. Notes. (Updated 10-8-14)

Lecture 9, 10/6/14: Dimension and smoothness. Projective algebraic varieties. Notes.

Lecture 10, 10/9/14: Affine coverings of projective varieties and projectivization of affine varieties. Rational maps. Notes.

Lectures 11 & 12, 10/16/14 & 10/20/14 : Algebraic curves. Notes.

Lecture 13, 10/23/14 : Differentials. Bezout's theorem. Elliptic curves over general fields. Notes. (Updated 10/28/14)

Lecture 14, 10/27/14 : Algebraic theory of elliptic curves. Notes.

Lecture 15, 10/30/14 : Algebraic theory of elliptic curves II. Notes.

Lecture 16, 11/3/14 : Complex multiplication over the comlex numbers. Reference: Chapter II.1 of [Sil] = Silverman: Advanced Topics in the Arithmetic of Elliptic Curves.

Lecture 17, 11/6/14 : Complex multiplication algebraically. Reference: Chapter II.2 of [Sil].

Lecture 18, 11/10/14: Algebraic construction of the action of the class group on CM j-invariants. Summary of CFT. Reference: [Sil] Proposition II.2.5, Chapter II.3.

**Note:** In the proof of Proposition II.2.5 in [Sil], the map $Hom( \mathfrak a, \mathbb C) \to Hom (\mathfrak a, E)$ in the first row of diagram (iii) is in fact surjective, because $\mathfrak a$ is a projective $R_K$-module.
Therefore $ Hom (\mathfrak a, E) \cong \mathbb C / \mathfrak a^{-1}\Lambda$ in the complex topology. Thus in the argument the phrase "identity compoenent of" could be deleted, and there's no need to write down the 3 times 3 diagram nor to use the snake lemma.
Lecture 19, 11/13/14: Cancelled due to Ahlfors Lecture. Please read: A brief summary of CFT.

Lecture 20, 11/17/14: Generating the Hilbert class field of $K$.

Lecture 21, 11/20/14: Generating Ray class fields of $K$.

Lecture 22, 11/24/14: Main Theorem of CM, analogue for $\mathbb Q$. Please read: CFT for the rationals. (Updated 11/30/2014)

## Problem Sets

The Tutorial will be graded solely on your final projects. The problem sets are designed to complement the lectures and you are encouraged to do them as they will facilitate your understanding of the material. Some of
the problems are already mentioned in class and also contained in the notes.
Problem Set 1, 9/11/14.

Problem Set 2, 9/18/14.

Problem Set 3, 9/29/14. Problems contained in the notes for Lectures 5-7.

Problem Set 4, 10/13/14.