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

**Taught by **

Rong Zhou, rzhou@math.harvard.edu

Yihang Zhu, yihang@math.harvard.edu

**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)

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.

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