website for
Complex Multiplication
(2013, L24)

This is a website for the Part III (non-examinable, graduate) course Complex Multiplication (2013, L24); here are some material for this course.

(from the official website of the department)

Meetings: Mon. / Wed. / Fri. 9am, at MR 14.
First lecture: Fri., 18th January, 2013.
Last lecture: Wed., 13th March, 2013.

  • p.32 of the [pdf] Mathematical Tripos Part III Lecture Courses.


    I may post some typed/scanned lecture notes here. There will be some overlap with my notes from the Postech Summer School:

  • [pdf] Notes for Postech Summer School 2010.

    The goal would be to cover the proofs of main theorems of complex multiplication for abelian varieties, probably over CM fields containing imaginary quadratic fields. We will refer to the course Class Field Theory (2012, L24) when needed, but I'll try to review the relevant material as much as possible.

  • Lecture 1 (Fr. 18/1/13) [scan]: 1.1 Local CFT / Global CFT
  • Lecture 2 (Mo. 21/1/13) [scan]: 1.3 The Langlands correspondence for GL(1)
  • Lecture 3 (We. 23/1/13) [scan]: 1.4 Algebraic Hecke characters
  • 2012 Handout: weak approximation for ideles
  • Lecture 4 (Fr. 25/1/13) [scan]: 1.4 Algebraic Hecke characters (continued) / 1.5 CM fields
  • Lecture 5 (Mo. 28/1/13) [scan]: 1.5 CM fields (continued)
  • Lecture 6 (We. 30/1/13) [scan]: 1.5 CM fields (continued) / 1.6 l-adic Galois characters
  • Lecture 7 (Fr. 1/2/13) [scan]: 1.7 CM Hecke characters
  • Lecture 8 (Mo. 4/2/13) [scan]: 1.8 Automorphic representations of GU(1) / 1.9 Algebraic groups


    Here are some references - they are not essential at all.

  • J.P. Serre, Complex Multiplication, in: Cassels-Frohlich, ed., Algebraic Number Theory, Academic Press, 1967.
  • G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971.
  • G. Shimura, Abelian Varieties with Complex Multiplication and Modular Functions, Princeton, 1998.
  • J.S. Milne, Complex Multiplication, available here.

    Lectures 1-6 are a modern adaptation of:

  • A. Weil, On a certain type of characters of the id`ele class group of an algebraic number field, 1956.
  • C. Chevalley, Deux Th'eor`emes d'Arithm'etique, 1951.

    with a bit of help from

  • J.-P. Serre, Abelian l-adic Representations and Elliptic Curves, 1968/1989.

  • Last modified: 12th February, 2013.