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Topics in Algebra and Number Theory: Complex Multiplication
Math 596
This course has been cancelled
Instructor: Eyal Goren   (eyal.goren@mcgill*dot*ca)

The theory of complex multiplication was the fertile ground from which class field theory had sprung. Special values of the j-function, a special complex analytic function, at complex numbers of the form t = a + b\sqrt{-d} (a, b, d rational, d>0) were found to generate abelian Galois extensions of Q(\sqrt{-d}), later to be called ring class fields. At the same time the lattice L = Z + Zt was associated with an elliptic curve C/L.
        The understanding of the hidden forces behind this phenomenon and it vast generalization took time: the first decisive step was taken by Shimura and Taniyama that replaced elliptic curves by abelian varieties and studied their fields of definition as a generalization of j(t). Shimura’s series of works on eponymous varieties allowed replacing the function j by suitable so-called modular functions and his reciprocity law allowed to calculate the Galois action on special values of these functions, thereby describing the fields that they generate. Deligne’s interpretation of Shimura’s work gave a new perspective and strengthening to these results.
        The work of Gross and Zagier revealed a remarkable factorization formula for quantities such as j(t) - j(s), where both t and s are imaginary quadratic arguments. This work was also understood as an arithmetic intersection formula for two arithmetic curves on the moduli stack of elliptic curves. Much more recently, the work of Gross and Zagier was extended in several directions where the original quantities j(t) are replaced by special cycles on Shimura varieties and their factorization by an arithmetic intersection formula.
        Elliptic curves with complex multiplication, as well as abelian varieties of higher dimension (notably 2 and 3), are also playing an important role in cryptography. The main reason being that one can calculate their group structure modulo a prime very efficiently and so generate groups suitable for cryptography. At the same time, the theory of complex multiplication is helpful in computational number theory questions such as finding the endomorphism ring of a given elliptic curve over a finite field.

To the extent I am able to, I will turn this resume into a coherent story. I will assume that people had taken a first course in algebraic geometry and in algebraic number theory. I will provide “crash-course” style background on other more specialized pre-requisites: class field theory, elliptic curves and abelian varieties, and Shimura varieties. So this you don’t have to know in advance. If you wish to discuss whether you have adequate preparation for the course, email me!

References:
- Conrad: Main theorem of complex multiplication (online note)
- Lang: Complex multiplication
- Milne: Complex multiplication (online lecture notes)
- Shimura and Taniyama: Complex multiplication of abelian varieties and applications to number theory
- Serre: Complex multiplication (in Algebraic Number Theory 1967)
- Sutherland: Isogeny volcanoes