Complex multiplication of abelian surfaces
David Kohel, Ronald van Luijk, Marco Streng · 2010
In this chapter, we give an introduction to the theory of complex multiplication. We define notions like CM-fields, CM-types, and the reflex type that occur in every chapter of this thesis, and we state the ‘main theorem’ of complex multiplication. We show in Theorem 10.3 that a general result of Shimura [76] can be improved for the case of CM-fields of degree 4 1 Kronecker’s Jugendtraum The following classical result describes all finite abelian extensions of Q via Galois theory. Theorem 1.1 (Kronecker-Weber Theorem). Let K/Q be a finite abelian Galois extension. Then there is a positive integer n such that we have an embedding K → Q(ζn) = Q(t : t ∈ Gm(Q)[n]) = Q(exp( 2πi n )). The Galois group of Q(ζn)/Q is (Z/nZ)∗, where (k mod n) maps ζn to ζ n. Kronecker’s Jugendtraum (a.k.a. Hilbert’s twelfth problem) is to find an analogue of this result when Q is replaced by an arbitrary number field F . 18 Chapter I. Complex multiplication Class field theory implicitly describes all finite abelian extensions of F and their Galois groups in terms of certain groups of equivalence classes of ideals. These groups are called class groups, and to each class group of F , there corresponds an abelian extension of F , which we call the class field corresponding to the group. The Galois group of an abelian extension of F is isomorphic to the corresponding class group via the Artin map. All class fields can be constructed from their class groups using Kummer theory. Suppose we want to construct the finite abelian extension M of F corresponding to a class group G. If e is the exponent of G, then by Kummer theory, we find that M is a subfield of F (ζe)( e √ S) for some finite set S ⊂ F (ζe). As the Artin map tells us much about the decomposition of primes in M/F , this allows us to find M . For details, see Cohen and Stevenhagen [18]. The approach of finding the abelian extensions of F via Kummer theory is arguably not in the spirit of Kronecker’s Jugendtraum, since it is not of the form of a single function that parametrizes generators of the abelian extensions of F , like the analytic map z 7→ exp(2πiz) for F = Q. If F is imaginary quadratic, then the theory of complex multiplication of elliptic curves does provide a complete solution to Kronecker’s Jugendtraum in terms of the j-invariant and the coordinates of torsion points. These torsion points can be parametrized by a normalized version of the Weierstrass ℘-function, or ‘better’ modular functions as in [18]. This approach does not suffer from the need for extra roots of unity ζe, as Kummer theory does. With the theory of complex multiplication of abelian varieties, Shimura and Taniyama [78] generalized the full answer for imaginary quadratic fields to a partial answer for CM-fields. For a CM-field F , we obtain many abelian extensions of F by replacing Gm above by an abelian variety that has complex multiplication by the reflex field K of F . Which abelian extensions are obtained is expressed in terms of the reflex type. We will first define these notions.