Section: Scientific Foundations
Complex multiplication
Participants : Karim Belabas, Henri Cohen, Andreas Enge.
Complex multiplication provides a link between number fields and
algebraic curves; for a concise introduction in the elliptic curve case,
see [9] , for more background material,
[8] . In fact, for most curves 
over a
finite field, the endomorphism ring of 
, which determines
its L -function and thus its cardinality, is an order in a special
kind of number field K , called CM field . The CM field
of an elliptic curve is an imaginary-quadratic field 
with D<0 , that of a hyperelliptic curve of genus g is an
imaginary-quadratic extension of a totally real number field of
degree g . Deuring's lifting theorem ensures that is the reduction
modulo some prime of a curve with the same endomorphism ring, but defined
over the Hilbert class field HK of K .
Algebraically, HK is defined as the maximal unramified abelian
extension of K ; the Galois group of HK/K is then precisely the
class group ClK . A number field extension H/K is called
Galois if ![Im32 ${H\#8771 K[X]/(f)}$](32.png){kind=small}
and H contains all
complex roots of f . For instance, 
is Galois since it contains not only 
, but also the second
root 
of X2-2 , whereas 
is not
Galois, since it does not contain the root 
of X3-2 . The Galois group GalH/K is the group of
automorphisms of H that fix K ; it permutes the roots of f . Finally,
an abelian extension is a Galois extension with abelian Galois
group.
Analytically, in the elliptic case HK may be obtained by adjoining to
K the singular value j(
) for a complex valued, so-called
modular function j in some 
; the correspondence
between GalH/K and ClK allows to obtain the different roots
of the minimal polynomial f of j() and finally f itself.
A similar, more involved construction can be used for hyperelliptic curves.
This direct application of complex multiplication yields algebraic
curves whose L -functions are known beforehand; in particular, it is
the only possible way of obtaining ordinary curves for pairing-based
cryptosystems.
The same theory can be used to develop algorithms that, given an arbitrary curve over a finite field, compute its L -function.
A generalisation is provided by ray class fields ; these are still abelian, but allow for some well-controlled ramification. The tools for explicitly constructing such class fields are similar to those used for Hilbert class fields.
