## 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* H_{K} of K .

Algebraically, H_{K} is defined as the maximal unramified abelian
extension of K ; the Galois group of H_{K}/K is then precisely the
class group Cl_{K} . A number field extension H/K is called
*Galois* if and H contains all
complex roots of f . For instance,
is Galois since it contains not only , but also the second
root of X^{2}-2 , whereas is not
Galois, since it does not contain the root
of X^{3}-2 . The *Galois group* Gal_{H/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 H_{K} 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 Gal_{H/K} and Cl_{K} 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.