Section:
Research Program
Complex multiplication
Participants :
Jared Guissmo Asuncion, Karim Belabas, Henri Cohen, Jean-Marc Couveignes, Andreas Enge, Fredrik Johansson, Chloe Martindale, Damien Robert.
Complex multiplication provides a link between number fields and
algebraic curves; for a concise introduction in the elliptic curve case,
see [38], for more background material,
[37]. In fact, for most curves over a
finite field, the endomorphism ring of , which determines
its -function and thus its cardinality, is an order in a special
kind of number field , called CM field. The CM field
of an elliptic curve is an imaginary-quadratic field
with , that of a hyperelliptic curve of genus is an
imaginary-quadratic extension of a totally real number field of
degree . 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 of .
Algebraically, is defined as the maximal unramified abelian
extension of ; the Galois group of is then precisely the
class group . A number field extension is called
Galois if and contains all
complex roots of . For instance,
is Galois since it contains not only , but also the second
root of , whereas is not
Galois, since it does not contain the root
of . The Galois group is the group of
automorphisms of that fix ; it permutes the roots of . Finally,
an abelian extension is a Galois extension with abelian Galois
group.
Analytically, in the elliptic case may be obtained by adjoining to
the singular value for a complex valued, so-called
modular function in some ; the correspondence
between and allows to obtain the different roots
of the minimal polynomial of and finally itself.
A similar, more involved construction can be used for hyperelliptic curves.
This direct application of complex multiplication yields algebraic
curves whose -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 -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.