Section: Research Program
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 , for more background material, . 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.