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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 Im20 $\#119966 $ over a finite field, the endomorphism ring of Im17 $Jac_\#119966 $ , 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 Im31 ${\#8474 (\sqrt D)}$ 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 Im20 $\#119966 $ 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 ${H\#8771 K[X]/(f)}$ and H contains all complex roots of f . For instance, Im33 ${\#8474 (\sqrt 2)}$ is Galois since it contains not only Im34 $\sqrt 2$ , but also the second root Im35 ${-\sqrt 2}$ of X2-2 , whereas Im36 ${\#8474 (\mroot 23)}$ is not Galois, since it does not contain the root Im37 ${e^{2\#960 i/3}\mroot 23}$ 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($ \tau$) for a complex valued, so-called modular function j in some Im38 ${\#964 \#8712 \#119978 _K}$ ; the correspondence between GalH/K and ClK allows to obtain the different roots of the minimal polynomial f of j($ \tau$) 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.


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