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LFANT - 2018

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 [41], for more background material, [40]. In fact, for most curves 𝒞 over a finite field, the endomorphism ring of Jac𝒞, 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 (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 𝒞 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 HK[X]/(f) and H contains all complex roots of f. For instance, (2) is Galois since it contains not only 2, but also the second root -2 of X2-2, whereas (23) is not Galois, since it does not contain the root e2πi/323 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(τ) for a complex valued, so-called modular function j in some τ𝒪K; the correspondence between GalH/K and ClK allows to obtain the different roots of the minimal polynomial f of j(τ) 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.