The group law on the jacobian of a curve of genus 2.

Abstract

In [8] an explicit embedding in P 15 is described of the Jacobian of a curve of genus 2 over an arbitrary ground field. The defining equations are 72 quadratic forms defined over the ground field ([8], Appendix A). It is the main aim of this paper to describe the biquadratic equations which define the group law on the Jacobian. Although these equations are too large to be written down, we shall give explicit bilinear forms (in Appendix B) relating to the Kummer surface from which the required biquadratic forms may be derived for any given specialisation. These forms, together with the defining equations of [8] will complete the explicit description of the Jacobian of a general curve of genus 2 äs an Abelian variety. Considerable care will be taken to ensure that all maps and equations are defined over the ground field, and # will have the general sextic (rather than quintic) form. This has the drawback of increasing the sheer size of the algebra; however, in compensation, the results will have both a more general applicability and a more pleasing symmetry of structure. An example of this additional symmetry is given by the transformation " in equation (12) of Section 3, which does not exist in the quintic Situation (where the Weierstrass point at infinity is given an artificial special Status). Before describing our forms, we shall first provide some motivation for our generalisations and give a few examples by way of Illustration.A general curve of genus 2 can be written in the form: Y 2 = sextic in X, and such a curve is reducible to the form Y 2 = quintic in X if and only if the original sextic has a rational root. In the general sextic case, a P 15 embedding of the Jacobian variety is required, whereas in the quintic case, one can use a P 8 embedding [10]. Our reasons for developing an explicit P 15 theory fall into two categories: first, most of the examples likely to arise in the near future will not have a rational Weierstrass point (and so will require the P 15 development); second, the P 15 development gives rise to structures which are more natural in appearance (regardless of whether the curve is in quintic or sextic form), and *) The author thanks SERC for financial support. 4 Journal für Mathematik. Band 439 Brought to you by | University of Iowa Libraries Authenticated Download Date | 6/13/15 3:49 PM Flynn, The Jacobian ofa curve ofgenus 2which will provide a better base for the further development of an explicit theory in genus 2 (and higher).Bost and Mestre in [2] have recently publicised curves of the form;where g(X) 9 h(X) 9 i(X) are quadratics. It is almost certain that most arithmetic progress in the near future will be on curves such äs:where g(X), h (X) 9 i(X) are defined over Q. Such a curve is only reducible (over O) to quintic form if one of the quadratic factors splits onto two linear factors (for example, (*) is not reducible to quintic form). A curve such äs (*) is of particular interest since its Jacobian is isogenous to that of a related curve of the same form. For example, (*) has J...

Kummer surfaces for primality testing

“…The curve H embeds into P 8 , see [16] for explicit formulas. (If H did not have a rational Weierstrass point, one would even need P 15 , see [15].) Unfortunately, this large number of coordinates turns out to be impractical computationally.…”