Computational Aspects of Curves of Genus At

This survey discusses algorithmsand explicitcalculationsfor curves of genus at least 2 and their Jacobians, mainly over number elds and nite elds. Miscellaneous examples and a list of possible future projects are given at the end. 1. Introduction An enormous number of people have performed an enormous number of computations on elliptic curves, as one can see from even a perfunctory glance at 29]. A few years ago, the same could not be said for curves of higher genus, even though the theory of such curves had been developed in detail. Now, however, polynomial-time algorithms and sometimes actual programs are available for solving a wide variety of problems associated with such curves. The genus 2 case especially is becoming accessible: in light of recent work, it seems reasonable to expect that within a few years, packages will be available for doing genus 2 computations analogous to the elliptic curve computations that are currently possible in PARI, MAGMA, SIMATH, apecs, and the \Elliptic Curve Calculator." As evidence of the growth of the literature, we note that the rst book devoted to the explicit study of genus 2 curves has just appeared 22]. Applications requiring computations with curves of genus at least 2 have existed for well over a century. The oldest (but which has also acquired new relevance since the advent of symbolic integration packages) is that of the integration of algebraic functions: according to a theorem of Risch, the problem of deciding whether the integral of an algebraic function is elementary can be reduced to the problem of deciding whether divisors on algebraic curves represent torsion points on the Jaco-bian. (See 30] for a detailed discussion.) More recently, the ability to deal with curves of large genus explicitly has had applications in coding theory: to construct eecient algebraic-geometric codes, one needs curves over nite elds having many points 46], 113]. Also, algorithmic aspects of Jacobians of genus 2 curves play an important role in Adleman and Huang's proof that the primes are recognizable in random polynomial time 3]. Finally, Jacobians of hyperelliptic curves over nite elds have been suggested for use in cryptosystems 56]. The security of such systems is dependent on the alleged diiculty of solving the discrete logarithm problem in these algebraic groups.