A REMARK CONCERNING m-DIVISIBILITY AND THE DISCRETE LOGARITHM IN THE DIVISOR CLASS GROUP OF CURVES

The aim of this paper is to show that the computation of the discrete logarithm in the m-torsion part of the divisor class group of a curve X over a finite field ko (with char(/co) prime to m), or over a local field k with residue field /co , can be reduced to the computation of the discrete logarithm in /co(Cm)* • For this purpose we use a variant of the (tame) Täte pairing for Abelian varieties over local fields. In the same way the problem to determine all linear combinations of a finite set of elements in the divisor class group of a curve over k or k$ which are divisible by m is reduced to the computation of the discrete logarithm in /co(im)* . 1. Results Let fco be a finite field with q elements and Xo a projective irreducible nonsingular curve of genus g over ko. For simplicity we assume that the curve Xo has a point Pq which is rational over ko. Let Divo(Xo) be the group of divisors of degree 0 on Xo. In particular, the set of divisors of functions on Xo is a subgroup of this group. The quotient group, i.e., the group of divisor classes of degree 0, is denoted by Picn(A'o). We consider a positive integer m which divides q 1. Then m is prime to the characteristic of ko and the mth roots of unity are contained in ko . We denote by Pico(X0)m the group of divisor classes whose m-fold is zero. We want to treat_the problem of the discrete logarithm in the group Pic0(X0)m : Let Dx and D2 be given elements in Pico(^o)m with D2 = pDx and /íéN; then evaluate the integer p (notice that the group law in Pico(-Yo) *s written additively, contrary to the notation "discrete logarithm"). In particular, we want to reduce this problem to the corresponding one in the multiplicative group k^ : Given elements n and C of A;0* with an integer p such that C = nß ', determine this element p. It is not our aim to give explicit formulas for the addition law in Pico(-Yn) ■ We want to assume that the elements in Pico(^o) are represented in the following way: The theorem of Riemann-Roch asserts that each class of Pico(-Yo) contains a divisor of the form A gP0 , where A is a positive divisor on Xo of degree g (without mentioning it explicitly, we mean that the divisor A is rational over ko). If A is given as A = Ylf=x ?> > men tne points P¡ on Xo are rational over a finite extension of ko of degree g\. Notice that this degree is Received by the editor July 5, 1991 and, in revised form, December 30, 1992. 1991 Mathematics Subject Classification. Primary 11G20, 11Y99. ©1994 American Mathematical Society 0025-5718/94 $1.00+ $.25 per page