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)* .

Abstract.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)* . ResultsLet 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

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We discuss the asymptotic behaviour of the genus and the number of rational places in towers of function fields over a finite field.Both authors were partially supported by GMD/CNPq and the first author also by PRONEX K41.96.0883.00 (Brazil).Brought to you by | The University of York Authenticated Download Date | 7/7/15 8:01 AM 2 ! .The proof of Property (1) is due to H. G. Rü ck; it is given in an appendix of this paper. It was noticed by M. Zieve that Property (1) also follows from Property (2) and the fact that the roots of HðX Þ are in F p 2 . Garcia, Stichtenoth and Rü ck, Tame towers over finite fields 54 Brought to you by | The University of York Authenticated Download Date | 7/7/15 8:01 AM

Abstract. A hyperelliptic function field can be always be represented as a real quadratic extension of the rational function field. If at least one of the rational prime divisors is rational over the field of constants, then it also can be represented as an imaginary quadratic extension of the rational function field. The arithmetic in the divisor class group can be realized in the second case by Cantor's algorithm. We show that in the first case one can compute in the divisor class group of the function field using reduced ideals and distances of ideals in the orders involved. Furthermore, we show how the two representations are connected and compare the computational complexity.