$ S$-units in hyperelliptic fields

In this note we present some results about computation of the group of S-units in hyperelliptic fields. Let k = Fq(x) be the field of rational functions of one variable over a finite field Fq of characteristic p > 2, and let d(x) = a0x 2n+1 +a1x 2n + · · ·+a2n+1 ∈ Fq[x] be a square-free polynomial, a0 = 0. Let K = k( √ d ). For an irreducible polynomial v ∈ Fq[x], we denote by | · |v the corresponding valuation on k. Let (α, β) be a point on the curve y = d(x), β = 0. Then the valuation | · |x−α has two extensions to K. These extensions will be denoted by | · |1 and | · |2. The non-Archimedean valuation | · |∞ has a unique extension to K, which also will be denoted by | · |∞. The following two cases are the basic cases for investigation of S-units: 1) S = {| · |∞, | · |1}; 2) S = {| · |∞, | · |1, | · |2}. Let OS be the ring of S-integers in K, that is, the elements y ∈ K such that |y|v > 0 for all valuations | · |v on K which do not belong to S. The set US of all invertible elements of OS is called the group of S-units of the field K. By the generalized Dirichlet theorem on units (see [1], Chap. IV, Theorem 9), the group US is the direct product of the group F ∗ q and the free Abelian group G of rank |S| − 1. The independent generators of the group G are called fundamental S-units. In the classical case of a quadratic extension L = Q( √ d ) of Q one can find a fundamental unit of the field L by using the continued fraction expansion of √ d [2]. However the method of continued fractions does not work properly for function fields. A goal of this note is to find an algorithm for computing the fundamental S-units in a hyperelliptic field K in the two basic cases above. The first proposition is of a technical character.