On value groups and residue fields of some valued function fields

Abstract

Let K = K0(x, y) be a function field of transcendence degree one over a field K0 with x, y satisfying y2 = F(x), F(x) being any polynomial over K0. Let υ0 be a valuation of K0 having a residue field k0 and υ be a prolongation of υ to K with residue field k. In the present paper, it is proved that if G0⊆G are the value groups of υ0 and υ, then either G/G0 is a torsion group or there exists an (explicitly constructible) subgroup G1 of G containing G0 with [G1:G0]<∞ together with an element γ of G such that G is the direct sum of G1 and the cyclic group ℤγ. As regards the residue fields, a method of explicitly determining k has been described in case k/k0 is a non-algebraic extension and char k0≠2. The description leads to an inequality relating the genus of K/K0 with that of k/k0: this inequality is slightly stronger than the one implied by the well-known genus inequality (cf. [Manuscripta Math.65 (1989), 357–376’, [Manuscripta Math.58 (1987), 179–214]).