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 v₀ be a valuation of K₀ having a residue field K₀ and v be a prolongation of v₀ to K with residue field k. In the present paper, it is proved that if G₀⊆G are the value groups of v₀ and v, then either G/G₀ is a torsion group or there exists an (explicitly constructive) subgroup G₁ of G containing G₀ with [G₁:G₀]<∞ together with an element ϒ of G such that G is the direct sum of G₁ and the cyclic group Z_ϒ. As regards the residue fields, a method of explicitly determining A: 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]).