Nonsplit Conics in the Reduction of an Arithmetic Curve

For a function field in one variable F/K and a discrete valuation v of K with perfect residue field k, we bound the number of discrete valuations on F extending v whose residue fields are non-ruled function fields in one variable over k. Assuming that K is relatively algebraically closed in F, we find that the number of non-ruled residually transcendental extensions of v to F is bounded by g+ 1 where g is the genus of F/K. An application to sums of squares in function fields of curves over R((t)) is outlined. © 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.