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For an algebraic function field $F/K$ 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 algebraic function fields of genuszero over $k$ but not ruled. Assuming that $K$ is relatively algebraicallyclosed in $F$, we find that the number of nonruled residually transcendentalextensions of $v$ to $F$ is bounded by $\mathfrak{g}+1$ where $\mathfrak{g}$ isthe genus of $F/K$. An application to sums of squares in function fields ofcurves over $\mathbb{R}(\!(t)\!)$ is presented.

Nonsplit conics in the reduction of an arithmetic curve