Witt equivalence of algebraic function fields over real closed fields.(English)Zbl 1067.11020

Two fields $$K$$ and $$L$$ are called Witt equivalent if their Witt rings are isomorphic. Harrison’s criterion states that two fields are Witt equivalent iff there exists an isomorphism $$t$$ between the respective square class groups sending $$-1$$ to $$-1$$ and such that the quadratic form $$\langle a,b\rangle$$ over $$K$$ represents $$1$$ iff the form $$\langle t(a),t(b)\rangle$$ over $$L$$ represents $$1$$. $$t$$ is also called a Harrison map. Witt equivalence of global fields has been studied earlier by R. Perlis et al. [Contemp. Math. 155, 365–387 (1994; Zbl 0807.11024)]. In the present paper, the author studies the case of function fields of transcendence degree one over a fixed real closed field.
The first main result is that two such function fields $$K$$ and $$L$$ are Witt equivalent iff they are both real or both nonreal. In the real case, the author considers a refined version of Witt equivalence which he calls tame equivalence. A tame equivalence is a Harrison isomorphism which induces a map from so-called $$1$$-pt fans of $$K$$ onto $$1$$-pt fans of $$L$$. If $$\gamma (K)$$ (resp. $$\gamma (L)$$) denotes the set of all real $$K$$-places (resp. all real $$L$$-places), equipped with the so-called strong topology, then the author shows that $$K$$ and $$L$$ are tamely equivalent iff $$\gamma (K)$$ and $$\gamma (L)$$ have the same number of semi-algebraically connected components. As a corollary, he obtains that there are at most $$g+1$$ tame classes of such real function fields of transcendence degree $$1$$ and genus not exceeding $$g$$ over a fixed real closed field of constants. As an example, the author gives representatives for all tame equivalence classes for genus $$0$$ and $$1$$ for the field of constants $${\mathbb R}$$.

MSC:

 11E81 Algebraic theory of quadratic forms; Witt groups and rings 11E10 Forms over real fields 14H05 Algebraic functions and function fields in algebraic geometry

Zbl 0807.11024
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