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**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 \({\mathbb R}\).

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 \({\mathbb R}\).

Reviewer: Detlev Hoffmann (Nottingham)

### 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 |