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**Finiteness theorems for forms over global fields.**
*(English)*
Zbl 0724.11021

We are interested in determining when global fields (number fields or function fields in one variable over finite fields) have isomorphic Witt rings. Baeza and Moresi settled this question for global fields of characteristic 2. Namely, any two such fields are Witt equivalent (i.e., have isomorphic Witt rings). For global fields of characteristic different from 2, R. Perlis, K. Szymiczek, P. E. Conner, and R. Litherland [Matching Witts with global fields, preprint 1989, see also the preceding review of K. Szymiczek’s paper] proved that two Witt equivalent global fields are either both number fields or both function fields. Furthermore, they introduced the concept of small equivalence, a finite but technical set of conditions, and reduced the Witt equivalence problem to that of small equivalence.

This paper is devoted to analyzing the notion of small equivalence and simplifying the conditions. Essentially, for number fields they reduce to degree over \(Q_ 2\), the square class of -1 and properties at the dyadic and real archimedean places; for function fields, it is simply the level.

These new conditions can be checked so readily in principle that one can promptly draw several corollaries. For example, given any finite degree n, the number of Witt equivalence classes of number fields of degree n must be finite. Perlis, Szymiczek, Conner, and Litherland showed that a Witt ring isomorphism produces a family of local isomorphisms called a reciprocity equivalence; these come in two types, tame and wild. Using the new conditions, one can prove that there always exists a Witt ring isomorphism which is tame at the place P, with finitely many exceptions.

This paper is devoted to analyzing the notion of small equivalence and simplifying the conditions. Essentially, for number fields they reduce to degree over \(Q_ 2\), the square class of -1 and properties at the dyadic and real archimedean places; for function fields, it is simply the level.

These new conditions can be checked so readily in principle that one can promptly draw several corollaries. For example, given any finite degree n, the number of Witt equivalence classes of number fields of degree n must be finite. Perlis, Szymiczek, Conner, and Litherland showed that a Witt ring isomorphism produces a family of local isomorphisms called a reciprocity equivalence; these come in two types, tame and wild. Using the new conditions, one can prove that there always exists a Witt ring isomorphism which is tame at the place P, with finitely many exceptions.

Reviewer: J.P.Carpenter (Ruston)

### MSC:

11E81 | Algebraic theory of quadratic forms; Witt groups and rings |

11R99 | Algebraic number theory: global fields |

11R58 | Arithmetic theory of algebraic function fields |

### Keywords:

global fields; Witt rings; number fields; function fields; Witt equivalence problem; small equivalence; level; Witt ring isomorphism### References:

[1] | [B-M] Baeza, R., Moresi, R.: On the Witt-equivalence of fields of characteristic 2. J. Algebra92, 446–453 (1985) · Zbl 0553.10016 |

[2] | [Ca] Carpenter, J.: Finiteness theorems for forms over number fields. Dissertation, LSU, Baton Rouge, LA (1989) |

[3] | [Cz] Czogala, A.: On reciprocity equivalence of quadratic number fields. Acta. Arith. (to appear) |

[4] | [I-R] Ireland, K., Rosen, M.: A classical introduction to modern number theory, New York: Springer 1982 · Zbl 0482.10001 |

[5] | [K] Kaplansky, I.: Linear algebra and geometry. New York: Chelsea 1974 · Zbl 0294.17003 |

[6] | [M] Marcus, D.: Number fields. New York: Springer 1977 · Zbl 0383.12001 |

[7] | [P] Palfrey, T.: Density theorems for reciprocity equivalences. Dissertation, LSU, Baton Rouge, LA (1989) · Zbl 0923.11066 |

[8] | [P-S-C-L] Perlis, R. Szymiczek, K., Conner, P.E., Litherland, R.: Matching Witts with global fields. (Preprint) · Zbl 0807.11024 |

[9] | [Ws] Weiss, E.: Algebraic number theory, New York: Chelsea 1976 |

[10] | [W1] Weil, A.: Basic Number Theory. New York: Springer 1974 |

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