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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.

MSC:

11E81 Algebraic theory of quadratic forms; Witt groups and rings
11R99 Algebraic number theory: global fields
11R58 Arithmetic theory of algebraic function fields
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References:

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