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Levels of Function Fields of Surfaces over Number Fields

✍ Scribed by U. Jannsen; R. Sujatha

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
87 KB
Volume
251
Category
Article
ISSN
0021-8693

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✦ Synopsis

We study the level of nonformally real function fields of surfaces over number fields and show that it is at most 4 for a large class of surfaces.  2002 Elsevier Science (USA)

The level of a field F is the least integer n such that -1 is expressible as a sum of n squares in F. If -1 is not a sum of squares in F, then the level is infinite. A field is nonformally real (resp. formally real) if it has finite (resp. infinite) level. The level of any field is known to be a power of 2, by results of Pfister [Sch, Chap. II, Theorem 10.8].

Let X be a geometrically integral variety over a number field k and let k X be its function field. If k X is nonformally real with tr β€’ deg k k X = 2 (resp. 1), then it is known (cf. [K, Appendix, J, CTJ]) that level k X ≀ 8 (resp. ≀4). It is easy to produce curves over k whose function fields are nonformally real with levels 2, 4. In this article, we study the level of nonformally real function fields of surfaces over number fields. We prove that the level of the function field is at most 4 for a class of 350

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