Period-index and u-invariant questions for function fields over complete discretely valued fields

• 成果类型: Article
• 署名作者: Parimala, R.; Suresh, V.
• 署名单位: Emory University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-013-0483-y
• 发表日期: 2014
• 页码: 215-235
• 关键词: galois cohomology; quadratic-forms; CURVES

摘要

Let K be a complete discretely valued field with residue field kappa and F the function field of a curve over K. Let p be the characteristic of kappa and a a prime not equal to p. If the Brauer a-dimensions of all finite extensions of kappa are bounded by d and the Brauer a-dimensions of all extensions of kappa of transcendence degree at most 1 are bounded by d+1, then it is known that the Brauer a-dimension of F is at most d+2 (Lieblich in J. Reine Angew. Math. 659:1-41, 2011; Saltman in J. Ramanujan Math. Soc. 12:25-47, 1997; Harbater et al. in Invent. Math. 178:231-263, 2009). In this paper we give a bound for the Brauer p-dimension of F in terms of the p-rank of kappa. As an application, we show that if kappa is a perfect field of characteristic 2, then any quadratic form over F in at least 9 variables is isotropic. This leads to the fact that every element in is a symbol. If kappa is a finite field of characteristic 2, u(F)=8 is a result of Heath-Brown/Leep (Heath-Brown in Compos. Math. 146:271-287, 2010; Leep in J. Reine Angew. Math., 2013, to appear).