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

Abstract

Let K be a complete discretely valued field with residue field κ and F the function field of a curve over K. Let p be the characteristic of κ and ℓ a prime not equal to p. If the Brauer ℓ-dimensions of all finite extensions of κ are bounded by d and the Brauer ℓ-dimensions of all extensions of κ of transcendence degree at most 1 are bounded by d+1, then it is known that the Brauer ℓ-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 κ. As an application, we show that if κ 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 $H^{3}(F,\mathbb{Z}/2\mathbb{Z})$ is a symbol. If κ 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).