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We give upper bounds for the level and the Pythagoras number of function fields over fraction fields of integral Henselian excellent local rings. In particular, we show that the Pythagoras number of $mathbb{R}((x_1,dots,x_n))$ is $leq 2^{n-1}$, which answers positively a question of Choi, Dai, Lam and Reznick.
This is the final version, to appear in Commentarii Mathematici Helvetici.
We give a definition of Cox rings and Cox sheaves for varieties over nonclosed fields that is compatible with torsors under quasitori, including universal torsors. We study their existence and classification, we make the relation to torsors precise, and we present arithmetic applications.
These are notes of my lectures at the summer school Higher-dimensional geometry over finite fields in Goettingen, June--July 2007. We present a proof of Tates theorem on homomorphisms of abelian varieties over finite fields (including the $ell=p$ c
In this paper we construct a noncommutative space of ``pointed Drinfeld modules that generalizes to the case of function fields the noncommutative spaces of commensurability classes of Q-lattices. It extends the usual moduli spaces of Drinfeld module
We compute the Chow group of zero-cycles on certain Ch{^a}telet surfaces over local fields.