Witt groups and torsion Picard groups of smooth real curves

The Witt group of smooth real projective curves was first computed by Knebusch in [Kn]. If the curve is not complete but still smooth, the Witt group is also studied in [Kn] but not explicitely calculated. However, for some precise examples of smooth affine curves, we may find explicit calculations ([Knu] and [Ay-Oj]). In this paper, the Witt group of a general smooth curve is explicitely calculated in terms of topological and geometrical invariants of the curve. My method is strongly inspired by Sujatha’s calculation of the Witt group of a smooth projective real surface and uses a comparison theorem between the graded Witt group and the tale cohomology groups established in [Mo]. In the second part of the paper, we are interested in the torsion subgroup of the Picard group (denoted by Pictors(X)) of a smooth geometrically connected (non complete) curve X over a real closed field R. Let C be the algebraic closure of R and XC := X ×SpecR SpecC. We compute Pictors(X) and Pictors(XC) using the Kummer exact sequence for tale cohomology. These calculations depend on a new invariant η(X) ∈ N (resp. η(XC)) which we introduce in this note. We study relations between η(X), η(XC) and the level and Pythagoras number of curves using new results of Huisman and Mah [Hu-Ma]. The last part is devoted to the study of smooth affine hyperelliptic curves. For such curves we calculate the Witt group and the torsion Picard group determining the invariant η.