Ivan Panin (PDMI).

"On a conjecture of Colliot-The'l`ene concerning quadratic spaces"

Abstract: Let X be an affine smooth complex algebraic variety and let q be a quadratic space over the ring of regular function on X. Let u be an invirtible regular function on X. Assume that q represents u over the rational functions on X. We will prove that for any point x in X the space q represents the function u in the local ring O_{X,x} of the point x. This solves in affirmative the conjecture of Colliot-The'l`ene mentioned in the title.