**Ivan Panin**

Petersburg Department of Steklov Institute of Math

December 5, 2013

**On a conjecture of Colliot-Th'el`ene concerning quadratic spaces**:
Let $X$ be an affine smooth complex algebraic variety and let $(V,q)$
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-Th'el`ene mentioned in the title.