Higher-dimensional Contou-Carrère symbol, I

The talk is based on a common work with Denis Osipov. Contou-Carrére symbol in dimension $n$ is a way to construct an invertible element of an arbitrary commutative ring $A$ from $n+1$ Laurent series in $n$ variables over $A$. This symbol arises when considering families of $n$-dimensional varieties and chains of irreducible subvarieties on them. The higher-dimensional Contou-Carrere symbol satisfies many fundamental properties, among them, a higher-dimensional reciprocity law, which implies basically all known reciprocity laws. In our survey talk, we will discuss all these phenomena starating from the Weil reciprocity law on a curve.