Covering data and higher dimensional global class field theory

TL;DR

This paper constructs a reciprocity homomorphism for certain schemes over Spec(Z), linking algebraic data to the abelianized fundamental group, extending class field theory to higher dimensions.

Contribution

It introduces an explicit reciprocity homomorphism for regular schemes over Spec(Z), generalizing class field theory to higher-dimensional schemes.

Findings

Constructed a surjective reciprocity homomorphism _X: C_X bb _1^{ab}(X)

Kernel of _X is the connected component of the identity

Results extend to smooth varieties over finite fields

Abstract

For a connected regular scheme X, flat and of finite type over Spec(Z), we construct a reciprocity homomorphism \rho_X: C_X --> \pi_1^\ab(X), which is surjective and whose kernel is the connected component of the identity. The (topological) group C_X is explicitly given and built solely out of data attached to points and curves on X. A similar but weaker statement holds for smooth varieties over finite fields. Our results are based on earlier work of G. Wiesend.