L-series and homomorphisms of number fields

TL;DR

This paper explores how $L$-series associated with Dirichlet characters can uniquely determine and relate number fields, establishing a correspondence between field homomorphisms and group homomorphisms of character groups under certain conditions.

Contribution

It extends the understanding of the relationship between number fields and their $L$-series by characterizing homomorphisms via $l$-torsion subgroup isomorphisms with divisibility conditions.

Findings

Number fields are determined by their $L$-series of Dirichlet characters.

Isomorphisms between number fields correspond to $L$-series preserving isomorphisms of $l$-torsion subgroups.

Homomorphisms between number fields correspond to group homomorphisms of $l$-torsion subgroups under divisibility conditions.

Abstract

While the zeta function does not determine a number field uniquely, the $L$-series of a well-chosen Dirichlet character does. Moreover, isomorphisms between two number fields are in natural bijection with $L$-series preserving isomorphisms of $l$-torsion subgroups of the Dirichlet character groups. We extend this by showing that homomorphisms between number fields are in natural bijection with group homomorphisms between $l$-torsion subgroups of the Dirichlet character groups abiding a divisibility condition on the $L$-series when $l$ is sufficiently large.