On the Schertz conjecture

TL;DR

This paper provides a conditional proof of Schertz's conjecture, asserting that finite abelian extensions of imaginary quadratic fields can be generated by Siegel-Ramachandra invariants, using class group characters and the Kronecker limit formula.

Contribution

The paper offers the first conditional proof of Schertz's conjecture leveraging advanced number theory tools.

Findings

Conditional proof of Schertz's conjecture established

Connection between class group characters and field generation demonstrated

Utilization of the second Kronecker limit formula in the proof

Abstract

Schertz conjectured that every finite abelian extension of imaginary quadratic fields can be generated by the norm of the Siegel-Ramachandra invariants. We shall present a conditional proof of his conjecture by means of the characters on class groups and the second Kronecker limit formula.