A proof of Sugawara's conjecture on Hasse-Weber ray class invariants

TL;DR

This paper proves Sugawara's 1936 conjecture that the ray class field of an imaginary quadratic field is generated by a specific division value of the tau-function, advancing the understanding of complex multiplication.

Contribution

It provides a rigorous proof of Sugawara's conjecture, establishing a new link between ray class fields and special values of the tau-function.

Findings

Proof of Sugawara's conjecture from 1936

Ray class field generated by a single tau-division value

Enhances understanding of complex multiplication and class field theory

Abstract

In this paper a proof is given of Sugawara's conjecture from 1936, that the ray class field of conductor $\mathfrak{f}$ over an imaginary quadratic field $K$ is generated over $K$ by a single primitive $\mathfrak{f}$-division value of the $\tau$-function, first defined by Weber and then modified by Hasse in his 1927 paper giving a new foundation of complex multiplication.