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

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 $τ$-function, first defined by Weber and then modified by Hasse in his 1927 paper giving a new foundation of complex multiplication.