A note on Shintani's invariants

TL;DR

This paper presents new expressions for Shintani's invariants, which are conjectured to generate abelian extensions of real quadratic fields, by generalizing Yamamoto's observation relating these invariants to the q-Pochhammer symbol.

Contribution

It introduces generalized formulas for Shintani's invariants, connecting them to the q-Pochhammer symbol, advancing understanding of their structure and potential role in class field theory.

Findings

Derived new formulas expressing Shintani's invariants via q-Pochhammer symbols

Extended Yamamoto's observation to a broader class of invariants

Provides insights into the algebraic properties of Shintani's invariants

Abstract

Shintani's celebrated invariants are conjectured to generate abelian extensions of real quadratic number fields, offering a potential solution to Hilbert's 12th problem in that setting. In this note, we derive new expressions for Shintani's invariants by generalizing an observation of Yamamoto, who showed that these invariants - originally formulated using the double sine function - can be expressed in terms of the q-Pochhammer symbol.