Elliptic Units Above Fields With Exactly One Complex Place

TL;DR

This paper investigates the construction of abelian extensions of number fields with exactly one complex place using special values of multivariate elliptic Gamma functions, proposing a conjecture about algebraic units related to Stark units.

Contribution

It introduces geometric variants of elliptic Gamma functions with transformation properties under SL_d(Z) and conjectures their special values produce algebraic units in abelian extensions.

Findings

Numerical evidence supports the conjecture for cubic, quartic, and quintic fields.

Construction of functions with SL_d(Z) symmetry properties.

Proposal that these special values yield algebraic units related to Stark units.

Abstract

In this work we explore the construction of abelian extensions of number fields with exactly one complex place using multivariate analytic functions in the spirit of Hilbert's 12th problem. To this end we study the special values of the multiple elliptic Gamma functions introduced in the early 2000s by Nishizawa following the work of Felder and Varchenko on Ruijsenaars' elliptic Gamma function. We construct geometric variants of these functions enjoying transformation properties under an action of $\mathrm{SL}_{d}(\mathbb{Z})$ for $d \geq 2$. The evaluation of these functions at points of a degree $d$ field $\mathbb{K}$ with exactly one complex place following the scheme of a recent article by Bergeron, Charollois and Garc\'ia (arXiv:2311.04110) seems to produce algebraic numbers. More precisely, we conjecture that such infinite products yield algebraic units in abelian extensions of $\mathbb{K}$ related to conjectural Stark units and we provide numerical evidence to support this conjecture for cubic, quartic and quintic fields.