  In this work we explore the construction of abelian extensions of 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 the elliptic Gamma function of
Ruijsenaars. 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.
