Geometric families of multiple elliptic Gamma functions and arithmetic applications, I

Abstract

This is the first paper in a series where we study arithmetic applications of the multiple elliptic Gamma functions originated from mathematical physics. The main purpose of this paper is the introduction of a framework for applications of these functions to Hilbert's 12th problem for general number fields with exactly one complex place following recent work by Bergeron, Charollois and Garc\'ia. Namely, we define geometric families of the multiple elliptic Gamma functions, upgrading the construction carried out by Felder, Henriques, Rossi and Zhu for rank $3$ lattices to lattices of higher ranks. These functions enjoy transformation properties under an action of the special linear group $\mathrm{SL}_n(\mathbb{Z})$ for $n \geq 2$ involving some Bernoulli rational functions as their so-called modularity defect. A second purpose of this paper is to use this collection of Bernoulli rational functions to construct $(n-1)$-cocycles for specific subgroups of $\mathrm{SL}_n(\mathbb{Z})$ associated to units groups in totally real number fields and use these cocycles to compute partial zeta values at $s=0$.