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

TL;DR

This work develops a geometric framework for the arithmetic applications of the multi-variable elliptic Gamma family by constructing geometric $G_{n-2, a_1, \dots, a_{n-1}}$ and Bernoulli rational functions $B_{n, a_1, \dots, a_n}$ that transform under $\mathrm{SL}_n(\mathbb{Z})$ with a modular defect. It proves modularity and cocycle relations through higher Bernoulli polynomials, and defines (n-1)-cocycles $\phi_{n,a}$ for unit subgroups, providing a bridge from analytic structures to arithmetic invariants. The framework then expresses partial zeta values at $s=0$ for totally real fields as traces in the field, via Shintani-type cone decompositions and Bernoulli data, with explicit cubic-field examples illustrating the method. Overall, the paper lays a comprehensive groundwork for constructing arithmetic invariants and elliptic-unit-type objects in totally real fields, aiming to advance Hilbert’s 12th problem in higher degree contexts.

Abstract

This is the first paper in a series where we study arithmetic applications of the multiple elliptic Gamma functions originated from theoretical 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 García. 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 $s=0$.