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

TL;DR

This work develops and analyzes smoothed, geometric families of multivariate elliptic Gamma functions to construct partial modular symbols for congruence subgroups $\Gamma_0(N,n)\subset SL_n(\mathbb{Z})$ with $n\ge 2$. The authors show that smoothing converts modular defects into cocycle relations and prove that the resulting smoothed Bernoulli rational functions are rational with denominators uniformly bounded by $\mathcal{D}(N,n)=\prod_{p|N} p^{\lfloor n/(p-1)\rfloor}$, by expressing them via traces of cyclotomic units. They provide explicit cone and sign analyses to derive a cocycle formula for the smoothed $G_{n-2}$-functions, and establish a cohomological interpretation in terms of coboundaries and cocycles, which connects to higher Dedekind sums and the theory of elliptic units. The results pave the way for interpreting these smoothed objects as partial modular symbols on tori arising from totally positive units in number fields, contributing to a framework aimed at generalizing elliptic units and Hilbert-type constructions in the broader setting of higher rank groups and one complex place number fields.

Abstract

This is the second paper in a series where we study arithmetic applications of the multiple elliptic Gamma functions originated in mathematical physics. In the first article in this series we defined geometric families of these functions and proved that these families satisfied coboundary relations involving an attached collection of Bernoulli rational functions. The main purpose of the present paper is to show that smoothed versions of our geometric elliptic Gamma functions give rise to partial modular symbols for congruence subgroups of $\mathrm{SL}_{n}(\mathbb{Z})$ for $n \geq 2$ which restrict to $(n-2)$-cocycles on tori in $\mathrm{SL}_{n}(\mathbb{Z})$ coming from groups of totally positive units in number fields. To achieve this, we show that the associated smoothed Bernoulli rational functions reduce to smoothed higher Dedekind sums with uniformly bounded denominators.