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

TL;DR

This paper explores the arithmetic properties of geometric families of multiple elliptic Gamma functions, demonstrating their connection to modular symbols and Dedekind sums in number theory.

Contribution

It introduces smoothed versions of elliptic Gamma functions that produce partial modular symbols and relates Bernoulli functions to Dedekind sums with bounded denominators.

Findings

Smoothed elliptic Gamma functions yield partial modular symbols.

Bernoulli rational functions reduce to higher Dedekind sums.

Bounded denominators in Dedekind sums are established.

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.