Computations of higher elliptic units

TL;DR

This paper proposes a conjecture for constructing generalized elliptic units over certain number fields using multiple elliptic Gamma functions, aiming to address Hilbert's 12th problem.

Contribution

It extends previous work by formulating a new conjecture for elliptic units in number fields with one complex place, based on multivariate elliptic Gamma functions.

Findings

Supports conjecture with examples in cubic, quartic, and quintic fields.

Extends previous constructions to more general number fields.

Provides evidence for elliptic units generating specific abelian extensions.

Abstract

In this paper we present a conjecture on the construction of generalised elliptic units above number fields with exactly one complex place. These elliptic units obtained as values of multiple elliptic Gamma functions. These form a collection of multivariate meromorphic functions which were studied in the late 1990s and early 2000s in mathematical physics. Our construction extends the scheme of a recent article by Bergeron, Charollois and Garc\'ia where they constructed conjectural elliptic units above complex cubic fields using the elliptic Gamma function. The elliptic units we construct are expected to generate specific abelian extensions of the base field where they are evaluated, thus giving a conjectural solution to Hilbert's 12th problem for the number fields with exactly one complex place. We provide several examples to support our conjecture in optimal cases for cubic, quartic and quintic fields.