Computations of higher elliptic units
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.
Cite
@article{arxiv.2601.11961,
title = {Computations of higher elliptic units},
author = {Pierre L. L. Morain},
journal= {arXiv preprint arXiv:2601.11961},
year = {2026}
}
Comments
20 pages. Submitted. This paper presents a simplified version of a conjecture presented in the series of papers "Geometric families of multiple elliptic Gamma functions and applications" which supersedes the working paper arXiv:2406.06094 (see arXiv:2510.16515 for the first paper in this series). In this paper, we focus on optimal examples in the conjecture and how we find them algorithmically