Computations of higher elliptic units in optimal settings

Abstract

In this paper we present a simplified form of a conjecture on the construction of generalised elliptic units above number fields with exactly one complex place. They are conjectural algebraic numbers which are obtained as special values of higher elliptic Gamma functions. These functions 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 García where they constructed conjectural elliptic units above complex cubic fields using the elliptic Gamma function. The higher 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 settings for number fields of degree 3, 4, 5 and 6.

24 pages + references. 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 arXiv:2406.06094 (see arXiv:2510.16515 and arXiv:2602.06561 for the first two papers in this series). Changes: better clarity in the presentation