Computations of higher elliptic units

TL;DR

Morain proposes a conjectural framework to construct generalized elliptic units over number fields with exactly one complex place by evaluating smoothed higher elliptic Gamma functions $G_{n-2}$ at carefully chosen field points. The approach extends the Bergeron–Charollois–García construction for complex cubic fields and posits that products of $G_{n-2}$-values produce algebraic units in the narrow ray class field $ ext{K}^+(rak f)$, obeying a Kronecker-type limit relation with partial zeta derivatives. The paper develops a computational toolkit for evaluating $G_r$, defines optimality criteria to minimize the product size, and provides substantial numerical evidence in degrees $n=3,4,5$ (and beyond) using fields from the LMFDB, including detailed cubic, quartic, and quintic examples. This work advances explicit class field theory and Hilbert’s 12th problem by offering an analytic construction of abelian extensions via higher elliptic functions and by tying the results to Stark-type conjectures through smoothed units and Kronecker-type relations.

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 García 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.