发表机构
Sorbonne Université; Université Paris Cité; CNRS; INRIA; IMJ-PRG（索邦大学; 巴黎西岱大学; 法国国家科学研究中心; 法国国家信息与自动化研究所; 雅克-路易·利翁斯数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文作为高阶椭圆伽马函数算术应用系列第三篇，升级了前人的椭圆单位构造，推测仅含一个复 place 的数域上的高阶椭圆单位为代数单位并满足克罗内克极限公式，且通过多次数域示例验证该推测。
AI 中文摘要
本文是研究高阶椭圆伽马函数算术应用系列论文的第三篇。在该系列的前两篇文章中，我们定义了这些函数的几何族，并证明它们在特殊线性群及其同余子群作用下满足上同调边界与上闭链关系。本文的主要目的是构造仅含一个复 place 的数域上的推测性高阶椭圆单位，将其作为高阶椭圆伽马函数的特殊值，这是对 Bergeron、Charollois 和 García 针对复三次域所做构造的升级。这些高阶椭圆单位通过将由高阶椭圆伽马函数构造的乘法 (n-2)-上闭链，与仅含一个复 place 的 n 次数域的某个全正单位群相关联的 (n-2)-闭链配对得到。我们推测这些高阶椭圆单位是代数单位，属于其被求值的基域的指定阿贝尔扩张，且满足克罗内克极限公式，该公式将其模的对数与基域上 s=0 处部分 ζ 函数的导数值相关联。我们针对次数为 3、4、5 的各类数域示例展示了我们的推测。
英文摘要
This is the third paper in a series where we study arithmetic applications of the higher elliptic Gamma functions. In the first two articles in this series, we defined geometric families of these functions and proved that they satisfy coboundary and cocycle relations under the action of special linear groups and associated congruence subgroups. The main purpose of the present paper is to present a construction of conjectural higher elliptic units above number fields with exactly one complex place as special values of higher elliptic Gamma functions, upgrading the construction carried out by Bergeron, Charollois and García for complex cubic fields. These higher elliptic units are obtained by evaluating a multiplicative $(n-2)$-cocycle built from higher elliptic Gamma functions against a $(n-2)$-cycle associated to some group of totally positive units of a given number field of degree $n$ with exactly one complex place. We conjecture that these higher elliptic units are algebraic units which belong to prescribed abelian extensions of the base field where they are evaluated and that they satisfy a Kronecker limit formula which relates the logarithm of their modulus to values of derivatives of partial zeta functions at $s = 0$ in the base field. We showcase our conjecture on various examples for number fields of degree 3, 4 and 5.
Comments52 pages plus references. This is the third paper in a series of 3 which supersede arXiv:2406.06094. The first articles in this series are arXiv:2510.16515 and arXiv:2602.06561
