AI 中文总结
本文利用平滑椭圆 Gamma 函数的上循环关系，证明了复三次域椭圆单位猜想解析表达式不依赖于挠点 $h$ 的选择，从而支持了显式类域论框架下的猜想。
AI 中文摘要
在 Bergeron-Charollois-García 最近的工作中，提出了复三次域椭圆单位的猜想解析表达式，即作为椭圆 Gamma 函数平滑商的值。该复数依赖于导子理想、射线类群元素、平滑理想以及一个“挠点”$h$。若此解析函数要纳入显式类域论的框架，则如同 CM 椭圆曲线上的挠点情形，我们对 $h$ 的选择应仅依赖于其模某个格 $L$ 的同余类。该猜想已得到大量数值证据支持。本文利用平滑椭圆 Gamma 函数满足的上循环关系，证明了该构造确实不依赖于 $h$。
英文摘要
In the recent work of Bergeron-Charollois-Garc\'ıa \cite{BCG23}, a conjectural analytic expression for elliptic units for complex cubic fields was proposed: namely, as the value of a smoothed quotient of elliptic Gamma functions. The complex number depended on the conductor ideal, ray class group element, smoothing ideal, and a `torsion point' $h$. If this analytic function is to fit into the framework of explicit class field theory, then, as in the case of torsion points on CM elliptic curves, our choice of $h$ should only depend on its congruence class modulo some lattice $L$. This conjecture has been supported by much numerical evidence. In this paper, we use the cocycle relations satisfied by the smoothed elliptic Gamma function to prove that the construction is indeed independent of $h$.
Comments57 pages, comments welcome!
