The exceptional zero phenomenon has been widely studied in the realm of $p$-adic $L$-functions, where the starting point lies in the foundational works of Ferrero–Greenberg, Katz, Gross, or Mazur–Tate–Teitelbaum, among others. This phenomenon also appears in the study of Euler systems, and in this case one is led to study higher order derivatives of cohomology classes in order to extract the arithmetic information which is usually encoded in the explicit reciprocity laws which make the connection with $p$-adic $L$-funtions. In this work, we focus on the elliptic units of an imaginary quadratic field and study this exceptional zero phenomenon, proving an explicit formula relating the logarithm of a derived elliptic unit either to special values of Katz’s two variable $p$-adic $L$-function or to its derivatives. Further, we interpret this fact in terms of an $\mathcal L$-invariant and relate this result to other approaches to the exceptional zero phenomenon, most notably to the work of Bley, and we also compare this setting with other scenarios concerning Heegner points and Beilinson–Flach elements.

中文翻译：

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椭圆单位的异常零现象

异常零现象已在 $p$-adic $L$-函数领域得到广泛研究，其起点在于 Ferrero-Greenberg、Katz、Gross 或 Mazur-Tate-Teitelbaum 等人的基础工作. 这种现象也出现在欧拉系统的研究中，在这种情况下，人们需要研究上同调类的高阶导数，以提取通常编码在与 $p$ 相关的显式互易律中的算术信息-adic $L$-功能。在这项工作中，我们专注于虚二次场的椭圆单位并研究这种特殊的零现象，证明了一个与导出的对数相关的明确公式椭圆单位要么是 Katz 的两个变量 $p$-adic $L$-function 的特殊值，要么是它的导数。此外，我们用 $\mathcal L$ 不变量来解释这个事实，并将这个结果与其他处理异常零现象的方法联系起来，最显着的是 Bley 的工作，我们还将这个设置与其他关于 Heegner 点的场景进行了比较和Beilinson-Flach 元素。