A remark on elliptic units
Let \(p\equiv 3(4)\) be a prime \(>3\) and let \(K={\mathbb{Q}}(\sqrt{-p})\). Let H be the absolute class field of K. In a paper published in 1982, \textit{B. H. Gross} [Prog. Math. 26, 219-236 (1982; Zbl 0506.14040)] defined units \(u_{\sigma}(\sigma \in Gal(H/K))\) in a class field of HT of a CM-field T containing K and asked a question regarding a property of these units. In the present paper, the author gives an explicit method of calculating \(u_{\sigma}\) following a result of \textit{G. Robert} [Ann. Sci. Ecole Norm. Sup., IV. Sér. 11, 297-389 (1978; Zbl 0409.12008)]. An interesting concrete example (the case \(p=23)\) is considered in detail at the end, where the answer to Gross' question is seen to be in the affirmative in this case.
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