Upper bounds for the Euclidean minima of abelian fields (Q5963340)

Let \(K\) be a number field with ring of integers \({\mathcal O}_K\). The Euclidean minimum \(M(K)\) is defined as NEWLINE\[NEWLINE M(K) = \sup \inf N(x-c), NEWLINE\]NEWLINE where the supremum is over all \(x\in K\), the infimum over all \(c \in {\mathcal O}_K\), and where \(N\) is the absolute value of the norm map. It is easy to see that \(K\) is Euclidean with respect to the norm if \(M(K) < 1\). The central conjecture about Euclidean minima is Minkowski's conjecture that \(M(K) \leq 2^{-n} \sqrt{D}\) for totally real fields \(K\) with degree \(n\) and discriminant \(D\). It is known that Minkowski's conjecture is true for fields with degree \(n \leq 8\).NEWLINENEWLINE NEWLINEThe main result of this article is that Minkowski's conjecture is true for all totally real abelian fields with prime power conductor \(p^r\); the case of odd primes \(p\) had already been taken care of by \textit{E. Bayer-Fluckiger} and \textit{P. Maciak} in [Math. Ann. 357, No. 3, 1071--1089 (2013; Zbl 1278.11098)]. The last section presents a brief survey on Euclidean minima of cyclotomic fields and their maximal real subfields.