On \(p\)-adic absolute CM-periods. II
Let \(F\) be a totally real number field and \(K\) be a CM-field that is abelian over \(F\). For a prime \(p\), fix an embedding \(F\) into \(\mathbb C_p\) and let \(\mathfrak p\) be the prime ideal corresponding to this embedding. For \(\tau\in\mathrm{Gal}(K/F)\), the \(p\)-adic absolute CM-period symbol is defined by \[ lg_{p,K/F}(\mathrm{id},\tau):=-\frac{\mu(\tau)}{2h_0}\log_p\alpha_0 +\frac1{|G|}\sum_{\chi\in\widehat G_-}\frac{\chi(\tau)}{L(0,\chi)} \sum_{c\in\mathfrak C_{\mathfrak f(\chi)\mathfrak p}}\chi(c)X_p(c). \] Here \(h_0\) is the narrow class number of \(F\), \(\alpha_0\) is a totally positive element of \(F\) such that \(\mathfrak p^{h_0}=(\alpha_0)\), \(\mu(\tau)=1,-1,0\) if \(\tau=\mathrm{id}\), complex conjugation, or other elements of \(\mathrm{Gal}(K/F)\), respectively, \(G=\mathrm{Gal}(K/F)\), \(\widehat G_-\) denotes the set of all odd characters of \(G\), \(\mathfrak f(\chi)\) is the finite part of the conductor of \(\chi\), \(\mathfrak C_{\mathfrak f(\chi)\mathfrak p}\) denotes the ideal class group of \(F\) modulo \(\mathfrak f(\chi)\mathfrak p\) times the product of all archimedean primes, \(X_p(c)\) is a certain class invariant constructed using the division values of the \(p\)-adic logarithmic multiple gamma function. In the first part of the work [Am. J. Math. 130, No. 6, 1629--1685 (2008; Zbl 1233.11066)], the authors presented a conjecture that predicts the exact value of this symbol in the completely split cases and provided some supporting evidences. In the paper under review, which constitutes the second part of the work, the authors study properties of the symbols in general cases and clarify their relations to \(p\)-adic periods.
scientific article; zbMATH DE number 5551351