Partial zeta values, Gross's tower of fields conjecture, and Gross-Stark units
Let \(G\) be the Galois group of a finite abelian extension \(K/F\) of a totally real field, and let \(S,T\) be disjoint sets of finite places of \(F\), all places in \(S\) ramified in \(K\). Define a \(C[G]\)-valued function \(\zeta_{S,T}(K/F;s)\) by \[ \zeta_{S,T}(K/F;s) = \prod_{\mathfrak q\in T}(1-N(\mathfrak q))^{1-s}\left[\sigma_{\mathfrak q}^{-1}\right]\sum_{\sigma\in G}\zeta_S(\sigma,s)\left[\sigma^{-1}\right], \] where \(\sigma_{\mathfrak q}\) is the value of \(\mathfrak q\) under the Artin map and \[ \zeta_S(\sigma,s) = \sum_{(\mathfrak a,S)=1,\sigma_{\mathfrak q}=\sigma}\frac{1}{N(\mathfrak a)^s}. \] Since it has been shown by \textit{P. Cassou-Nogues} [Invent. Math. 51, 29--59 (1979; Zbl 0408.12015)] and \textit{P. Deligne} and \textit{K. A. Ribet} [Invent. Math. 59, 227--286 (1980; Zbl 0434.12009)] that for integers \(k\le 0\) one has \(\zeta_{S,T}(\sigma,k)\in \mathbb Z\), the Stickelberger element \[ \zeta_{S.T}(K/F,k)=\sum_{\sigma\in G}\zeta_{S,T}(\sigma,k)\left[\sigma^{-1}\right] \] lies in \(\mathbb Z[G]\). It has been conjectured by \textit{B. H. Gross} [J. Fac. Sci., Univ. Tokyo, Sect. I A 35, No. 1, 177--197 (1988; Zbl 0681.12005)] that if \(F\subset E\subset K\), \(H=\mathrm{Gal}(E/F)\), \(E\) is not totally real and \(I\) is the kernel of \(\mathbb Z[G]\longrightarrow \mathbb Z[H]\), then \[ \zeta_{S,T}(K/F,0)\in I^r, \] and predicted also a stronger result for totally real \(E\). The author establishes this conjecture (Theorem 1) in the case when \(E\) is not totally real, and in the remaining case he proves \[ \zeta_{S,T}(K/F,0)\in I^{r-1},\quad 2\zeta_{S,T}(K/F,0)\in I^r. \] To prove it the author studies a version of the Eisenstein cocycle lying in a cohomology group of \(\mathrm{GL}_{\deg F}(\mathbb Q)\). In an earlier paper [Duke Math. J. 143, 225--279 (2008; Zbl.1235.11102)], the first author presented a conjectural construction of a unit \(u\) (the \textit{Gross-Stark unit}) (see the Conjecture 7.6. in [Gross, loc. cit.], and now the Eisenstein cycle is applied to simplify this construction.
scientific article; zbMATH DE number 6965901