On a Conjecture for Rubin-Stark Elements in a Special Case

Abstract

A conjecture on a relation between two different Rubin-Stark elements was recently proposed by the author, and also by Mazur and Rubin. For a tower of finite extensions of global fields $K/L/k$ such that $K/k$ is abelian, this conjecture gives a relation between Rubin-Stark elements $\varepsilon_{K,S,T,V}$ and $\varepsilon_{L,S,T,V'}$, where $S, T, V$ and $V'$ are suitable sets of places of $k$. In this paper, we prove this conjecture under the following three assumptions: (i) $V$ contains all infinite places of $k$; (ii) all $v\in S$ split completely in $L$; (iii) $\mathrm{Gal}(K/L)$ is the direct product of the inertia groups at $v\in S\setminus V$.