Relative Tate-Shafarevich kernel of a reductive group over a global field

Abstract

We describe an algorithm and a computer program computing the relative Tate-Shafarevich kernel $Sha^S(K,G)$ of a reductive group $G$ over a global field $K$ with respect to a finite set $S$ of places of $K$ containing no archimedean places. We assume that we know the algebraic fundamental group $Λ=π_1^{alg}(G)$, the Galois group $Θ=Gal(L/K)$ acting on $Λ$, where $L$ is a finite Galois extension of $K$ having no real places, the decomposition subgroups $Θ_v\subseteq Θ$ for all places $v$ in $S$, and all non-cyclic decomposition subgroups $Θ_v$ for $v$ outside $S$. We also compute $Sha^S(K,G)$ in some cases where $S$ contains archimedean places.