Hasse norm principle for $M_{11}$ and $J_1$ extensions

Abstract

Let k be a field and T be an algebraic k-torus. In 1969, over a global field k, Voskresenskii proved that there exists an exact sequence 0 → A(T ) → H 1 (k, Pic X) ∨ → X(T ) → 0 where A(T ) is the kernel of the weak approximation of T , X(T ) is the Shafarevich-Tate group of T , X is a smooth k-compactification of T , Pic X is the Picard group of X = X × k k and ∨ stands for the Pontryagin dual. On the other hand, in 1963, Ono proved that for the norm one torus T = R(1) K/k ( m) of K/k, X(T ) = 0 if and only if the Hasse norm principle holds for K/k. We determine H 1 (k, Pic X) for norm one tori T = R(1) K/k ( m) when the Galois group Gal(L/k) of the Galois closure L/k of K/k is isomorphic to the Mathieu group M 11 of degree 11. We also give a necessary and sufficient condition for the Hasse norm principle for such extensions K/k with Gal(L/k) ≃ M 11 .