Hasse principle for a group $\DeclareMathOperator\PSL{PSL}$In the paper Ono - "Hasse principle" for $\PSL_2 (\mathbb Z)$ and $\PSL_2(\mathbb F)$ there's a definition of a Hasse principle for a group $G$, but I don't completely get it. Is there a more motivated reformulation of this definition?
Why I am interested: local–global principles are often very interesting in arithmetic geometry, so when I noticed a paper with this title I looked at it to see whether this proves something geometric.
As said below, one can formulate a problem of computing a group $\mathrm{Sha}$ defined by a $g$-module $G$ and a family of subgroups $h_i\in G$, but the actual computation in the paper is for a specific choice of $h_i$, and I can't parse if there is an application of interest. Is it so?
(I suspect this problem arises when you try to prove Hasse principle for equations, like $x^n = a$ but with different Galois groups, see Ono and Terasoma - On Hasse principle for $x^n = a$, though the results there have a mistake, corrected in the next one)
 A: Given a group $g$, a possibly nonabelian $g$-module $G$, and a family of subgroups $(h_i)_{i \in I}$ of $g$, Ono defines a pointed Shafarevich-Tate set, say $(S,0)$.  He says the Hasse Principle holds for $G$ (with respect to the data of the $g$-action and the set of subgroups ${h_i}$) if the $S = {0}$.
Namely, for each subgroup $h_i$ of $g$ there is a restriction map in Galois cohomology 
$r_i: H^1(g,G) \rightarrow H^1(h_i,G)$.
(The restriction map is defined on one-cocycles merely by pulling back by the inclusion map $h_i \hookrightarrow g$.)  Restriction carries the distinguished (trivial) class to the trivial class.  
Then $S$ is defined as the intersection of the kernels of all the $r_i$'s.  Evidently the 
trivial class $0$ lies in $S$, so $(S,0)$ is a pointed set.
All of the above was just a detailed review of the beginning of Ono's paper.  Now let me explain why this generalizes the notion of the Shafarevich-Tate group of an abelian variety A over $\mathbb{Q}$ [or take a more general global field, if you like].
The Shafarevich-Tate group $Sha(A,\mathbb{Q})$ is the set of all principal homogeneous spaces (henceforth phs) $X$ under $A$ which have $\mathbb{Q}_p$-points for every prime $p$ and also $\mathbb{R}$-points.  Because the automorphism group of a phs under a group $A$ is just $A$ itself, by [what I call] the first principle of Galois cohomology, the pointed set of all phs under A is isomorphic to the Galois cohomology set
$H^1(\mathbb{Q},A) = H^1(\mathfrak{g}_{\mathbb{Q}},A(\overline{\mathbb{Q}}))$,
where $\mathfrak{g}_{\mathbb{Q}} = Aut(\overline{\mathbb{Q}}/\mathbb{Q})$ is the absolute Galois group of $\mathbb{Q}$.  [Since $A$ is commutative, the $H^1$ is itself a commutative group, whereas for nonabelian $A$ it would in general be only a pointed set.]
Thus here we have $G = A(\overline{\mathbb{Q}})$ and $g = \mathfrak{g}_{\mathbb{Q}}$. What are the $h_i$'s?  For each prime $p$, $h_p$ is the Galois group of $\mathbb{Q}_p$, 
viewed as a subgroup of $\mathbb{Q}$ (i.e., as a decomposition group at $p$) via choosing an embedding of the algebraic closure of $\mathbb{Q}$ into the algebraic closure of $Q_p$; also we define $h_{\infty}$ to be the restriction to the subgroup generated by a complex conjugation, i.e., a group isomorphic to $Aut(\mathbb{C}/\mathbb{R})$.  A cohomology class lies in the kernel of $h_p$ iff the corresponding phs acquires a point after base extension to $Q_p$ (and similarly for $h_{\infty}$.  
Thus the Shafarevich-Tate pointed set of $A$ is indeed a special case of Ono's construction.
That's the motivation I can give you.  As to exactly why Ono's particular choice of Shafarevich-Tate set for an arbitrary group $G$ -- namely take $g = G$ with the conjugation action, and let $(h_i)_{i \in I}$ be the family of cyclic subgroups of $G$ -- is interesting and natural...I can't help you there, and I'd like to know myself.
Why are you interested in this paper?
