nt.number theory - Hasse principle for a group

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?

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