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Let $K$ be a number field, $\bar{K}$ a fixed algebraic closure, $G_K$ the absolute galois group, $O_\bar{K}$ the ring of integers of the algebraic closure, $\mathfrak{p}$ runs over prime ideals and subscripts mean p-adic completions.

Define the "Integral Shafarevich-Tate group" of $GL_2(O_\bar{K})$, a $G_K$-module, to be (normal galois cohomology follows): $$\mathrm{III}^{int}(GL_2) = Ker\\ \lgroup\\ H^1(G_K, GL_2(O_\bar{K}) \rightarrow \prod_{\mathfrak{p}} H^1(G_{K_{\mathfrak{p}}},GL_2(O_\bar{K_\mathfrak{p}})) \rgroup$$

For $GL_1(O_\bar{K})$ it is well known to be isomorphic to the class group of $K$. My question is: $$\text{What is }\mathrm{III}^{int}(GL_2)?$$

It is not clear that the above defines a group if $G$ is not abelian. The article "Abelianization of the First Galois Cohomology of Reductive Groups", M. Borovoi, shows that reductive groups over a number field have abelian Sha (over the field - not "integral" Sha). It might not be related, but I suspect something similar would work for the above.

The same question for $GL_n$, quaternion units, etc. is also very interesting.

Let $K$ be a number field, $\bar{K}$ a fixed algebraic closure, $G_K$ the absolute galois group, $O_\bar{K}$ the ring of integers of the algebraic closure, $\mathfrak{p}$ runs over prime ideals and subscripts mean p-adic completions.

Define the "Integral Shafarevich-Tate group" of $GL_2(O_\bar{K})$, a $G_K$-module, to be (normal galois cohomology follows): $$\mathrm{III}^{int}(GL_2) = Ker\\ \lgroup\\ H^1(G_K, GL_2(O_\bar{K}) \rightarrow \prod_{\mathfrak{p}} H^1(G_{K_{\mathfrak{p}}},GL_2(O_\bar{K_\mathfrak{p}})) \rgroup$$

For $GL_1(O_\bar{K})$ it is known^* to be isomorphic to the class group of $K$. My question is: $$\text{What is }\mathrm{III}^{int}(GL_2)?$$

It is not clear that the above defines a group since $GL_2$ is not abelian. The article "Abelianization of the First Galois Cohomology of Reductive Groups", M. Borovoi, shows that reductive groups over a number field have abelian Sha (over the field - not "integral" Sha). It might not be related, but I suspect something similar would work for the above.

The same question for $GL_n$, quaternion units, etc. is also very interesting.

(*) The case of $GL_1$ is proved in "Visibility of Ideal Classes" by Schoof and Washington (arxiv:0809.5209).

Let $K$ be a number field, $\bar{K}$ a fixed algebraic closure, $G_K$ the absolute galois group, $O_\bar{K}$ the ring of integers of the algebraic closure, $\mathfrak{p}$ runs over prime ideals and subscripts mean p-adic completions.

Define the "Shafarevich-Tate group" of $GL_2(O_\bar{K})$, a $G_K$-module, to be (normal galois cohomology follows): $$\mathrm{III}(GL_2) = Ker\\ \lgroup\\ H^1(G_K, GL_2(O_\bar{K}) \rightarrow \prod_{\mathfrak{p}} H^1(G_{K_{\mathfrak{p}}},GL_2(O_\bar{K_\mathfrak{p}})) \rgroup$$

For $GL_1(O_\bar{K})$ it is well known to be isomorphic to the class group of $K$. My question is: $$\text{What is }\mathrm{III}(GL_2)?$$

It is not clear that the above defines a group if $G$ is not abelian. The article "Abelianization of the First Galois Cohomology of Reductive Groups", M. Borovoi, shows that reductive groups over a number field have abelian Sha. It might not be related, but I suspect something similar would work for the above.

The same question for $GL_n$, quaternion units, etc. is also very interesting.

Let $K$ be a number field, $\bar{K}$ a fixed algebraic closure, $G_K$ the absolute galois group, $O_\bar{K}$ the ring of integers of the algebraic closure, $\mathfrak{p}$ runs over prime ideals and subscripts mean p-adic completions.

Define the "Integral Shafarevich-Tate group" of $GL_2(O_\bar{K})$, a $G_K$-module, to be (normal galois cohomology follows): $$\mathrm{III}^{int}(GL_2) = Ker\\ \lgroup\\ H^1(G_K, GL_2(O_\bar{K}) \rightarrow \prod_{\mathfrak{p}} H^1(G_{K_{\mathfrak{p}}},GL_2(O_\bar{K_\mathfrak{p}})) \rgroup$$

For $GL_1(O_\bar{K})$ it is well known to be isomorphic to the class group of $K$. My question is: $$\text{What is }\mathrm{III}^{int}(GL_2)?$$

It is not clear that the above defines a group if $G$ is not abelian. The article "Abelianization of the First Galois Cohomology of Reductive Groups", M. Borovoi, shows that reductive groups over a number field have abelian Sha (over the field - not "integral" Sha). It might not be related, but I suspect something similar would work for the above.

The same question for $GL_n$, quaternion units, etc. is also very interesting.

Let $K$ be a number field, $\bar{K}$ a fixed algebraic closure, $G_K$ the absolute galois group, $O_\bar{K}$ the ring of integers of the algebraic closure, $\mathfrak{p}$ runs over prime ideals and subscripts mean p-adic completions.

Define the Shafarevich-Tate group of $G$, a $G_K$-module, to be: $$\mathrm{III}(G) = Ker\\ \lgroup\\ H^1(G_K, G(O_\bar{K}) \rightarrow \prod_{\mathfrak{p}} H^1(G_{K_{\mathfrak{p}}},G(O_\bar{K_\mathfrak{p}})) \rgroup$$

For $G=GL_1$ it is well known to be isomorphic to the class group of $K$. My question is: $$\text{What is }\mathrm{III}(GL_2)?$$

It is not clear that the above defines a group if $G$ is not abelian. The article "Abelianization of the First Galois Cohomology of Reductive Groups", M. Borovoi, shows that if instead of $G(O_\bar{K})$ we put $G$, a reductive group over $K$, then above has in fact a canonical natural abelian group structure. I suspect this remains true if we restrict the group to $O_\bar{K}$.

The same question for $GL_n$, quaternion units, etc. is also very interesting.

Let $K$ be a number field, $\bar{K}$ a fixed algebraic closure, $G_K$ the absolute galois group, $O_\bar{K}$ the ring of integers of the algebraic closure, $\mathfrak{p}$ runs over prime ideals and subscripts mean p-adic completions.

Define the "Shafarevich-Tate group" of $GL_2(O_\bar{K})$, a $G_K$-module, to be (normal galois cohomology follows): $$\mathrm{III}(GL_2) = Ker\\ \lgroup\\ H^1(G_K, GL_2(O_\bar{K}) \rightarrow \prod_{\mathfrak{p}} H^1(G_{K_{\mathfrak{p}}},GL_2(O_\bar{K_\mathfrak{p}})) \rgroup$$

For $GL_1(O_\bar{K})$ it is well known to be isomorphic to the class group of $K$. My question is: $$\text{What is }\mathrm{III}(GL_2)?$$

It is not clear that the above defines a group if $G$ is not abelian. The article "Abelianization of the First Galois Cohomology of Reductive Groups", M. Borovoi, shows that reductive groups over a number field have abelian Sha. It might not be related, but I suspect something similar would work for the above.

The same question for $GL_n$, quaternion units, etc. is also very interesting.