In this short paper of Sameer Kailasa, he reviews a curious phenomenon by which the class group of a number field $K$ appears as the exact kernel of the following morphism:
$$H^1(Spec(K),\mathcal{O}_{\bar{K}}^\times)\rightarrow\prod_{\nu\in N_K}H^1(Spec(K_\nu),\mathcal{O}_{\bar{K_\nu}}^\times)$$
where $N_K$ is the set of (non-archimedian) places of $K$.
As you may know, the Tate-Shafarevich group $Ш(E/K)$ of an elliptic curve (or more generally any abelian variety) $E$ defined over $K$ is the kernel of this:
$$H^1(Spec(K), E(\bar{K}))\rightarrow\prod_{\nu\in M_K}H^1(Spec(K_\nu),E(\bar{K}))$$
with $M_K$ the set of all places of $K$. This leads one to the hypothesis of a supposed single calculation which provides both the class group of the number field and the Tate-Shafarevich group of an abelian variety, perhaps with some correction terms for the archimedean places. In this question, I will consider Tate-Shafarevich to be the group formed by restricting only to non-archimedean places; if the elliptic curve is real-connected, there is no difference, and even if it is not then it has the usual Tate-Shafarevich as a subgroup of finite order.
This question on MSE asks for a single reason behind both of these formulations. The most conclusive answer I could find was Kevin Buzzard's paper (in a comment) that explains one way to derive the ideal class group in a manner similarly to Tate-Shafarevich. However, I'm not completely satisfied; to me, this paper describes how $\mathcal{O}_\bar{K}^\times$ and $E(\bar{K})$ are cognates as Galois-modules, without very much describing some sort of deeper cognates between the group of units and an elliptic curve.
Another answer was simply that Tate-Shafarevich is $H^1(X,A)$ for $X=Spec(\mathcal{O_K})$ and $A$ the Néron model. This answer was much more tempting to me; after all, $H^1(X,\mathbb{G}_m)$ is the ideal class group. Unfortunately, this is oversimplified; while we do see that $Cl$ and $Ш$ are both related to $H^1$ computed with coefficients in different abelian sheaves, both can be described as the images of some morphisms $H^1(X,F)\rightarrow H^1(Spec(K),G)$. In the case of $Cl$, this morphism is injective, providing an isomorphism with the source, and hence the famous realization that the Picard group of $X$ is the class group of $K$.
To extrapolate on this cohomological interpretation of the groups, we end up constructing a small dictionary:
- Rational functions $\leftrightarrow$ Néron model $A$
- $\bar{K}^\times$ $\leftrightarrow$ $E(\bar{K})$
- Group of units $\mathbb{G}_m$ $\leftrightarrow$ open subscheme $A^0$
- $\mathcal{O}_{\bar{K}}^\times$ $\leftrightarrow$ global sections $A^0(X)$
- Divisors $\leftrightarrow$ global sections of $\Phi_A=A/A^0$
However, in the analogy posited by the adelic kernel formulation, we see that $\mathcal{O}_{\bar{K}}^\times$ should be analogous to $E(\bar{K})$. But the Néron model dictionary posits that $E(\bar{K})$ is actually analogous to $\bar{K}^\times$; which makes a simultaneous proof of both adelic kernels seem impossible! In fact, in order to prove that $H^1(X,A^0)\rightarrow H^1(Spec(K),E(\bar{K}))$ gives Tate-Shafarevich, you must construct a morphism $H^1(X,A^0)\rightarrow H^1(X,A)$ which it factors through; but the analogue of $H^1(X,A)$ is $H^1(X,\mathcal{M}^\times)$ which is known to be trivial!
What is going on here? Perhaps this is near the correct analogy, or this is nowhere close. It seems however like the adelic kernel formulations remain a complete mystery, and the dictionary resolves nothing about it.
Below I give a more detailed explanation of the above dictionary.
Consider $X=Spec(\mathcal{O}_K)$. Let $j$ be the inclusion of the generic point, and $i_p$ be the inclusion of a closed point $Spec(\mathbb{Z}/p\mathbb{Z})$. Consider the following short exact sequence of abelian sheaves in the smooth topology:
$$1\rightarrow\mathbb{G}_m\rightarrow j_\ast\bar{K}^\times\rightarrow\bigoplus_{p\in|X|}i_{p\ast}\mathbb{Z}\rightarrow 1$$
Taking sheaf cohomology, we find that $H^1(X,j_\ast\bar{K}^\times) = H^1(Spec(K),\bar{K}^\times)$ which vanishes by Theorem 90; hence we get the following exact sequence used as the basis of Kailasa's proof:
$$1\rightarrow\mathcal{O}_K^\times\rightarrow K^\times\rightarrow \text{Div}(X)\rightarrow H^1(X,\mathbb{G}_m)\rightarrow 1$$
where $\text{Div}(X)$ is the group of fractional divisors on $\mathcal{O}_K$. In other words, $Cl(K)=H^1(X,\mathbb{G}_m)$, the Picard group of the scheme. This is a well-known relationship.
Less well-known is the analogue for a Néron model; given the Néron model $A$ of some abelian variety on $K$, the sheaf $A(U)=\text{Hom}_X(U, A)$ on the smooth topology of $X$ satisfies $j_* E(\bar{K})\cong A$ (as an abelian sheaf).
We can construct a smooth subgroup-scheme $A^0$ of $A$; the construction can be found in Mazur's paper (page 200). He also describes the following short exact sequence of sheaves:
$$1\rightarrow A^0\rightarrow j_*E(\bar{K})\rightarrow\bigoplus_{p\in|X|}i_{p^*}F_p\rightarrow 1$$
where $F_p$ refers to a particular $\mathbb{Z}/p\mathbb{Z}$-module having to do with the construction of $A$.