**Introduction:**

Let $K$ be a number field and let's denote with $\mathbb A_K$ the ring of adeles which is also a locally compact group (with respect to the addition). Remember that the topology is the restricted product topology with respect to the rings $\mathcal O_v$ for any nonarchimedean place $v$.

By taking the standard character $\psi_v$ on each local fied $K_v$ (here $v$ ranges on all places of $K$, also the archimedean ones), we can obtain tha standard character $$\psi=\prod \psi_v:\mathbb A_K\to S^1\,.$$

Tate in his famous thesis proved that $\psi$ composed with the multiplication induces an isomorphism of topological groups $\mathbb A_K\cong\widehat{\mathbb A_K}$ (here $\widehat{\mathbb A_K}$ is the Pontryagin dual).

Furthermore we have the following symmetric pairing: $$\left<,\right>:\mathbb A_K\times\mathbb A_K\to S^1$$ $$(a,b)\mapsto\psi(ab)$$

and for any subset $H\subseteq\mathbb A_K$ we can define the annihilator (or the orthogonal complement):

$$H^\perp:=\{b\in\mathbb A_K\colon\left<b,H\right>=1\}$$

which is always a closed subgroup of $\mathbb A_K$.

Another important result in Tate's thesis is the following: If we consider the embedding $K\subset \mathbb A_K$, then $K^\perp=K$. In this post I'd like to know if it is possible to calculate explicitly $H^\perp$ for a particular $H$.

**Question:**

Suppose that $H=\prod H_v\subset \mathbb A_K$ is an additive subgroup constructed in the following way:

- For all but finitely many nonarchimedean $v$, we have $H_v=\mathcal O_v$.
- For all nonarchimedean $v$ there is $n_v\in\mathbb Z$ such that $H_v=\mathfrak p_v^{n_v}\mathcal O_v$. Here $\mathfrak p_v$ is the maximal ideal of $\mathcal O_v$. Note that due to property $1$, all but finitely many $n_v$ are equal to $0$.
- If $v$ is archimedean then $H_v=K_v$ (namely $H_v$ is equal to $\mathbb C$ or to $\mathbb R$ depending on the nature of $v$).

What is the explicit expression of $H^\perp$? Is there a way to calculate it?

Is it true in general that $H^\perp=\prod H_v^\perp$? Here by $H_v^\perp$ I mean the annihilator of $H_v$ in $K_v$ with respect to the "local pairing".