$Q$ is thus defined over $k$, and hence it gives a section $s$ of $\T\to C$ such that $s(P)=Q$. Hence any adelic point on $C$ lifts to an adelic point on $s(C)\subset \T$. Since $\operatorname{Br}(C)=\operatorname{Br}(k)$ we conclude that $C({\mathbb{A}}_k)$ is contained in, and hence is equal to the étale Brauer–Manin set $C({\mathbb{A}}_k)^{\operatorname{\acute{e}t},\operatorname{Br}}$.

Let $k$ be a number field, and let $f(x,y)$ be a separable homogeneous polynomial such that its zero locus $Z^f \subseteq {\mathbb{P}}^1_{k}$ is a $0$-dimensional scheme violating the Hasse principle. It is easy to see that such a polynomial exists for any number field $k$. For example, for $k={\mathbb{Q}}$ one can take $$f(x,y) = (x^2-2y^2)(x^2-17y^2)(x^2-34y^2).\label{e1}$$ For an arbitrary number field $k$ the following polynomial will do: $$f(x,y) = (x^2-ay^2)(x^2-by^2)(x^2-aby^2)(x^2-cy^2),$$ where $a,b,c\in k^*\setminus k^{*2}$ are such that $ab\notin k^{*2}$, whereas $c\in k_v^{*2}$ for all places $v$ such that $\val_v(a)\not=0$ or $\val_v(b)\not=0$, and also for the archimedean places, and the places with residual characteristic 2. (For fixed $a$ and $b$ one finds $c$ using weak approximation.) Let $d=\deg(f)$.

Let $C^f \subseteq {\mathbb{P}}^2_k$ be the curve with the equation $f(x,y) = 0$. It is geometrically connected and has a unique singular point $P = (0:0:1) \in C^f \subset {\mathbb{P}}^2_k$ which is 