Irreducible Plane Curves Over the Algebraic $p$-adic Numbers

Question

Let $A=\overline{\mathbb{Q}}\cap \mathbb{Q}_p$ be the relative algebraic closure of $\mathbb{Q}$ in the field of $p$-adics. Is there an example of an irreducible plane curve $C$ given by $f(x,y)=0$ for $f(x,y)\in A[x,y]$ such that $C$ has a smooth $\mathbb{Q}_p$-point, but no smooth $A$-point?

Answer 1

Your ring $A$ is, I believe, a p-adically closed field.

For each polynomial $f$, the statement that the curve $f(x,y)$=0 has a smooth point can be expressed in the first-order language of arithmetic. Consequently, it has such an $A$-point if and only if it has a $\mathbb{Q}_p$-point.