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Given a $G$-Galois branched cover of the projective line over a number field$K$, we study whether there exists a closed point of $\mathbb{P}^1_K$ with aconnected fiber such that the $G$-Galois field extension induced byspecialization "arithmetically descends" to $\mathbb{Q}$ (i.e., there exists a$G$-Galois field extension of $\mathbb{Q}$ whose compositum with the residuefield of the point is equal to the specialization). We prove that the answer isfrequently positive (whenever $G$ is regularly realizable over $\mathbb{Q}$) ifone first allows a base change to a finite extension of $K$. If one does notallow base change, we prove that the answer is positive when $G$ is cyclic.Furthermore, we provide an explicit example of a Galois branched cover of$\mathbb{P}^1_K$ with no $K$-rational points of arithmetic descent.

Arithmetic descent of specializations of Galois covers