$$ We give a criterion, based on the automorphism group, for certain cyclic covers of the projective line to be defined over their field of moduli. An example of a cyclic cover of the complex projective line with field of moduli $ℝ$ that can not be defined over $ℝ$ is also given.

We continue the examination of the stable reduction and fields of moduli of $G$-Galois covers of the projective line over a complete discrete valuation field of mixed characteristic $(0,p)$, where $G$ has a cyclic$p$-Sylow subgroup $P$ of order $p^n$. Suppose further that the normalizer of $P$ acts on $P$ via an involution. Under mild assumptions, if $f:Y\to {\mathbb{P}}^1$ is a three-point $G$-Galois cover defined over $\overline{\mathbb{Q}}$, then the $n$th higher ramification groups above $p$ for the upper numbering of the (Galois closure of the) extension $K/\mathbb{Q}$ vanish,...

For each integer s ≥ 1, we present a family of curves that are ${}_q$-Frobenius nonclassical with respect to the linear system of plane curves of degree s. In the case s=2, we give necessary and sufficient conditions for such curves to be ${}_q$-Frobenius nonclassical with respect to the linear system of conics. In the ${}_q$-Frobenius nonclassical cases, we determine the exact number of ${}_q$-rational points. In the remaining cases, an upper bound for the number of ${}_q$-rational points will follow from Stöhr-Voloch...