| We examine the action of the absolute Galois group of K on prime-to-p etale fundamental groups of the projective line P1K minus finitely many points, where K is either a strictly henselian field with mixed characteristic (0, p), or a finite field with characteristic p. In the former case, we give an explicit description of this action. This allows us to compute prime-to- p etale fundamental groups explicitly. E.g., for p ≠ 2, the prime-to-p etale fundamental group p'1&parl0;P 1Qunp {0, pm, 1, 2}) is isomorphic to the maximal prime-to-p quotient of the profinite completion of ⟨alpha1, alpha2, alpha3, alpha 4, delta|alpha1···alpha4 = 1, ad1=a&parl0;a 1a2&parr0;m1 ,ad2=a&parl0; a1a2&parr0;m 2,ad3=a 3,ad4=a 4 ⟩. Furthermore, given a G-Galois branched cover of P1Q we examine the relationship between its fields of definition and its field of moduli. For example, we prove that there is always a field of definition which is Galois over the field of moduli with group a subgroup of Aut(G). Lastly, given a field K, we explore which finite groups may appear as Galois groups of regular branched covers of P1K , ramified only over K-rational points. |
