Field of definition and Galois orbits for the Macbeath-Hurwitz curves

Abstract.

Hurwitz curves are Riemann surfaces with 84(g-1) automorphisms, g the genus. Defined over some number field they permit an obvious ${\rm Gal} (\overline {{\Bbb Q}}/{\Bbb Q})$ action. We investigate this action for the first known infinite series of Hurwitz curves, due to Macbeath, using the canonical model of the curves. As a result we obtain the minimal field of definition for these curves. The method can be extended to some other infinite series of modular curves for non-congruence subgroups.