On the fields of definition of genus-one covers of P^1

I will discuss some of my recent results related to genus one covers of the projective line over global fields and their fields of definition. The absolute Galois group acts on the set of equivalence classes of such covers in a natural way. The fixed field of the stabilizer with respect to this action is called the field of moduli of a cover and it is expected to be the minimal filed of definition. Covers without automorphisms as well as regular covers are known to be defined over the field of moduli. However, there are examples of covers (and even of Belyi pairs) which are not defined over the field of moduli. We will construct a series of such examples where the degree of a minimal field of definition over the field of moduli is unbounded. If time permits I will also discuss counterexamples to the local-global principle for covers.