The field of definition of point sets in [formula omitted

Abstract

Let K be a perfect field of characteristic not equal to two, K¯ an algebraic closure of K and let GK be the Galois group of the extension K¯/K. Let T be an n-point set in P1(K¯). The field of moduli of T is contained in each field of definition but it is not necessarily a field of definition. In this paper we show that point sets of odd cardinality n⩾5 in P1(K¯) with field of moduli K are defined over their field of moduli. We, also, show that, except for the special case of the 4-point sets, this does not hold in general for point sets of even cardinality n⩾6. Finally we prove that the following local-to-global principle holds for point sets of cardinality n⩾5: if an n-point set T in P1(Q¯) is defined over Qp for each prime p (including the prime at ∞), then it is defined over Q. From this result we derive an analogue local-to-global principle for hyperelliptic curves.