Erratum to: Fields of moduli and definition of hyperelliptic covers

Abstract

where r1, r2, r3 are the roots of the polynomial x − 3x + 1 (or any other degree 3 polynomial p(x)∈Q [x] whose Galois group has order 3) and the parameters qi are distinct rational numbers q4, . . . , q2g+2 chosen so that Aut(C) ∼= Z2⊕Z2⊕Z2. Then Theorem 1. I) C is hyperelliptically defined over Q (r1). II) The field of moduli of C is Q. III) Let k be a subfield of the reals and Ck a curve of the form Ck: y = q(x), where q(x) is a polynomial with coefficients in k without multiple roots. Suppose that there is a birational isomorphism f : C → Ck defined over the compositum of the fields Q (r1) and k, namely k(r1). Then k must contain the field Q (r1).