Effective descent of the field of definition of a Galois cover
Let \(k\) be a field of zero characteristic and let \(f:X\to \mathbb{P} {1\over k}\) be a cover defined over an algebraic closure \(\overline k\) of \(k\). Write \(f^\tau: X^\tau\to \mathbb{P}{1 \over k}\) for the cover obtained from \(f\) by the action of an element \(\tau\) of the Galois group \(G=\text{Gal}(\overline k/k)\). The field of moduli of \(f\) is \(k_0=\overline k^H\), where \(H\) is the subgroup of elements \(\tau\) of \(G\) such that \(f\simeq f^\tau\). This field is contained in any field of definition of the cover \(f\). When \(f\) is a Galois cover, \textit{K. R. Coombes} and \textit{D. Harbater} have proved that \(f\) is defined over \(k_0\) [Duke Math. J. 52, 821-839 (1985; Zbl 0601.14023)]. A new proof of this result is given by the author, it has the advantage to give an explicit model (with bounds for the degrees and the heights of its equations) of the Galois cover \(f\) over its fields of moduli \(k_0\).
scientific article; zbMATH DE number 1309268