Explicit Galois obstruction and descent for hyperelliptic curves with tamely cyclic reduced automorphism group (Q2796030)

Let \(K\) be the algebraic closure of a perfect field with \(\mathrm{char}(K)\neq 2\), and let \(F\) be a subfield of \(K\). Let \(X\) be a variety defined over \(F\), and \textit{the field of moduli \(k\) of \(X\)} is the intersection of all subfields \(L\) of \(K\) such that there is a variety \(Y\) defined over \(L\) and \(X_{K}\cong Y_K\). The field of moduli \(k\) of \(X\) has the property that \(X_K \cong X_K^\sigma\) for \(\sigma\in\mathrm{Aut}(K)\) if and only if \(\sigma\in \mathrm{Gal}(K/k)\). In [Am. J. Math. 78, 509--524 (1956; Zbl 0072.16001)], \textit{A. Weil} asked if there is a model of \(X\) over the field of moduli \(k\). Such a model, if exists, is called a \textit{descent} of \(X\), and we say, \(X\) has a descent for the extension \(K/k\). Weil showed that if \(\mathrm{Aut}(X_K)\) is trivial, then a descent of \(X\) exists.NEWLINENEWLINEIf \(X\) is a hyperelliptic curve, the presence of the hyperelliptic involution \(\iota\) makes \(\mathrm{Aut}(X_K)\) nontrivial. If we further require the existence of a model in the form of a hyperelliptic equation \(y^2=p(x)\) over \(k\), we call such a model a \textit{hyperelliptic descent} of \(X\). Let \(G=\mathrm{Aut}(X_K)\), and \(\overline{G}=G/\langle \iota \rangle\). In [\textit{B. Huggins}, Math. Res. Lett. 14, No. 2, 249--262 (2007; Zbl 1126.14036)], it was shown that \(X\) has a hyperelliptic descent if \(\overline{G}\) is not cyclic of order coprime to \(\mathrm{char}(K)\), and if it is, examples of \(X\) with no hyperelliptic descents were constructed in [\textit{C. J. Earle}, in: Adv. Theory Riemann Surfaces, Proc. 1969 Stony Brook Conf., 119--130 (1971; Zbl 0218.32010)] and [\textit{G. Shimura}, Nagoya Math. J. 45, 167--178 (1972; Zbl 0243.14012)]. Recently, found in [\textit{E. Bujalance} and \textit{P. Turbek}, Manuscr. Math. 108, No. 1, 1--11 (2002; Zbl 0997.14008)] is the full classification of hyperelliptic curves that has a hyperelliptic descent for \(\mathbb C/\mathbb R\).NEWLINENEWLINEIn this paper under review the authors present a complete answer to the descent problem for the case where \(K/k\) is any extenstion and \(\overline{G}\) is cyclic of order coprime to \(\mathrm{char}(K)\). They prove that there is always a field extension \(L\) of \(k\) with minimal degree \([L:k]\leq 2\) such that \(X\) has a hyperelliptic model over \(L\), and they give us explicit conditions on determining when \([L:k]=1\) or \(2\). The paper also presents how a descent can be effectively constructed, and given a quadratic extension \(L/k\), it gives an explicit description of the \(K\)-isomorphism classes of the curves which are defined over \(L\) and \(K\)-isomorphic to their conjugates, but do not descend to \(k\).