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Explicit Galois obstruction and descent for hyperelliptic curves with tamely cyclic reduced automorphism group. (English) Zbl 1343.14047
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 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)], A. Weil asked if there is a model of $$X$$ over the field of moduli $$k$$. Such a model, if exists, is called a 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.
If $$X$$ 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 hyperelliptic descent of $$X$$. Let $$G=\mathrm{Aut}(X_K)$$, and $$\overline{G}=G/\langle \iota \rangle$$. In [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 [C. J. Earle, in: Adv. Theory Riemann Surfaces, Proc. 1969 Stony Brook Conf., 119–130 (1971; Zbl 0218.32010)] and [G. Shimura, Nagoya Math. J. 45, 167–178 (1972; Zbl 0243.14012)]. Recently, found in [E. Bujalance and 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 $$\mathbb C/\mathbb R$$.
In 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 $$k$$.

##### MSC:
 14Q05 Computational aspects of algebraic curves 13A50 Actions of groups on commutative rings; invariant theory 14H10 Families, moduli of curves (algebraic) 14H25 Arithmetic ground fields for curves 14H37 Automorphisms of curves
Magma
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