zbMATH — the first resource for mathematics

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 \(k\).

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
Full Text: DOI arXiv
[1] Bosma, Wieb; Cannon, John; Playoust, Catherine, The Magma algebra system. I. The user language, J. Symbolic Comput., 24, 3-4, 235-265, (1997) · Zbl 0898.68039
[2] Brandt, Rolf; Stichtenoth, Henning, Die Automorphismengruppen hyperelliptischer Kurven, Manuscripta Math., 55, 1, 83-92, (1986) · Zbl 0588.14022
[3] Bujalance, Emilio; Turbek, Peter, Asymmetric and pseudo-symmetric hyperelliptic surfaces, Manuscripta Math., 108, 1, 1-11, (2002) · Zbl 0997.14008
[4] Cardona, Gabriel; Quer, Jordi, Field of moduli and field of definition for curves of genus 2. Computational Aspects of Algebraic Curves, Lecture Notes Ser. Comput. 13, 71-83, (2005), World Sci. Publ., Hackensack, NJ · Zbl 1126.14031
[5] D\`ebes, Pierre; Emsalem, Michel, On fields of moduli of curves, J. Algebra, 211, 1, 42-56, (1999) · Zbl 0934.14019
[6] Earle, Clifford J., On the moduli of closed Riemann surfaces with symmetries. Advances in The Theory of Riemann Surfaces, Proc. Conf., Stony Brook, N.Y., 1969, 119-130. Ann. of Math. Studies, No. 66, (1971), Princeton Univ. Press, Princeton, N.J.
[7] Gutierrez, J.; Shaska, T., Hyperelliptic curves with extra involutions, LMS J. Comput. Math., 8, 102-115, (2005) · Zbl 1110.14025
[8] [hidrey] R. A. Hidalgo and S. Reyes, \newblockA constructive proof of Weil’s Galois descent theorem. \newblock Preprint at \urlhttp://arxiv.org/abs/1203.6294.
[9] Huggins, Bonnie Sakura, Fields of Moduli and Fields of Definition of Curves, 156 pp., (2005), ProQuest LLC, Ann Arbor, MI
[10] Huggins, Bonnie, Fields of moduli of hyperelliptic curves, Math. Res. Lett., 14, 2, 249-262, (2007) · Zbl 1126.14036
[11] Lercier, Reynald; Ritzenthaler, Christophe, Hyperelliptic curves and their invariants: geometric, arithmetic and algorithmic aspects, J. Algebra, 372, 595-636, (2012) · Zbl 1276.14088
[12] [lrs] R. Lercier, C. Ritzenthaler, and J. Sijsling, \newblockFast computation of isomorphisms of hyperelliptic curves and explicit descent, \newblockANTS X: Proceedings of the Tenth Algorithmic Number Theory Symposium, pages 463–486. Mathematical Science Publishers, 2013. · Zbl 1344.11049
[13] Mestre, Jean-Fran\ccois, Construction de courbes de genre \(2\) \`a partir de leurs modules. Effective Methods in Algebraic Geometry, Castiglioncello, 1990, Progr. Math. 94, 313-334, (1991), Birkh\"auser Boston, Boston, MA · Zbl 0752.14027
[14] Sekiguchi, Tsutomu, On the fields of rationality for curves and for their Jacobian varieties, Nagoya Math. J., 88, 197-212, (1982) · Zbl 0473.14012
[15] Serre, Jean-Pierre, Cohomologie Galoisienne, Lecture Notes in Mathematics 5, x+181 pp., (1994), Springer-Verlag, Berlin · Zbl 0812.12002
[16] Shimura, Goro, On the field of rationality for an abelian variety, Nagoya Math. J., 45, 167-178, (1972) · Zbl 0243.14012
[17] Shioda, Tetsuji, On the graded ring of invariants of binary octavics, Amer. J. Math., 89, 1022-1046, (1967) · Zbl 0188.53304
[18] Wehlau, David L., Constructive invariant theory for tori, Ann. Inst. Fourier (Grenoble), 43, 4, 1055-1066, (1993) · Zbl 0789.14009
[19] Weil, Andr\'e, The field of definition of a variety, Amer. J. Math., 78, 509-524, (1956) · Zbl 0072.16001
[20] Xarles, Xavier, Trivial points on towers of curves, J. Th\'eor. Nombres Bordeaux, 25, 2, 477-498, (2013) · Zbl 1294.11109
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.