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Fields of moduli and definition of hyperelliptic curves of odd genus. (English) Zbl 1206.14053

A hyperelliptic curve is said to be hyperelliptically defined over a field \(k \subset \mathbb{C}\) if it is birationally equivalent to a curve of the form \(y^2=q(x)\), where \(q(x)\) is a polynomial with simple roots and coefficients in \(k\). In [Effective methods in algebraic geometry, Proc. Symp., Castiglioncello/Italy 1990, Prog. Math. 94, 313-334 (1991 ; Zbl 0752.14027)] it was shown that ‘being defined over \(k\)’ and ‘being hyperelliptically defined over \(k\)’ are equivalent when the genus is even. However, in this article, the author constructs an explicit hyperelliptic curve defined over \(\mathbb{Q}\) which cannot be hyperelliptically defined over \(\mathbb{Q}\) for all \(g>1\) with \(g \equiv 1 \pmod{4}\).
The proof goes as follows. In [Arch. Math. 86, No. 5, 398–408 (2006; Zbl 1095.14028)], the authors showed that a certain hyperelliptic curve \(C\) of genus \(g \equiv 1 \pmod{4}\) could not be hyperelliptically defined over \(\mathbb{Q}\). However, in the present article, it is proved that the field of moduli of \(C\) is \(\mathbb{Q}\). Since this curve has automorphism group \((\mathbb{Z}/2\mathbb{Z})^2\), which is not cyclic, [Math. Res. Lett. 14, No. 2, 249–262 (2007; Zbl 1126.14036)] shows that the field of moduli is a field of definition.

MSC:

14H37 Automorphisms of curves
14H45 Special algebraic curves and curves of low genus
14G99 Arithmetic problems in algebraic geometry; Diophantine geometry
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References:

[1] C. Earle, On the moduli of closed Riemann surfaces with symmetry, Advances in the theory of Riemann surfaces, Ann. of Math. Studies 66, Princeton 1971, pp.\(\sim\)119–130. · Zbl 0218.32010
[2] C. Earle, Diffeomorphisms and automorphisms of compact hyperbolic 2-orbifolds, in: Proc. of the Conference on Geometry of Riemann surfaces at Anogia, 2007, F. Gardiner, G. González-Diez, and C. Kouranoitis (eds.), (to appear in 2009). · Zbl 1198.30039
[3] Fuertes Y., González-Diez G.: Smooth double coverings of hyperelliptic curves, Proceedings of the III Iberoamerican Congress on Geometry. Contemp. Math. 397, 73–77 (2006) · Zbl 1101.14044
[4] Fuertes Y., González-Diez G.: On unramified normal coverings of hyperelliptic curves. J. Pure Appl. Algebra, 208, 1063–1070 (2007) · Zbl 1123.14019 · doi:10.1016/j.jpaa.2006.05.008
[5] Fuertes Y., González-Diez G.: Fields of moduli and definition of hyperelliptic covers. Arch. Math. 86, 398–408 (2006) · Zbl 1095.14028 · doi:10.1007/s00013-005-1433-8
[6] González-Diez G.: Variations on Belyi’s Theorem. Quart. J. Math. 57, 339–354 (2006) · Zbl 1123.14016 · doi:10.1093/qmath/hai021
[7] Huggins B.: Fields of moduli of hyperelliptic curves. Math. Res. Lett. 14, 249–262 (2007) · Zbl 1126.14036
[8] J.-F. Mestre, Construction de courbes de genre 2 à partir de leur modules, Effective methods in algebraic geometry (Castiglionello, 1990), 313–334. Prog. Math. 94, Birkhäuser Boston, MA, 1991.
[9] Shimura G.: On the field of rationality for an abelian variety. Nagoya Math. J. 45, 167–178 (1972) · Zbl 0243.14012
[10] Weil A.: The field of definition of a variety, Amer. J. Math. 78, 509–524 (1956) · Zbl 0072.16001 · doi:10.2307/2372670
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