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Hyperelliptic curves and their invariants: geometric, arithmetic and algorithmic aspects. (English) Zbl 1276.14088
This paper contains a thorough treatment of geometric, arithmetic and algorithmic aspects of hyperelliptic curves of small genus with special emphasis on genus $$3$$ curves.
Shioda’s computation of the graded ring $$\mathcal{I}_8$$ of invariants of binary octics is reviewed and shown to be valid for any algebraically closed field of positive characteristic $$p>7$$. The ring is generated by $$9$$ fundamental invariants $$J_2,\dots,J_{10}$$ which satisfy $$5$$ relations. This yields a representation of the coarse moduli space of genus $$3$$ hyperelliptic curves as a projective variety defined by the five Shioda relations on a weighted projective space of dimension $$9$$ whose points are of the form $$(J_2: J_3:\dots : J_{10})$$ and the coordinates have weights $$2,3,\dots,10$$. This description of the moduli space facilitates the enumeration of rational points in the strata determined by the automorphism group, over a finite field. The paper contains an exhaustive description of these strata and their characterization in terms of equations defined by the invariants, and it corrects several mistakes that occur in previous monographs on this topic.
The construction of curves with prefixed values of the fundamental invariants is also tackled. The authors are inspired by the general method of J.-F. Mestre [Effective methods in algebraic geometry, Proc. Symp., Castiglioncello/Italy 1990, Prog. Math. 94, 313–334 (1991; Zbl 0752.14027)], based on the computation of the eight points of intersection of a non-singular conic $$\mathcal{Q}$$ with a degree $$4$$ curve $$\mathcal{H}$$ whose coefficients are invariants. The construction depends on the stratum of the moduli space to which the point $$(J_2: J_3:\dots : J_{10})$$ belongs. Algorithms for the computation of the coefficients of $$\mathcal{H}$$ as polynomials in the Shioda invariants are developed and specific tricks are used in the cases where the conic $$\mathcal{Q}$$ is singular.
Finally, several arithmetic questions are addressed, again by working specifically on the different strata of the moduli space. Among others, the following problems are solved for each stratum: Is the field of moduli automatically a field of definition? Can the curve be always hyperelliptically defined over the field of moduli? Can one construct a model over the field of moduli when there is no obstruction?
A Magma code containing the various algorithms to check the computational assertions, calculate invariants and construct curves with prescribed invariants is available on the web page of the authors.

##### MSC:
 14Q05 Computational aspects of algebraic curves 13A50 Actions of groups on commutative rings; invariant theory 14H10 Families, moduli of curves (algebraic) 14H37 Automorphisms of curves
Magma
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##### References:
 [1] () [2] Shioda, T., On the graded ring of invariants of binary octavics, Amer. J. math., 89, 1022-1046, (1967) · Zbl 0188.53304 [3] Igusa, J.-I., Arithmetic variety of moduli for genus two, Ann. math., 72, 612-649, (1960) · Zbl 0122.39002 [4] Mestre, J.-F., Construction de courbes de genre 2 à partir de leurs modules, (), 313-334 · Zbl 0752.14027 [5] Clebsch, A., Theorie der binären algebraischen formen, (1872), Verlag von B.G. Teubner Leipzig · JFM 04.0047.02 [6] Bolza, O., On binary sextics with linear transformations into themselves, Amer. J. math., 10, 47-70, (1887) · JFM 19.0119.04 [7] Cardona, G.; Quer, J., Field of moduli and field of definition for curves of genus 2, (), 71-83 · Zbl 1126.14031 [8] Lercier, R.; Ritzenthaler, C., Invariants and reconstructions for genus 2 curves in any characteristic, 2008; available in MAGMA 2.15 and later [9] Shimura, G., On the field of rationality for an abelian variety, Nagoya math. J., 45, 167-178, (1972) · Zbl 0243.14012 [10] B. Huggins, Fields of moduli and fields of definition of curves, PhD thesis, University of California, Berkeley, Berkeley, California, 2005, http://arxiv.org/abs/math.NT/0610247. [11] R. Lercier, C. Ritzenthaler, J. Sijsling, Fast computation of isomorphisms of hyperelliptic curves and explicit descent, in: K. Kedlaya (Ed.), Proceedings of the Tenth Algorithmic Number Theory Symposium ANTS-X, Mathematical Sciences Publishers, 2012, in press. · Zbl 1344.11049 [12] R. Lercier, C. Ritzenthaler, J. Sijsling, Explicit descent obstruction for genus 3 hyperelliptic curves, preprint, 2012. · Zbl 1343.14047 [13] Nart, E., Counting hyperelliptic curves, Adv. math., 221, 774-787, (2009) · Zbl 1214.11078 [14] R. Lercier, C. Ritzenthaler, Equations for the automorphism group stratification of the coarse moduli space of genus 3 hyperelliptic curves, supplementary material to this article, please visit http://dx.doi.org/10.1016/j.jalgebra.2012.07.054. [15] Couveignes, J.-M., Calcul et rationalité de fonctions de belyĭen genre 0, Ann. inst. Fourier (Grenoble), 44, 1-38, (1994) · Zbl 0791.11059 [16] Procesi, C., Lie groups: an approach through invariants and representations, Universitext, (2007), Springer New York · Zbl 1154.22001 [17] Gordan, P., Beweis, dass jede covariante und invariante einer binären form eine ganze function mit numerischen coefficienten einer endlichen anzahl solcher formen ist, J. reine angew. math., 323-354, (1868) · JFM 01.0046.01 [18] Grace, J.; Young, A., The algebra of invariants, (1903), Chelsea Publishing Company New York · JFM 34.0114.01 [19] Dixmier, J., Quelques aspects de la théorie des invariants, Gaz. math., soc. math. fr., 43, 39-64, (1990) · Zbl 0708.14008 [20] Von Gall, F., Das vollständige formensystem der binären form 7ter ordnung, Math. ann., 31, 318-336, (1888) · JFM 20.0128.01 [21] Dixmier, J.; Lazard, D., Le nombre minimum dʼinvariants fondamentaux pour LES formes binaires de degré 7, Port. math., 43, 377-392, (1985/1986) · Zbl 0602.15022 [22] Bedratyuk, L., On complete system of invariants for the binary form of degree 7, J. symbolic comput., 42, 935, (2007) · Zbl 1141.13005 [23] H. Croeni, Zur Berechnung von Kovarianten von Quantiken, PhD thesis, Univ. des Saarlandes, Saarbrücken, 2002. [24] Brouwer, A.; Popoviciu, M., The invariants of the binary nonic, (2010), available at · Zbl 1189.13005 [25] Brouwer, A.; Popoviciu, M., The invariants of the binary decimic, (2010), available at · Zbl 1192.13005 [26] Mumford, D.; Fogarty, J., Geometric invariant theory, Ergeb. math. grenzgeb., vol. 34, (1982), Springer-Verlag Berlin · Zbl 0504.14008 [27] Bogomolov, F.A.; Katsylo, P.I., Rationality of some quotient varieties, Math. USSR-sb., 54, 571-576, (1986) · Zbl 0591.14040 [28] Maeda, T., On the invariant field of binary octavics, Hiroshima math. J., 20, 619-632, (1990) · Zbl 0741.14004 [29] Geyer, W.D., Invarianten binärer formen, (), 36-69 · Zbl 0298.14017 [30] Derksen, H.; Kemper, G., Computational invariant theory, () · Zbl 1011.13003 [31] Weber, H., Lehrbuch der algebra, vol. II, (1899), Vieweg Braunshweig · JFM 30.0093.01 [32] R. Brandt, Über die Automorphismengruppen von algebraischen Funktionenkörpern, PhD thesis, Universität Essen, 1988. [33] Brandt, R.; Stichtenoth, H., Die automorphismengruppen hyperelliptischer kurven, Manuscripta math., 55, 83-92, (1986) · Zbl 0588.14022 [34] Singerman, D., Finitely maximal Fuchsian groups, J. lond. math. soc. (2), 6, 29-38, (1972) · Zbl 0251.20052 [35] Ries, J.F.X., Subvarieties of moduli space determined by finite groups acting on surfaces, Trans. amer. math. soc., 335, 385-406, (1993) · Zbl 0784.32017 [36] Magaard, K.; Shaska, T.; Shpectorov, S.; Völklein, H., The locus of curves with prescribed automorphism group, Communications in arithmetic fundamental groups, Kyoto, 1999/2001, Sūrikaisekikenkyūsho Kōkyūroku, 112-141, (2002) [37] Cardona, G., On the number of curves of genus 2 over a finite field, Finite fields appl., 9, 505-526, (2003) · Zbl 1091.11023 [38] Cardona, G.; Nart, E.; Pujolàs, J., Curves of genus two over fields of even characteristic, Math. Z., 250, 177-201, (2005) · Zbl 1097.11033 [39] Gutierrez, J.; Sevilla, D.; Shaska, T., Hyperelliptic curves of genus 3 with prescribed automorphism group, (), 109-123 · Zbl 1121.14021 [40] Roquette, P., Abschätzung der automorphismenanzahl von funktionenkörpern, Math. Z., 117, 157-163, (1970) · Zbl 0194.35302 [41] Grothendieck, A., Revêtements étales et géométrie algébrique (SGA 1), Lecture notes in math., vol. 224, (1971), Springer-Verlag Heidelberg [42] Nart, E.; Sadornil, D., Hyperelliptic curves of genus three over finite fields of characteristic two, Finite fields appl., 10, 198-220, (2004) · Zbl 1049.11063 [43] Babu, H.; Venkataraman, P., Group action on genus 3 curves and their Weierstrass points, (), 264-272 · Zbl 1126.14041 [44] Koizumi, S., The fields of moduli for polarized abelian varieties and for curves, Nagoya math. J., 48, 37-55, (1972) · Zbl 0246.14006 [45] Dèbes, P.; Emsalem, M., On fields of moduli of curves, J. algebra, 211, 42-56, (1999) · Zbl 0934.14019 [46] Huggins, B., Fields of moduli of hyperelliptic curves, Math. res. lett., 14, 249-262, (2007) · Zbl 1126.14036 [47] Weil, A., The field of definition of a variety, Amer. J. math., 78, 509-524, (1956) · Zbl 0072.16001 [48] Serre, J.-P., Corps locaux, (1968), Hermann Paris, Publications de lʼUniversité de Nancago, No. VIII [49] Fuertes, Y.; González-Diez, G., Fields of moduli and definition of hyperelliptic covers, Arch. math. (basel), 86, 398-408, (2006) · Zbl 1095.14028 [50] Fuertes, Y., Fields of moduli and definition of hyperelliptic curves of odd genus, Arch. math. (basel), 95, 15-81, (2010) · Zbl 1206.14053 [51] Brock, B.W.; Granville, A., More points than expected on curves over finite field extensions, Finite fields appl., 7, 70-91, (2001), dedicated to Professor Chao Ko on the occasion of his 90th birthday · Zbl 1023.11029
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