Jacobian Nullwerte, periods and symmetric equations for hyperelliptic curves
Annales de l'Institut Fourier, Volume 57 (2007) no. 4, p. 1253-1283
We propose a solution to the hyperelliptic Schottky problem, based on the use of Jacobian Nullwerte and symmetric models for hyperelliptic curves. Both ingredients are interesting on its own, since the first provide period matrices which can be geometrically described, and the second have remarkable arithmetic properties.
Nous proposons une solution au problème de Schottky hyperelliptique. Celle-ci est basée sur l’utilisation de matrices jacobiennes de fonctions thêta et de modèles symétriques pour les courbes hyperelliptiques. Ces ingrédients sont intéressants en eux-mêmes  : le premier fournit des matrices de périodes qui peuvent être décrites géométriquement et le second possède de remarquables propriétés arithmétiques.
DOI : https://doi.org/10.5802/aif.2293
Classification:  11G30,  14H42
Keywords: Hyperelliptic curves, periods, Jacobian Nullwerte
@article{AIF_2007__57_4_1253_0,
     author = {Gu\`ardia, Jordi},
     title = {Jacobian Nullwerte, periods and symmetric equations for hyperelliptic curves},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {57},
     number = {4},
     year = {2007},
     pages = {1253-1283},
     doi = {10.5802/aif.2293},
     zbl = {pre05176621},
     mrnumber = {2339331},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2007__57_4_1253_0}
}
Jacobian Nullwerte, periods and symmetric equations for hyperelliptic curves. Annales de l'Institut Fourier, Volume 57 (2007) no. 4, pp. 1253-1283. doi : 10.5802/aif.2293. https://aif.centre-mersenne.org/item/AIF_2007__57_4_1253_0/

[1] Arbarello, E.; Cornalba, M.; Griffiths, P. A.; Harris, J. Geometry of algebraic curves. Vol. I, Springer-Verlag, New York, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Tome 267 (1985) | MR 770932 | Zbl 0559.14017

[2] Bayer, Pilar; Guàrdia, Jordi Hyperbolic uniformization of the Fermat curves, Ramanjujan J., Tome 12 (2006), pp. 207-223 | Article | MR 2286246 | Zbl 05119608

[3] Birch, B. J.; Kuyk, W. Modular functions of one variable. IV, Springer-Verlag, Berlin (1975) (Lecture Notes in Mathematics, Vol. 476) | MR 376533

[4] Cardona, Gabriel; Quer, Jordi Field of moduli and field of definition for curves of genus 2, Computational aspects of algebraic curves, World Sci. Publ., Hackensack, NJ (Lecture Notes Ser. Comput.) Tome 13 (2005), pp. 71-83 | MR 2181874 | Zbl 1126.14031

[5] Cremona, J. E. Algorithms for modular elliptic curves, Cambridge University Press, Cambridge (1992) | MR 1201151 | Zbl 0758.14042

[6] Frobenius, Ferdinand Georg Über die constanten Factoren der Thetareihen, J. reine angew. Math., Tome 98 (1885), pp. 241-260

[7] González, Josep; Guàrdia, Jordi; Rotger, Victor Abelian surfaces of GL 2 -type as Jacobians of curves, Acta Arith., Tome 116 (2005) no. 3, pp. 263-287 | Article | MR 2114780 | Zbl 02169948

[8] González-Jiménez, Enrique; González, Josep Modular curves of genus 2, Math. Comp., Tome 72 (2003) no. 241, p. 397-418 (electronic) | Article | MR 1933828 | Zbl 1081.11042

[9] González-Jiménez, Enrique; González, Josep; Guàrdia, Jordi Computations on modular Jacobian surfaces, Algorithmic number theory (Sydney, 2002), Springer, Berlin (Lecture Notes in Comput. Sci.) Tome 2369 (2002), pp. 189-197 | MR 2041083 | Zbl 1055.11038

[10] Guàrdia, Jordi Jacobian nullwerte and algebraic equations, J. Algebra, Tome 253 (2002) no. 1, pp. 112-132 | Article | MR 1925010 | Zbl 1054.14041

[11] Guàrdia, Jordi Jacobi Thetanullwerte, periods of elliptic curves and minimal equations, Math. Res. Lett., Tome 11 (2004) no. 1, pp. 115-123 | MR 2046204 | Zbl 02104761

[12] Guàrdia, Jordi; Torres, Eugenia; Vela, Montserrat Stable models of elliptic curves, ring class fields, and complex multiplication, Algorithmic number theory, Springer, Berlin (Lecture Notes in Comput. Sci.) Tome 3076 (2004), pp. 250-262 | MR 2137358 | Zbl 02194079

[13] Igusa, Jun-Ichi On Jacobi’s derivative formula and its generalizations, Amer. J. Math., Tome 102 (1980) no. 2, pp. 409-446 | Article | Zbl 0433.14033

[14] Igusa, Jun-Ichi On the nullwerte of Jacobians of odd theta functions, Symposia Mathematica, Vol. XXIV (Sympos., INDAM, Rome, 1979), Academic Press, London (1981), pp. 83-95 | MR 619242

[15] Igusa, Jun-Ichi Problems on abelian functions at the time of Poincaré and some at present, Bull. Amer. Math. Soc. (N.S.), Tome 6 (1982) no. 2, pp. 161-174 | Article | MR 640943 | Zbl 0484.14015

[16] Igusa, Jun-Ichi Multiplicity one theorem and problems related to Jacobi’s formula, Amer. J. Math., Tome 105 (1983) no. 1, pp. 157-187 | Article | Zbl 0527.14037

[17] Lockhart, P. On the discriminant of a hyperelliptic curve, Trans. Amer. Math. Soc., Tome 342 (1994) no. 2, pp. 729-752 | Article | MR 1195511 | Zbl 0815.11031

[18] Magma http://magma.math.usyd.edu.au/magma/ (2004) (University of Sydney)

[19] Mckean, Henry; Moll, Victor Elliptic curves, Cambridge University Press, Cambridge (1997) (Function theory, geometry, arithmetic) | MR 1471703 | Zbl 0895.11002

[20] Mestre, Jean-François Construction de courbes de genre 2 à partir de leurs modules, Effective methods in algebraic geometry (Castiglioncello, 1990), Birkhäuser Boston, Boston, MA (Progr. Math.) Tome 94 (1991), pp. 313-334 | MR 1106431 | Zbl 0752.14027

[21] Mumford, David Tata lectures on theta. II, Birkhäuser Boston Inc., Boston, MA, Progress in Mathematics, Tome 43 (1984) (Jacobian theta functions and differential equations, With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman and H. Umemura) | MR 742776 | Zbl 0549.14014

[22] Rosenhain, G. Mémoire sur les fonctions de deux variables et à quatre périodes qui sont les inverses des intégrales ultra-elliptiques de la première classe, Mémoires des savants étrangers, Tome XI (1851), pp. 362-468

[23] Shimura, Goro Abelian varieties with complex multiplication and modular functions, Princeton University Press, Princeton, NJ, Princeton Mathematical Series, Tome 46 (1998) | MR 1492449 | Zbl 0908.11023

[24] Silverman, Joseph H. The arithmetic of elliptic curves, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 106 (1992) (Corrected reprint of the 1986 original) | MR 1329092 | Zbl 0585.14026

[25] Takase, Koichi A generalization of Rosenhain’s normal form for hyperelliptic curves with an application, Proc. Japan Acad. Ser. A Math. Sci., Tome 72 (1996) no. 7, pp. 162-165 | Article | Zbl 0924.14016

[26] Thomae, J. Beitrag zur Bestimmung von θ(0,0,...,0) durch die Klassenmoduln algebraischer Funktionen, J. reine angew. Math., Tome 71 (1870), pp. 201-222 | Article

[27] Van Wamelen, Paul Examples of genus two CM curves defined over the rationals, Math. Comp., Tome 68 (1999) no. 225, pp. 307-320 | Article | MR 1609658 | Zbl 0906.14025

[28] Wang, Xiang Dong 2-dimensional simple factors of J 0 (N), Manuscripta Math., Tome 87 (1995) no. 2, pp. 179-197 | Article | MR 1334940 | Zbl 0846.14007

[29] Weber, Hermann-Josef Hyperelliptic simple factors of J 0 (N) with dimension at least 3, Experiment. Math., Tome 6 (1997) no. 4, pp. 273-287 | MR 1606908 | Zbl 1115.14304

[30] Weil, André Sur les périodes des intégrales abéliennes, Comm. Pure Appl. Math., Tome 29 (1976) no. 6, pp. 813-819 | Article | MR 422164 | Zbl 0342.14020

[31] Weng, Annegret A class of hyperelliptic CM-curves of genus three, J. Ramanujan Math. Soc., Tome 16 (2001) no. 4, pp. 339-372 | MR 1877806 | Zbl 1066.11028