Jacobian Nullwerte, periods and symmetric equations for hyperelliptic curves
[Matrices jacobiennes de fonctions thêta, périodes et équations symétriques pour les courbes hyperelliptiques]
Annales de l'Institut Fourier, Tome 57 (2007) no. 4, pp. 1253-1283.

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.

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.

DOI : 10.5802/aif.2293
Classification : 11G30, 14H42
Keywords: Hyperelliptic curves, periods, Jacobian Nullwerte
Mot clés : courbes hyperelliptiques, periods, thetanullwerte
Guàrdia, Jordi 1

1 Escola Politècnica Superior d’Enginyeria de Vilanova i la Geltrú Departament de Matemàtica Aplicada IV Avinguda Víctor Balaguer s/n 08800 Vilanova i la Geltrú (Spain)
@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},
     pages = {1253--1283},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {57},
     number = {4},
     year = {2007},
     doi = {10.5802/aif.2293},
     mrnumber = {2339331},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2293/}
}
TY  - JOUR
AU  - Guàrdia, Jordi
TI  - Jacobian Nullwerte, periods and symmetric equations for hyperelliptic curves
JO  - Annales de l'Institut Fourier
PY  - 2007
SP  - 1253
EP  - 1283
VL  - 57
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2293/
DO  - 10.5802/aif.2293
LA  - en
ID  - AIF_2007__57_4_1253_0
ER  - 
%0 Journal Article
%A Guàrdia, Jordi
%T Jacobian Nullwerte, periods and symmetric equations for hyperelliptic curves
%J Annales de l'Institut Fourier
%D 2007
%P 1253-1283
%V 57
%N 4
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2293/
%R 10.5802/aif.2293
%G en
%F AIF_2007__57_4_1253_0
Guàrdia, Jordi. Jacobian Nullwerte, periods and symmetric equations for hyperelliptic curves. Annales de l'Institut Fourier, Tome 57 (2007) no. 4, pp. 1253-1283. doi : 10.5802/aif.2293. http://www.numdam.org/articles/10.5802/aif.2293/

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

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

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

[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.), Volume 13, World Sci. Publ., Hackensack, NJ, 2005, pp. 71-83 | MR | Zbl

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

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

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

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

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

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

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

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

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

[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

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

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

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

[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 | Zbl

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

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

[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, Volume XI (1851), pp. 362-468

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

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

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

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

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

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

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

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

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

Cité par Sources :