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Jordi Guàrdia
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, 57 no. 4 (2007), p. 1253-1283, doi: 10.5802/aif.2293
Article PDF | Reviews MR 2339331 | Zbl pre05176621
Class. Math.: 11G30, 14H42
Keywords: Hyperelliptic curves, periods, Jacobian Nullwerte
[1] E. Arbarello, M. Cornalba, P. A. Griffiths & J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 267, Springer-Verlag, New York, 1985 MR 770932 | Zbl 0559.14017
[2] Pilar Bayer & Jordi Guàrdia, “Hyperbolic uniformization of the Fermat curves”, Ramanjujan J. 12 (2006), p. 207-223 Article | MR 2286246 | Zbl 05119608
[3] B. J. Birch & W. Kuyk (ed.), Modular functions of one variable. IV, Springer-Verlag, Berlin, 1975, Lecture Notes in Mathematics, Vol. 476 MR 376533
[4] Gabriel Cardona & Jordi Quer, Field of moduli and field of definition for curves of genus 2, Computational aspects of algebraic curves, Lecture Notes Ser. Comput. 13, World Sci. Publ., Hackensack, NJ, 2005, p. 71–83 MR 2181874 | Zbl 1126.14031
[5] J. E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, 1992 MR 1201151 | Zbl 0758.14042
[6] Ferdinand Georg Frobenius, “Über die constanten Factoren der Thetareihen”, J. reine angew. Math. 98 (1885), p. 241-260
[7] Josep González, Jordi Guàrdia & Victor Rotger, “Abelian surfaces of ${\rm GL}_2$-type as Jacobians of curves”, Acta Arith. 116 (2005) no. 3, p. 263-287 Article | MR 2114780 | Zbl 02169948
[8] Enrique González-Jiménez & Josep González, “Modular curves of genus 2”, Math. Comp. 72 (2003) no. 241, p. 397-418 (electronic) Article | MR 1933828 | Zbl 1081.11042
[9] Enrique González-Jiménez, Josep González & Jordi Guàrdia, Computations on modular Jacobian surfaces, Algorithmic number theory (Sydney, 2002), Lecture Notes in Comput. Sci. 2369, Springer, 2002, p. 189–197 MR 2041083 | Zbl 1055.11038
[10] Jordi Guàrdia, “Jacobian nullwerte and algebraic equations”, J. Algebra 253 (2002) no. 1, p. 112-132 Article | MR 1925010 | Zbl 1054.14041
[11] Jordi Guàrdia, “Jacobi Thetanullwerte, periods of elliptic curves and minimal equations”, Math. Res. Lett. 11 (2004) no. 1, p. 115-123 MR 2046204 | Zbl 02104761
[12] Jordi Guàrdia, Eugenia Torres & Montserrat Vela, Stable models of elliptic curves, ring class fields, and complex multiplication, Algorithmic number theory, Lecture Notes in Comput. Sci. 3076, Springer, 2004, p. 250–262 MR 2137358 | Zbl 02194079
[13] Jun-ichi Igusa, “On Jacobi’s derivative formula and its generalizations”, Amer. J. Math. 102 (1980) no. 2, p. 409-446 Article | Zbl 0433.14033
[14] Jun-ichi Igusa, On the nullwerte of Jacobians of odd theta functions, Symposia Mathematica, Vol. XXIV (Sympos., INDAM, Rome, 1979), Academic Press, 1981, p. 83–95 MR 619242
[15] Jun-ichi Igusa, “Problems on abelian functions at the time of Poincaré and some at present”, Bull. Amer. Math. Soc. (N.S.) 6 (1982) no. 2, p. 161-174 Article | MR 640943 | Zbl 0484.14015
[16] Jun-ichi Igusa, “Multiplicity one theorem and problems related to Jacobi’s formula”, Amer. J. Math. 105 (1983) no. 1, p. 157-187 Article | Zbl 0527.14037
[17] P. Lockhart, “On the discriminant of a hyperelliptic curve”, Trans. Amer. Math. Soc. 342 (1994) no. 2, p. 729-752 Article | MR 1195511 | Zbl 0815.11031
[18] MAGMA, “http://magma.math.usyd.edu.au/magma/”, University of Sydney, 2004
[19] Henry McKean & Victor Moll, Elliptic curves, Cambridge University Press, Cambridge, 1997, Function theory, geometry, arithmetic MR 1471703 | Zbl 0895.11002
[20] Jean-François Mestre, Construction de courbes de genre $2$ à partir de leurs modules, Effective methods in algebraic geometry (Castiglioncello, 1990), Progr. Math. 94, Birkhäuser Boston, 1991, p. 313–334 MR 1106431 | Zbl 0752.14027
[21] David Mumford, 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 742776 | Zbl 0549.14014
[22] G. Rosenhain, “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 XI (1851), p. 362-468
[23] Goro Shimura, Abelian varieties with complex multiplication and modular functions, Princeton Mathematical Series 46, Princeton University Press, Princeton, NJ, 1998 MR 1492449 | Zbl 0908.11023
[24] Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics 106, Springer-Verlag, New York, 1992, Corrected reprint of the 1986 original MR 1329092 | Zbl 0585.14026
[25] Koichi Takase, “A generalization of Rosenhain’s normal form for hyperelliptic curves with an application”, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996) no. 7, p. 162-165 Article | Zbl 0924.14016
[26] J. Thomae, “Beitrag zur Bestimmung von $\theta (0,0,...,0)$ durch die Klassenmoduln algebraischer Funktionen”, J. reine angew. Math. 71 (1870), p. 201-222 Article | JFM 02.0244.01
[27] Paul van Wamelen, “Examples of genus two CM curves defined over the rationals”, Math. Comp. 68 (1999) no. 225, p. 307-320 Article | MR 1609658 | Zbl 0906.14025
[28] Xiang Dong Wang, “$2$-dimensional simple factors of $J_0(N)$”, Manuscripta Math. 87 (1995) no. 2, p. 179-197 Article | MR 1334940 | Zbl 0846.14007
[29] Hermann-Josef Weber, “Hyperelliptic simple factors of $J_0(N)$ with dimension at least $3$”, Experiment. Math. 6 (1997) no. 4, p. 273-287 Article | MR 1606908 | Zbl 1115.14304
[30] André Weil, “Sur les périodes des intégrales abéliennes”, Comm. Pure Appl. Math. 29 (1976) no. 6, p. 813-819 Article | MR 422164 | Zbl 0342.14020
[31] Annegret Weng, “A class of hyperelliptic CM-curves of genus three”, J. Ramanujan Math. Soc. 16 (2001) no. 4, p. 339-372 MR 1877806 | Zbl 1066.11028
Jordi Guàrdia
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, 57 no. 4 (2007), p. 1253-1283, doi: 10.5802/aif.2293
Article PDF | Reviews MR 2339331 | Zbl pre05176621
Class. Math.: 11G30, 14H42
Keywords: Hyperelliptic curves, periods, Jacobian Nullwerte
Résumé - Abstract
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.
Bibliography
[2] Pilar Bayer & Jordi Guàrdia, “Hyperbolic uniformization of the Fermat curves”, Ramanjujan J. 12 (2006), p. 207-223 Article | MR 2286246 | Zbl 05119608
[3] B. J. Birch & W. Kuyk (ed.), Modular functions of one variable. IV, Springer-Verlag, Berlin, 1975, Lecture Notes in Mathematics, Vol. 476 MR 376533
[4] Gabriel Cardona & Jordi Quer, Field of moduli and field of definition for curves of genus 2, Computational aspects of algebraic curves, Lecture Notes Ser. Comput. 13, World Sci. Publ., Hackensack, NJ, 2005, p. 71–83 MR 2181874 | Zbl 1126.14031
[5] J. E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, 1992 MR 1201151 | Zbl 0758.14042
[6] Ferdinand Georg Frobenius, “Über die constanten Factoren der Thetareihen”, J. reine angew. Math. 98 (1885), p. 241-260
[7] Josep González, Jordi Guàrdia & Victor Rotger, “Abelian surfaces of ${\rm GL}_2$-type as Jacobians of curves”, Acta Arith. 116 (2005) no. 3, p. 263-287 Article | MR 2114780 | Zbl 02169948
[8] Enrique González-Jiménez & Josep González, “Modular curves of genus 2”, Math. Comp. 72 (2003) no. 241, p. 397-418 (electronic) Article | MR 1933828 | Zbl 1081.11042
[9] Enrique González-Jiménez, Josep González & Jordi Guàrdia, Computations on modular Jacobian surfaces, Algorithmic number theory (Sydney, 2002), Lecture Notes in Comput. Sci. 2369, Springer, 2002, p. 189–197 MR 2041083 | Zbl 1055.11038
[10] Jordi Guàrdia, “Jacobian nullwerte and algebraic equations”, J. Algebra 253 (2002) no. 1, p. 112-132 Article | MR 1925010 | Zbl 1054.14041
[11] Jordi Guàrdia, “Jacobi Thetanullwerte, periods of elliptic curves and minimal equations”, Math. Res. Lett. 11 (2004) no. 1, p. 115-123 MR 2046204 | Zbl 02104761
[12] Jordi Guàrdia, Eugenia Torres & Montserrat Vela, Stable models of elliptic curves, ring class fields, and complex multiplication, Algorithmic number theory, Lecture Notes in Comput. Sci. 3076, Springer, 2004, p. 250–262 MR 2137358 | Zbl 02194079
[13] Jun-ichi Igusa, “On Jacobi’s derivative formula and its generalizations”, Amer. J. Math. 102 (1980) no. 2, p. 409-446 Article | Zbl 0433.14033
[14] Jun-ichi Igusa, On the nullwerte of Jacobians of odd theta functions, Symposia Mathematica, Vol. XXIV (Sympos., INDAM, Rome, 1979), Academic Press, 1981, p. 83–95 MR 619242
[15] Jun-ichi Igusa, “Problems on abelian functions at the time of Poincaré and some at present”, Bull. Amer. Math. Soc. (N.S.) 6 (1982) no. 2, p. 161-174 Article | MR 640943 | Zbl 0484.14015
[16] Jun-ichi Igusa, “Multiplicity one theorem and problems related to Jacobi’s formula”, Amer. J. Math. 105 (1983) no. 1, p. 157-187 Article | Zbl 0527.14037
[17] P. Lockhart, “On the discriminant of a hyperelliptic curve”, Trans. Amer. Math. Soc. 342 (1994) no. 2, p. 729-752 Article | MR 1195511 | Zbl 0815.11031
[18] MAGMA, “http://magma.math.usyd.edu.au/magma/”, University of Sydney, 2004
[19] Henry McKean & Victor Moll, Elliptic curves, Cambridge University Press, Cambridge, 1997, Function theory, geometry, arithmetic MR 1471703 | Zbl 0895.11002
[20] Jean-François Mestre, Construction de courbes de genre $2$ à partir de leurs modules, Effective methods in algebraic geometry (Castiglioncello, 1990), Progr. Math. 94, Birkhäuser Boston, 1991, p. 313–334 MR 1106431 | Zbl 0752.14027
[21] David Mumford, 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 742776 | Zbl 0549.14014
[22] G. Rosenhain, “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 XI (1851), p. 362-468
[23] Goro Shimura, Abelian varieties with complex multiplication and modular functions, Princeton Mathematical Series 46, Princeton University Press, Princeton, NJ, 1998 MR 1492449 | Zbl 0908.11023
[24] Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics 106, Springer-Verlag, New York, 1992, Corrected reprint of the 1986 original MR 1329092 | Zbl 0585.14026
[25] Koichi Takase, “A generalization of Rosenhain’s normal form for hyperelliptic curves with an application”, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996) no. 7, p. 162-165 Article | Zbl 0924.14016
[26] J. Thomae, “Beitrag zur Bestimmung von $\theta (0,0,...,0)$ durch die Klassenmoduln algebraischer Funktionen”, J. reine angew. Math. 71 (1870), p. 201-222 Article | JFM 02.0244.01
[27] Paul van Wamelen, “Examples of genus two CM curves defined over the rationals”, Math. Comp. 68 (1999) no. 225, p. 307-320 Article | MR 1609658 | Zbl 0906.14025
[28] Xiang Dong Wang, “$2$-dimensional simple factors of $J_0(N)$”, Manuscripta Math. 87 (1995) no. 2, p. 179-197 Article | MR 1334940 | Zbl 0846.14007
[29] Hermann-Josef Weber, “Hyperelliptic simple factors of $J_0(N)$ with dimension at least $3$”, Experiment. Math. 6 (1997) no. 4, p. 273-287 Article | MR 1606908 | Zbl 1115.14304
[30] André Weil, “Sur les périodes des intégrales abéliennes”, Comm. Pure Appl. Math. 29 (1976) no. 6, p. 813-819 Article | MR 422164 | Zbl 0342.14020
[31] Annegret Weng, “A class of hyperelliptic CM-curves of genus three”, J. Ramanujan Math. Soc. 16 (2001) no. 4, p. 339-372 MR 1877806 | Zbl 1066.11028
© Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310