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Original Articles

Computing the Modular Degree of an Elliptic Curve

Pages 487-502 | Published online: 03 Apr 2012

REFERENCES

  • Abbes , A. and Ullmo , E. 1996 . “À propos dela conjecture de Manin pour les courbes elliptiques modulaires” . Compositio Math. , 103 : 269 – 286 . [Abbes and Ullmo 96]
  • Atkin , A. and Lehner , J. 1970 . “Hecke operators on γ).” . Math. Ann. , 185 : 134 – 160 . [Atkin and Lehner 70]
  • Barban , M. 1966 . “The 'Large Sieve' Method and its Applications in the Theory of Numbers” (Russian) . Uspehki Mat. Nauk , 21 ( l (127) : 51 – 102 . [Barban 64], Russian Math Surveys 21:1 (1966), 49–103
  • Birch , B. 1991 . “Hecke Action on Classes of Ternary Quadratic Forms.”. ” . In Computational Number Theory, (Debrecen, 1989) Edited by: Pethö , A. , Pohst , M. , Williams , H. and Zimmer , H. 191 – 212 . Berlin : de Gruyter. . [Birch 91]
  • Birch , B. and Swinnerton-Dyer , H. 1975 . “Elliptic Curves and Modular Functions.”. ” . In Modular Functions of One variable IV Edited by: Birch , B. and Kuyk , W. Vol. 476 , 32 Berlin : Springer- Verlag. . [Birch and Swinnerton-Dyer 75], Lecture Notes in Math
  • Breuil , C. , Conrad , B. , Diamond , F. and Taylor , R. 2001 . “On the Modularity of Elliptic Curves over Q: Wild 3-Adic Exercises.” . J. Amer. Math. Soc. , 14 : 843 – 939 . [Breuil et al. 01]
  • Brumer , A. and McGuinness , O. 1990 . “The Behavior of the Mordell-Weil Group of Elliptic Curves.” . Bull. Amer. Math. Soc. (N.S.) , 23 : 375 – 382 . [Brumer and McGuinness 90]
  • Brumer , A. and Silverman , J. 1996 . “The Number of Elliptic Curves over Q with Conductor N.” . Manuscripta Math. , 91 : 95 – 102 . [Brumer and Silverman 96]
  • Buhler , J. and Gross , B. 1985 . “Arithmetic on Elliptic Curves with Complex Multiplication II.” . Invent. Math. , 79 : 11 – 29 . [Buhler and Gross 85]
  • Buhler , J. , Schoen , C. and Top , J. 1997 . “Cycles, -Functions and Triple Products of Elliptic Curves.” . J. Reine Angew. Math. , 492 : 93 – 133 . [Buhler et al. 97]
  • Chowla , S. and Erdös , P. 1951 . “A Theorem on the Distribution of Values of L-Series.” . Journ. Indian Math. Soc. , 15A : 11 – 18 . [Chowla and Erdös 51]
  • Coates , J. and Schmidt , C.-G. 1987 . “Iwasawa Theory for the Symmetric Square of an Elliptic Curve.” . J. Reine Angew. Math. , 375/376 : 104 – 156 . [Coates and Schmidt 87]
  • Cohen , H. 1993 . A Course in Computational Algebraic Number Theory New York : Springer-Verlag. . [Cohen 93], Graduate Texts in Mathematics 138
  • Cohen , H. 2000 . Advanced Topics in Computational Number Theory New York : Springer-Verlag. . [Cohen 00], Graduate Texts in Mathematics 193
  • Cohen , H. and Lenstra , H. 1984 . “Heuristics on Class Groups of Number Fields.”. ” . In Number Theory, Noordwijkerhout 1983 Edited by: Jager , H. 33 – 62 . Berlin : Springer-Verlag. . [Cohen and Lenstra 84], Lecture Notes in Math. 1068
  • Cremona , J. 1992 . Algorithms for Modular Elliptic Curves., , Second edition Cambridge : Cambridge University Press. . [Cremona 92], 1997
  • Cremona , J. 1995 . “Computing the Degree of the Modular Parametrization of a Modular Elliptic Curve.” . Math. Comp. , 64 ( 211 ) : 1235 – 1250 . [Cremona 95]
  • Cremona , J. 1997 . “Computing Periods of Cusp Forms and Modular Elliptic Curves.” . Experiment. Math. , : 97 – 107 . [Cremona 97]
  • Duke , W. and Kowalski , E. 2000 . “A problem of Linnik for Elliptic Curves and Mean-Value Estimates for Automorphic Representations.” . Invent. Math. , 139 : 39 [Duke and Kowalski 00]
  • Edixhoven , B. 1991 . “On the Manin Constants of Modular Elliptic Curves.”. ” . In Arithmetic Algebraic Geometry (Texel, 1989) Edited by: van der Geer , G. , Oort , F. and Steenbrink , J. 25 – 39 . Boston , MA : Birkhäuser Boston. . [Edixhoven 91], Progr. Math. 89
  • Eichler , M. 1973 . “The Basis Problem for Modular Forms and the Traces of the Hecke Operators.”. ” . In Modular functions of One Variable I Edited by: Kuyk , W. Vol. 320 , 75 – 151 . Berlin : Springer-Verlag. . [Eichler 73], Lecture Notes in Math
  • Elliott , P. 1980 . Probabilistic Number Theory II. Central Limit Theorems Vol. 240 , Berlin-New York : Springer–Verlag. . [Elliott 80], Grundlehren der Mathematischen Wissenschaften
  • Flach , M. 1993 . “On the Degree of Modular Parametrizations.”. ” . In Séminaire de Théorie des Nombres, Paris 1991–92 Edited by: David , S. 23 – 36 . Boston , MA : Birkhäuser Boston. . [Flach 93], Progr. Math. 116
  • Frey , G. 1987 . “Elliptic Curves and Solutions of A − B = C. ” . In Séminaire de Théorie des Nombres, Paris 1985–86 Edited by: Goldstein , C. 39 – 51 . Boston , MA : Birkhäuser Boston. . [Frey 87], Progr. Math. 71
  • Frey , G. and Müller , M. 1999 . “Arithmetic of Modular Curves and Applications.”. ” . In Algorithmic Algebra and Number Theory (Heidelberg, 1997) Edited by: Matzat , B. , Greuel , G.-M. and Hiss , G. 11 – 48 . Berlin : Springer-Verlag. . [Frey and Müller 99]
  • Gelbart , S. and Jacquet , H. 1978 . “A Relation between Automorphic Representations of GL(2) and GL(3).” . Ann. Sci. École Norm. Sup. (4) , 11 : 471 – 542 . [Gelbart and Jacquet 78]
  • Gross , B. and Zagier , D. 1986 . “Heegner Points and Derivatives of -Series.” . Invent. Math. , 84 : 225 – 320 . [Gross and Zagier 86]
  • Hoffstein , J. and Lockhart , P. 1994 . “Coefficients of Maass Forms and the Siegel Zero.” . Ann. of. Math. (2) , 140 : 161 – 181 . [Hoffstein and Lockhart 94], Appendix by D. Goldfeld, J. Hoffstein, and D. Lieman
  • Jacobson , M. 1998 . “Experimental Results on Class Groups of Real Quadratic Fields.”. ” . In Algorithmic Number Theory (Portland, OR, 1998) Edited by: Buhler , J. 463 – 474 . Berlin : Springer-Verlag. . [Jacobson 98], Lecture Notes in Comput. Sci. 1423
  • Katz , N. and Mazur , B. 1985 . Arithmetic Moduli of Elliptic Curves, Annals of Mathematics Studies 108 Princeton , NJ : Princeton University Press. . [Katz and Mazur 85]
  • Kohel , D. and Stein , W. 2000 . “Component Groups of Quotients of).”. ” . In Algorithmic Number Theory (Leiden, 2000) Edited by: Bosma , W. 405 – 412 . Berlin : Springer-Verlag. . [Kohel and Stein 00], Lecture Notes in Comput. Sci. 1838
  • Lavrik , A. 1967 . “Functional Equations of the Dirichlet Functions (Russian).” . Izv. Akad. Nauk SSSR Ser. Mat. , 31 : 431 – 442 . [Lavrik 67]
  • Luo , W. 1999 . “Values of Symmetric Square -Functions at 1.” . J. Reine Angew. Math. , 506 : 215 – 235 . [Luo 99]
  • Manin , J. 1972 . “Parabolic Points and Zeta Functions of Modular Curves (Russian).” . Izv. Akad. Nauk SSSR Ser. Mat. , 36 : 19 – 66 . [Manin 72]
  • Mazur , B. 1977 . “Modular Curves and the Eisenstein Ideal.” . Inst. Hautes Études Sci. Publ. Math. , 47 : 33 – 186 . [Mazur 77], (1978)
  • Merel , L. 1993 . “Intersections sur des courbes modularies.” . Manuscripta Math. , 80 : 283 – 289 . [Merel 93]
  • Merel , L. 1995 . “Homologie des courbes modulaires affines et paramétrisations modulaires.”. ” . In Elliptic Curves, Modular Forms, and Fermat's Last Theorem (Hong Kong, 1993) Edited by: Coates , J. and Yau , S.-T. 110 – 130 . Cambridge , MA : International Press. . [Merel 95], Series in Number Theory I
  • Mestre , J.-F. “La méthode des graphes. Exemples et applications.” . Proceedings of the International Conference on Class Numbers and Fundamental Units of Algebraic Number Fields (Katata, 1986) . pp. 217 – 242 . Nagoya : Nagoya Univ. . [Mestre 86]
  • Mestre , J.-F. and Oesterlé , J. 1989 . “Courbes de Weil semi-stables de discriminant une puissance m-ième.” . J. Reine Angew. Math. , 400 : 173 – 184 . [Mestre and Oesterlé 89]
  • Mestre , J.-F. and Schappacher , N. 1991 . “Séries de Kronecker et fonctions des puissances symétriques de courbes elliptiques sur Q.”. ” . In Arithmetic Algebraic Geometry (Texel, 1989) Edited by: van der Geer , G. , Oort , F. and Steenbrink , J. 209 – 245 . Boston , MA : Birkhäuser Boston. . [Mestre and Schappacher 91], Progr. Math., 89
  • Neumann , O. 1971 . “Elliptische Kurven mit vorgeschriebenem Reduktionsverhalten. I.” . Math. Nachr. , 49 : 107 – 123 . [Neumann 71]
  • Neumann , O. 1973 . “Elliptische Kurven mit vorgeschriebenem Reduktionsverhalten. II.” . Math. Nachr. , 56 : 269 – 280 . [Neumann 73]
  • Pizer , A. 1976 . “On the Arithmetic of Quaternion Algebras II.” . J. Math. Soc. Japan , 28 : 676 – 688 . [Pizer 76]
  • Royer , E. 2001 . “Statistiques de la variable aléatoire (Sym, 1).” . Mathematische Annalen , 321 : 667 – 687 . [Royer 01]
  • Schoof , R. 1985 . “Elliptic Curves over Finite Fields and the Computation of Square Roots mod p.” . Math. Comp. , 44 ( 170 ) : 483 – 494 . [Schoof 85]
  • Setzer , B. 1975 . “Elliptic Curves of Prime Conductor.” . J. London Math. Soc. (2) , 10 : 367 – 378 . [Setzer 75]
  • Shimura , G. 1975 . “On the Holomorphy of Certain Dirichlet Series.” . Proc. London Math. Soc. (3) , 31 : 79 – 98 . [Shimura 75]
  • Shimura , G. 1976 . “The Special Values of the Zeta Functions Associated with Cusp Forms.” . Comm. Pure Appl. Math , 29 : 783 – 804 . [Shimura 76]
  • Stein , W. and Watkins , M. 2002 . “A Database of Elliptic Curves–First Report.”. ” . In Algorithmic Number Theory (Sydney, 2002) 267 – 275 . Berlin : Springer-Verlag. . [Stein and Watkins 02], Lecture Notes in Comput. Sci. 2369
  • Stevens , G. 1989 . “Stickelberger Elements and Modular Parametrizations of Elliptic Curves.” . Invent. Math. , 98 : 75 – 106 . [Stevens 89]
  • Tate , J. 1975 . “Algorithm for Determining the Type of a Singular Fiber in an Elliptic Pencil.”. ” . In Modular Functions of One Variable IV Edited by: Birch , B. and Kuyk , W. Vol. 476 , 33 – 52 . Berlin : Springer-Verlag. . [Tate 75], Lecture Notes in Math
  • Wiles , A. 1995 . “Modular Elliptic Curves and Fermat's Last Theorem.” . Ann. of Math. (2) , 141 : 443 – 551 . [Wiles 95]
  • Womack , T. 2002 . [Womack 02], Available from World Wide Web: http://www.maths.nott.ac.uk/personal/jec/ftp/data/womack.data
  • Wong , S. 2001 . “Exponents of Class Groups and Elliptic Curves.” . J. Number Theory , 89 : 114 – 120 . [Wong 01]
  • Zagier , D. 1985 . “Modular Parametrizations of Elliptic Curves.” . Canad. Math. Bull. , 28 : 372 – 384 . [Zagier 85]
  • Zagier , D. and Gangl , H. 2000 . “Classical and Elliptic Polylogarithms and Special Values of -Series.”. ” . In The Arithmetic and Geometry of Algebraic Cycles (Banff, AB, 1998) Edited by: Gordon , B. , Lewis , J. , Müller-Stach , S. , Saito , S. and Yui , N. 561 – 615 . Dordrecht : Kluwer Acad. Publ. . [Zagier and Gangl 00], NATO Sci. Ser. C Math. Phys. Sci., 548

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