Skip to main content

Elliptic Curves over Finite Fields

  • Chapter
The Arithmetic of Elliptic Curves

Part of the book series: Graduate Texts in Mathematics ((GTM,volume 106))

  • 22k Accesses

Abstract

In this chapter we study elliptic curves defined over a finite fieldĀ \(\mathbb{F}_{q}\). The most important arithmetic quantity associated to such a curve is its number of rational points.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 59.95
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. M.Ā Abdalla, M.Ā Bellare, and P.Ā Rogaway. The oracle Diffie-Hellman assumptions and an analysis of DHIES. In Topics in cryptologyā€”CT-RSA 2001 (San Francisco, CA), volume 2020 of Lecture Notes in Comput. Sci., pages 143ā€“158. Springer, Berlin, 2001.

    Google ScholarĀ 

  2. D.Ā Abramovich. Formal finiteness and the torsion conjecture on elliptic curves. A footnote to a paper: ā€œRational torsion of prime order in elliptic curves over number fieldsā€ [AstĆ©risque No.Ā 228 (1995), 3, 81ā€“100] by S. Kamienny and B. Mazur. AstĆ©risque, (228):3, 5ā€“17, 1995. Columbia University Number Theory Seminar (New York, 1992).

    Google ScholarĀ 

  3. L.Ā V. Ahlfors. Complex analysis. McGraw-Hill Book Co., New York, third edition, 1978. An introduction to the theory of analytic functions of one complex variable, International Series in Pure and Applied Mathematics.

    Google ScholarĀ 

  4. T.Ā M. Apostol. Introduction to analytic number theory. Springer-Verlag, New York, 1976. Undergraduate Texts in Mathematics.

    Google ScholarĀ 

  5. T.Ā M. Apostol. Modular functions and Dirichlet series in number theory, volumeĀ 41 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1990.

    Google ScholarĀ 

  6. N.Ā Arthaud. On Birch and Swinnerton-Dyerā€™s conjecture for elliptic curves with complex multiplication. I. Compositio Math., 37(2):209ā€“232, 1978.

    Google ScholarĀ 

  7. E.Ā Artin. Galois theory. Dover Publications Inc., Mineola, NY, second edition, 1998. Edited and with a supplemental chapter by Arthur N. Milgram.

    Google ScholarĀ 

  8. M.Ā F. Atiyah and I.Ā G. Macdonald. Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, Mass.ā€“Londonā€“Don Mills, Ont., 1969.

    Google ScholarĀ 

  9. M.Ā F. Atiyah and C.Ā T.Ā C. Wall. Cohomology of groups. In Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), pages 94ā€“115. Thompson, Washington, D.C., 1967.

    Google ScholarĀ 

  10. A.Ā O.Ā L. Atkin and F.Ā Morain. Elliptic curves and primality proving. Math. Comp., 61(203):29ā€“68, 1993.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  11. A.Ā Baker. Transcendental number theory. Cambridge Mathematical Library. Cambridge University Press, Cambridge, second edition, 1990.

    MATHĀ  Google ScholarĀ 

  12. A.Ā Baker and J.Ā Coates. Integer points on curves of genus 1. Proc. Cambridge Philos. Soc., 67:595ā€“602, 1970.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  13. R.Ā Balasubramanian and N.Ā Koblitz. The improbability that an elliptic curve has subexponential discrete log problem under the Menezes-Okamoto-Vanstone algorithm. J. Cryptology, 11(2):141ā€“145, 1998.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  14. A.Ā F. Beardon. Iteration of Rational Functions, volume 132 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1991. Complex analytic dynamical systems.

    Google ScholarĀ 

  15. E.Ā Bekyel. The density of elliptic curves having a global minimal Weierstrass equation. J. Number Theory, 109(1):41ā€“58, 2004.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  16. D.Ā Bernstein and T.Ā Lange. Faster addition and doubling on elliptic curves. In Advances in cryptologyā€”ASIACRYPT 2007, volume 4833 of Lecture Notes in Comput. Sci., pages 29ā€“50. Springer, Berlin, 2007.

    Google ScholarĀ 

  17. B.Ā J. Birch. Cyclotomic fields and Kummer extensions. In Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), pages 85ā€“93. Thompson, Washington, D.C., 1967.

    Google ScholarĀ 

  18. B.Ā J. Birch. How the number of points of an elliptic curve over a fixed prime field varies. J. London Math. Soc., 43:57ā€“60, 1968.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  19. B.Ā J. Birch and W.Ā Kuyk, editors. Modular functions of one variable. IV. Springer-Verlag, Berlin, 1975. Lecture Notes in Mathematics, Vol. 476.

    Google ScholarĀ 

  20. B.Ā J. Birch and H.Ā P.Ā F. Swinnerton-Dyer. Notes on elliptic curves. I. J. Reine Angew. Math., 212:7ā€“25, 1963.

    Google ScholarĀ 

  21. B.Ā J. Birch and H.Ā P.Ā F. Swinnerton-Dyer. Elliptic curves and modular functions. In Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pages 2ā€“32. Lecture Notes in Math., Vol. 476. Springer, Berlin, 1975.

    Google ScholarĀ 

  22. I.Ā F. Blake, G.Ā Seroussi, and N.Ā P. Smart. Elliptic curves in cryptography, volume 265 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2000. Reprint of the 1999 original.

    Google ScholarĀ 

  23. D.Ā Boneh and M.Ā Franklin. Identity-based encryption from the Weil pairing. In Advances in Cryptologyā€”CRYPTO 2001 (Santa Barbara, CA), volume 2139 of Lecture Notes in Comput. Sci., pages 213ā€“229. Springer, Berlin, 2001.

    Google ScholarĀ 

  24. D.Ā Boneh, B.Ā Lynn, and H.Ā Shacham. Short signatures from the Weil pairing. In Advances in cryptologyā€”ASIACRYPT 2001 (Gold Coast), volume 2248 of Lecture Notes in Comput. Sci., pages 514ā€“532. Springer, Berlin, 2001.

    Google ScholarĀ 

  25. A.Ā I. Borevich and I.Ā R. Shafarevich. Number theory. Translated from the Russian by Newcomb Greenleaf. Pure and Applied Mathematics, Vol. 20. Academic Press, New York, 1966.

    Google ScholarĀ 

  26. A.Ā Bremner. On the equation Y 2ā€‰=ā€‰X(X 2 + p). In Number theory and applications (Banff, AB, 1988), volume 265 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., pages 3ā€“22. Kluwer Acad. Publ., Dordrecht, 1989.

    Google ScholarĀ 

  27. A.Ā Bremner and J.Ā W.Ā S. Cassels. On the equation Y 2ā€‰=ā€‰X(X 2 + p). Math. Comp., 42(165):257ā€“264, 1984.

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  28. C.Ā Breuil, B.Ā Conrad, F.Ā Diamond, and R.Ā Taylor. On the modularity of elliptic curves over Q: wild 3-adic exercises. J. Amer. Math. Soc., 14(4):843ā€“939 (electronic), 2001.

    Google ScholarĀ 

  29. F.Ā Brezing and A.Ā Weng. Elliptic curves suitable for pairing based cryptography. Des. Codes Cryptogr., 37(1):133ā€“141, 2005.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  30. M.Ā L. Brown. Note on supersingular primes of elliptic curves over Q. Bull. London Math. Soc., 20(4):293ā€“296, 1988.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  31. W.Ā D. Brownawell and D.Ā W. Masser. Vanishing sums in function fields. Math. Proc. Cambridge Philos. Soc., 100(3):427ā€“434, 1986.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  32. Y.Ā Bugeaud. Bounds for the solutions of superelliptic equations. Compositio Math., 107(2):187ā€“219, 1997.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  33. J.Ā P. Buhler, B.Ā H. Gross, and D.Ā B. Zagier. On the conjecture of Birch and Swinnerton-Dyer for an elliptic curve of rank 3. Math. Comp., 44(170):473ā€“481, 1985.

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  34. E.Ā R. Canfield, P.Ā Erdős, and C.Ā Pomerance. On a problem of Oppenheim concerning ā€œfactorisatio numerorum.ā€ J. Number Theory, 17(1):1ā€“28, 1983.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  35. H.Ā Carayol. Sur les reprĆ©sentations galoisiennes modulo l attachĆ©es aux formes modulaires. Duke Math. J., 59(3):785ā€“801, 1989.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  36. J.Ā W.Ā S. Cassels. A note on the division values of ā„˜(u). Proc. Cambridge Philos. Soc., 45:167ā€“172, 1949.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  37. J.Ā W.Ā S. Cassels. Arithmetic on curves of genus 1. III. The Tate-Å afarevič and Selmer groups. Proc. London Math. Soc. (3), 12:259ā€“296, 1962.

    Google ScholarĀ 

  38. J.Ā W.Ā S. Cassels. Arithmetic on curves of genus 1. IV. Proof of the Hauptvermutung. J. Reine Angew. Math., 211:95ā€“112, 1962.

    Google ScholarĀ 

  39. J.Ā W.Ā S. Cassels. Arithmetic on curves of genus 1. V. Two counterexamples. J. London Math. Soc., 38:244ā€“248, 1963.

    Google ScholarĀ 

  40. J.Ā W.Ā S. Cassels. Arithmetic on curves of genus 1. VIII. On conjectures of Birch and Swinnerton-Dyer. J. Reine Angew. Math., 217:180ā€“199, 1965.

    Google ScholarĀ 

  41. J.Ā W.Ā S. Cassels. Diophantine equations with special reference to elliptic curves. J. London Math. Soc., 41:193ā€“291, 1966.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  42. J.Ā W.Ā S. Cassels. Global fields. In Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), pages 42ā€“84. Thompson, Washington, D.C., 1967.

    Google ScholarĀ 

  43. J.Ā W.Ā S. Cassels. Lectures on elliptic curves, volumeĀ 24 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1991.

    Google ScholarĀ 

  44. T.Ā Chinburg. An introduction to Arakelov intersection theory. In Arithmetic geometry (Storrs, Conn., 1984), pages 289ā€“307. Springer, New York, 1986.

    Google ScholarĀ 

  45. D.Ā V. Chudnovsky and G.Ā V. Chudnovsky. PadĆ© approximations and Diophantine geometry. Proc. Nat. Acad. Sci. U.S.A., 82(8):2212ā€“2216, 1985.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  46. C.Ā H. Clemens. A scrapbook of complex curve theory, volumeĀ 55 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, 2003.

    Google ScholarĀ 

  47. L.Ā Clozel, M.Ā Harris, and R.Ā Taylor. Automorphy for some l-adic lifts of automorphic mod l representations. 2007. IHES Publ. Math., submitted.

    Google ScholarĀ 

  48. J.Ā Coates. Construction of rational functions on a curve. Proc. Cambridge Philos. Soc., 68:105ā€“123, 1970.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  49. J.Ā Coates and A.Ā Wiles. On the conjecture of Birch and Swinnerton-Dyer. Invent. Math., 39(3):223ā€“251, 1977.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  50. H.Ā Cohen. A Course in Computational Algebraic Number Theory, volume 138 of Graduate Texts in Mathematics. Springer-Verlag, Berlin, 1993.

    Google ScholarĀ 

  51. H.Ā Cohen, G.Ā Frey, R.Ā Avanzi, C.Ā Doche, T.Ā Lange, K.Ā Nguyen, and F.Ā Vercauteren, editors. Handbook of Elliptic and Hyperelliptic Curve Cryptography. Discrete Mathematics and Its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2006.

    Google ScholarĀ 

  52. D.Ā A. Cox. The arithmetic-geometric mean of Gauss. Enseign. Math. (2), 30(3-4):275ā€“330, 1984.

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  53. J.Ā Cremona. Elliptic Curve Data. http://sage.math.washington. edu/cremona/index.html , http://www.math.utexas.edu/users/ tornaria/cnt/cremona.html .

  54. J.Ā E. Cremona. Algorithms for modular elliptic curves. Cambridge University Press, Cambridge, second edition, 1997. available free online at www.warwick.ac.uk/ staff/J.E.Cremona/book/fulltext/index.html .

  55. J.Ā E. Cremona, M.Ā Prickett, and S.Ā Siksek. Height difference bounds for elliptic curves over number fields. J. Number Theory, 116(1):42ā€“68, 2006.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  56. L.Ā V. Danilov. The Diophantine equation x 3 āˆ’ y 2ā€‰=ā€‰k and a conjecture of M. Hall. Mat. Zametki, 32(3):273ā€“275, 425, 1982. English translation: Math. Notes Acad. Sci. USSR 32 (1982), no. 3ā€“4, 617ā€“618 (1983).

    Google ScholarĀ 

  57. H.Ā Davenport. On f 3ā€‰(t) āˆ’ g 2ā€‰(t). Norske Vid. Selsk. Forh. (Trondheim), 38:86ā€“87, 1965.

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  58. S.Ā David. Minorations de formes linĆ©aires de logarithmes elliptiques. MĆ©m. Soc. Math. France (N.S.), (62):iv+143, 1995.

    Google ScholarĀ 

  59. B.Ā M.Ā M. deĀ Weger. Algorithms for Diophantine equations, volumeĀ 65 of CWI Tract. Stichting Mathematisch Centrum Centrum voor Wiskunde en Informatica, Amsterdam, 1989.

    Google ScholarĀ 

  60. M.Ā Deuring. Die Typen der Multiplikatorenringe elliptischer Funktionenkƶrper. Abh. Math. Sem. Hansischen Univ., 14:197ā€“272, 1941.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  61. M.Ā Deuring. Die Zetafunktion einer algebraischen Kurve vom Geschlechte Eins. Nachr. Akad. Wiss. Gƶttingen. Math.-Phys. Kl. Math.-Phys.-Chem. Abt., 1953:85ā€“94, 1953.

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  62. M.Ā Deuring. Die Zetafunktion einer algebraischen Kurve vom Geschlechte Eins. II. Nachr. Akad. Wiss. Gƶttingen. Math.-Phys. Kl. IIa., 1955:13ā€“42, 1955.

    Google ScholarĀ 

  63. M.Ā Deuring. Die Zetafunktion einer algebraischen Kurve vom Geschlechte Eins. III. Nachr. Akad. Wiss. Gƶttingen. Math.-Phys. Kl. IIa., 1956:37ā€“76, 1956.

    Google ScholarĀ 

  64. M.Ā Deuring. Die Zetafunktion einer algebraischen Kurve vom Geschlechte Eins. IV. Nachr. Akad. Wiss. Gƶttingen. Math.-Phys. Kl. IIa., 1957:55ā€“80, 1957.

    Google ScholarĀ 

  65. W.Ā Diffie and M.Ā E. Hellman. New directions in cryptography. IEEE Trans. Information Theory, IT-22(6):644ā€“654, 1976.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  66. L.Ā Dirichlet. Ɯber den biquadratischen Charakter der Zahl ā€œZwei.ā€ J. Reine Angew. Math., 57:187ā€“188, 1860.

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  67. Z.Ā Djabri, E.Ā F. Schaefer, and N.Ā P. Smart. Computing the p-Selmer group of an elliptic curve. Trans. Amer. Math. Soc., 352(12):5583ā€“5597, 2000.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  68. D.Ā S. Dummit and R.Ā M. Foote. Abstract algebra. John Wiley & Sons Inc., Hoboken, NJ, third edition, 2004.

    MATHĀ  Google ScholarĀ 

  69. R.Ā Dupont, A.Ā Enge, and F.Ā Morain. Building curves with arbitrary small MOV degree over finite prime fields. J. Cryptology, 18(2):79ā€“89, 2005.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  70. B.Ā Dwork. On the rationality of the zeta function of an algebraic variety. Amer. J. Math., 82:631ā€“648, 1960.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  71. H.Ā M. Edwards. A normal form for elliptic curves. Bull. Amer. Math. Soc. (N.S.), 44(3):393ā€“422 (electronic), 2007.

    Google ScholarĀ 

  72. M.Ā Eichler. QuaternƤre quadratische Formen und die Riemannsche Vermutung fĆ¼r die Kongruenzzetafunktion. Arch. Math., 5:355ā€“366, 1954.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  73. D.Ā Eisenbud. Commutative algebra, volume 150 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995. With a view toward algebraic geometry.

    Google ScholarĀ 

  74. T.Ā ElGamal. A public key cryptosystem and a signature scheme based on discrete logarithms. IEEE Trans. Inform. Theory, 31(4):469ā€“472, 1985.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  75. N.Ā Elkies. List of integers x, y with \(x < 10^{18}\), \(0 < \vert x^{3} - y^{2}\vert < x^{1/2}\). www.math.harvard.edu/~elkies/hall.html.

  76. N.Ā Elkies. \(\mathbb{Z}^{28}\) in \(E(\mathbb{Q})\). Number Theory Listserver, May 2006.

    Google ScholarĀ 

  77. N.Ā D. Elkies. The existence of infinitely many supersingular primes for every elliptic curve over \(\mathbb{Q}\). Invent. Math., 89(3):561ā€“567, 1987.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  78. N.Ā D. Elkies. Distribution of supersingular primes. AstĆ©risque, (198-200):127ā€“132 (1992), 1991. JournĆ©es ArithmĆ©tiques, 1989 (Luminy, 1989).

    Google ScholarĀ 

  79. N.Ā D. Elkies. Elliptic and modular curves over finite fields and related computational issues. In Computational perspectives on number theory (Chicago, IL, 1995), volumeĀ 7 of AMS/IP Stud. Adv. Math., pages 21ā€“76. Amer. Math. Soc., Providence, RI, 1998.

    Google ScholarĀ 

  80. J.-H. Evertse. On equations in S-units and the Thue-Mahler equation. Invent. Math., 75(3):561ā€“584, 1984.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  81. J.-H. Evertse and J.Ā H. Silverman. Uniform bounds for the number of solutions to Y nā€‰=ā€‰f(X). Math. Proc. Cambridge Philos. Soc., 100(2):237ā€“248, 1986.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  82. G.Ā Faltings. EndlichkeitssƤtze fĆ¼r abelsche VarietƤten Ć¼ber Zahlkƶrpern. Invent. Math., 73(3):349ā€“366, 1983.

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  83. G.Ā Faltings. Calculus on arithmetic surfaces. Ann. of Math. (2), 119(2):387ā€“424, 1984.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  84. G.Ā Faltings. Finiteness theorems for abelian varieties over number fields. In Arithmetic geometry (Storrs, Conn., 1984), pages 9ā€“27. Springer, New York, 1986. Translated from the German original [Invent.Ā Math.Ā 73 (1983), no.Ā 3, 349ā€“366; ibid.Ā 75 (1984), no.Ā 2, 381; MR 85g:11026ab] by Edward Shipz.

    Google ScholarĀ 

  85. S.Ā Fermigier. Une courbe elliptique dĆ©finie sur Q de rangā€‰ā‰„ā€‰22. Acta Arith., 82(4):359ā€“363, 1997.

    MathSciNetĀ  Google ScholarĀ 

  86. E.Ā V. Flynn and C.Ā Grattoni. Descent via isogeny on elliptic curves with large rational torsion subgroups. J. Symbolic Comput., 43(4):293ā€“303, 2008.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  87. D.Ā Freeman. Constructing pairing-friendly elliptic curves with embedding degree 10. In Algorithmic number theory, volume 4076 of Lecture Notes in Comput. Sci., pages 452ā€“465. Springer, Berlin, 2006.

    Google ScholarĀ 

  88. G.Ā Frey. Links between stable elliptic curves and certain Diophantine equations. Ann. Univ. Sarav. Ser. Math., 1(1):iv+40, 1986.

    Google ScholarĀ 

  89. G.Ā Frey. Elliptic curves and solutions of A āˆ’ Bā€‰=ā€‰C. In SĆ©minaire de ThĆ©orie des Nombres, Paris 1985ā€“86, volumeĀ 71 of Progr. Math., pages 39ā€“51. BirkhƤuser Boston, Boston, MA, 1987.

    Google ScholarĀ 

  90. G.Ā Frey. Links between solutions of A āˆ’ Bā€‰=ā€‰C and elliptic curves. In Number theory (Ulm, 1987), volume 1380 of Lecture Notes in Math., pages 31ā€“62. Springer, New York, 1989.

    Google ScholarĀ 

  91. G.Ā Frey and H.-G. RĆ¼ck. A remark concerning m-divisibility and the discrete logarithm problem in the divisor class group of curves. Math. Comp., 62:865ā€“874, 1994.

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  92. A.Ā Frƶhlich. Local fields. In Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), pages 1ā€“41. Thompson, Washington, D.C., 1967.

    Google ScholarĀ 

  93. A.Ā Frƶhlich. Formal groups. Lecture Notes in Mathematics, No. 74. Springer-Verlag, Berlin, 1968.

    Google ScholarĀ 

  94. R.Ā Fueter. Ueber kubische diophantische Gleichungen. Comment. Math. Helv., 2(1):69ā€“89, 1930.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  95. W.Ā Fulton. Algebraic curves. Advanced Book Classics. Addison-Wesley Publishing Company Advanced Book Program, Redwood City, CA, 1989. An introduction to algebraic geometry, Notes written with the collaboration of Richard Weiss, Reprint of 1969 original.

    Google ScholarĀ 

  96. J.Ā Gebel, A.Ā Pethő, and H.Ā G. Zimmer. Computing integral points on elliptic curves. Acta Arith., 68(2):171ā€“192, 1994.

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  97. S.Ā Goldwasser and J.Ā Kilian. Almost all primes can be quickly certified. In STOC ā€™86: Proceedings of the eighteenth annual ACM symposium on Theory of computing, pages 316ā€“329, New York, 1986. ACM.

    Google ScholarĀ 

  98. R.Ā Greenberg. On the Birch and Swinnerton-Dyer conjecture. Invent. Math., 72(2):241ā€“265, 1983.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  99. P.Ā Griffiths and J.Ā Harris. Principles of algebraic geometry. Wiley Classics Library. John Wiley & Sons Inc., New York, 1994. Reprint of the 1978 original.

    Google ScholarĀ 

  100. B.Ā Gross, W.Ā Kohnen, and D.Ā Zagier. Heegner points and derivatives of L-series. II. Math. Ann., 278(1-4):497ā€“562, 1987.

    Google ScholarĀ 

  101. B.Ā Gross and D.Ā Zagier. Points de Heegner et dĆ©rivĆ©es de fonctions L. C. R. Acad. Sci. Paris SĆ©r. I Math., 297(2):85ā€“87, 1983.

    Google ScholarĀ 

  102. B.Ā H. Gross and D.Ā B. Zagier. Heegner points and derivatives of L-series. Invent. Math., 84(2):225ā€“320, 1986.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  103. R.Ā Gross. A note on Rothā€™s theorem. J. Number Theory, 36:127ā€“132, 1990.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  104. R.Ā Gross and J.Ā Silverman. S-integer points on elliptic curves. Pacific J. Math., 167(2):263ā€“288, 1995.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  105. K.Ā Gruenberg. Profinite groups. In Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), pages 116ā€“127. Thompson, Washington, D.C., 1967.

    Google ScholarĀ 

  106. M.Ā Hall, Jr. The Diophantine equation x 3 āˆ’ y 2ā€‰=ā€‰k. In Computers in number theory (Proc. Sci. Res. Council Atlas Sympos. No. 2, Oxford, 1969), pages 173ā€“198. Academic Press, London, 1971.

    Google ScholarĀ 

  107. D.Ā Hankerson, A.Ā Menezes, and S.Ā Vanstone. Guide to elliptic curve cryptography. Springer Professional Computing. Springer-Verlag, New York, 2004.

    MATHĀ  Google ScholarĀ 

  108. G.Ā H. Hardy and E.Ā M. Wright. An introduction to the theory of numbers. The Clarendon Press Oxford University Press, New York, fifth edition, 1979.

    MATHĀ  Google ScholarĀ 

  109. J.Ā Harris. Algebraic geometry, volume 133 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1992. A first course.

    Google ScholarĀ 

  110. M.Ā Harris, N.Ā Shepherd-Barron, and R.Ā Taylor. A family of Calabi-Yau varieties and potential automorphy. Ann. of Math. (2). to appear.

    Google ScholarĀ 

  111. R.Ā Hartshorne. Algebraic geometry. Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, No. 52.

    Google ScholarĀ 

  112. M.Ā Hazewinkel. Formal groups and applications, volumeĀ 78 of Pure and Applied Mathematics. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1978.

    Google ScholarĀ 

  113. M.Ā Hindry and J.Ā H. Silverman. The canonical height and integral points on elliptic curves. Invent. Math., 93(2):419ā€“450, 1988.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  114. M.Ā Hindry and J.Ā H. Silverman. Diophantine geometry, volume 201 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2000. An introduction.

    Google ScholarĀ 

  115. G.Ā Hochschild and J.-P. Serre. Cohomology of group extensions. Trans. Amer. Math. Soc., 74:110ā€“134, 1953.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  116. J.Ā Hoffstein, J.Ā Pipher, and J.Ā H. Silverman. An introduction to mathematical cryptography. Undergraduate Texts in Mathematics. Springer-Verlag, New York, 2008.

    MATHĀ  Google ScholarĀ 

  117. A.Ā Hurwitz. Ɯber ternƤre diophantische Gleichungen dritten Grades. Vierteljahrschrift d. Naturf. Ges. ZĆ¼rich, 62:207ā€“229, 1917.

    MATHĀ  Google ScholarĀ 

  118. D.Ā Husemƶller. Elliptic curves, volume 111 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 2004. With appendices by Otto Forster, Ruth Lawrence and Stefan Theisen.

    Google ScholarĀ 

  119. J.-I. Igusa. Class number of a definite quaternion with prime discriminant. Proc. Nat. Acad. Sci. U.S.A., 44:312ā€“314, 1958.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  120. A.Ā Joux. A one round protocol for tripartite Diffie-Hellman. In Algorithmic number theory (Leiden, 2000), volume 1838 of Lecture Notes in Comput. Sci., pages 385ā€“393. Springer, Berlin, 2000.

    Google ScholarĀ 

  121. S.Ā Kamienny. Torsion points on elliptic curves and q-coefficients of modular forms. Invent. Math., 109(2):221ā€“229, 1992.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  122. S.Ā Kamienny and B.Ā Mazur. Rational torsion of prime order in elliptic curves over number fields. AstĆ©risque, (228):3, 81ā€“100, 1995. With an appendix by A. Granville, Columbia University Number Theory Seminar (New York, 1992).

    Google ScholarĀ 

  123. N.Ā M. Katz. An overview of Deligneā€™s proof of the Riemann hypothesis for varieties over finite fields. In Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Vol. XXVIII, Northern Illinois Univ., De Kalb, Ill., 1974), pages 275ā€“305. Amer. Math. Soc., Providence, R.I., 1976.

    Google ScholarĀ 

  124. N.Ā M. Katz and B.Ā Mazur. Arithmetic moduli of elliptic curves, volume 108 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1985.

    Google ScholarĀ 

  125. M.Ā A. Kenku. On the number of Q-isomorphism classes of elliptic curves in each Q-isogeny class. J. Number Theory, 15(2):199ā€“202, 1982.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  126. J.-H. Kim, R.Ā Montenegro, Y.Ā Peres, and P.Ā Tetali. A birthday paradox for Markov chains, with an optimal bound for collision in Pollard rho for discrete logarithm. In Algorithmic number theory, volume 5011 of Lecture Notes in Comput. Sci., pages 402ā€“415. Springer, Berlin, 2008.

    MATHĀ  Google ScholarĀ 

  127. A.Ā W. Knapp. Elliptic curves, volumeĀ 40 of Mathematical Notes. Princeton University Press, Princeton, NJ, 1992.

    Google ScholarĀ 

  128. N.Ā Koblitz. Elliptic curve cryptosystems. Math. Comp., 48(177):203ā€“209, 1987.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  129. N.Ā Koblitz. Introduction to elliptic curves and modular forms, volumeĀ 97 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1993.

    Google ScholarĀ 

  130. V.Ā A. Kolyvagin. Finiteness of \(E(\mathbb{Q})\) and \((E, \mathbb{Q})\) for a subclass of Weil curves. Izv. Akad. Nauk SSSR Ser. Mat., 52(3):522ā€“540, 670ā€“671, 1988.

    Google ScholarĀ 

  131. S.Ā V. Kotov and L.Ā A. Trelina. S-ganze Punkte auf elliptischen Kurven. J. Reine Angew. Math., 306:28ā€“41, 1979.

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  132. D.Ā S. Kubert. Universal bounds on the torsion of elliptic curves. Proc. London Math. Soc. (3), 33(2):193ā€“237, 1976.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  133. E.Ā Kunz. Introduction to plane algebraic curves. BirkhƤuser Boston Inc., Boston, MA, 2005. Translated from the 1991 German edition by Richard G. Belshoff.

    Google ScholarĀ 

  134. M.Ā Lal, M.Ā F. Jones, and W.Ā J. Blundon. Numerical solutions of the Diophantine equation y 3 āˆ’ x 2ā€‰=ā€‰k. Math. Comp., 20:322ā€“325, 1966.

    Google ScholarĀ 

  135. S.Ā Lang. Elliptic curves: Diophantine analysis, volume 231 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1978.

    Google ScholarĀ 

  136. S.Ā Lang. Introduction to algebraic and abelian functions, volumeĀ 89 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1982.

    Google ScholarĀ 

  137. S.Ā Lang. Complex multiplication, volume 255 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York, 1983.

    Google ScholarĀ 

  138. S.Ā Lang. Conjectured Diophantine estimates on elliptic curves. In Arithmetic and geometry, Vol. I, volumeĀ 35 of Progr. Math., pages 155ā€“171. BirkhƤuser Boston, Boston, MA, 1983.

    Google ScholarĀ 

  139. S.Ā Lang. Fundamentals of Diophantine geometry. Springer-Verlag, New York, 1983.

    BookĀ  MATHĀ  Google ScholarĀ 

  140. S.Ā Lang. Elliptic functions, volume 112 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1987. With an appendix by J. Tate.

    Google ScholarĀ 

  141. S.Ā Lang. Number theory III, volumeĀ 60 of Encyclopedia of Mathematical Sciences. Springer-Verlag, Berlin, 1991.

    Google ScholarĀ 

  142. S.Ā Lang. Algebraic number theory, volume 110 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1994.

    Google ScholarĀ 

  143. S.Ā Lang. Algebra, volume 211 of Graduate Texts in Mathematics. Springer-Verlag, New York, third edition, 2002.

    Google ScholarĀ 

  144. S.Ā Lang and J.Ā Tate. Principal homogeneous spaces over abelian varieties. Amer. J. Math., 80:659ā€“684, 1958.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  145. S.Ā Lang and H.Ā Trotter. Frobenius distributions in \(\mathop{\mathrm{GL}}\nolimits _{2}\) -extensions. Springer-Verlag, Berlin, 1976. Distribution of Frobenius automorphisms in \(\mathop{\mathrm{GL}}\nolimits _{2}\)-extensions of the rational numbers, Lecture Notes in Mathematics, Vol. 504.

    Google ScholarĀ 

  146. M.Ā Laska. An algorithm for finding a minimal Weierstrass equation for an elliptic curve. Math. Comp., 38(157):257ā€“260, 1982.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  147. M.Ā Laska. Elliptic curves over number fields with prescribed reduction type. Aspects of Mathematics, E4. Friedr. Vieweg & Sohn, Braunschweig, 1983.

    Google ScholarĀ 

  148. D.Ā J. Lewis and K.Ā Mahler. On the representation of integers by binary forms. Acta Arith., 6:333ā€“363, 1960/1961.

    Google ScholarĀ 

  149. S.Ā Lichtenbaum. The period-index problem for elliptic curves. Amer. J. Math., 90:1209ā€“1223, 1968.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  150. C.-E. Lind. Untersuchungen Ć¼ber die rationalen Punkte der ebenen kubischen Kurven vom Geschlecht Eins. Thesis, University of Uppsala,, 1940:97, 1940.

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  151. J.Ā Liouville. Sur des classes trĆØs-Ć©tendues de quantitĆ©s dont la irrationalles algĆ©briques. C. R. Acad. Paris, 18:883ā€“885 and 910ā€“911, 1844.

    Google ScholarĀ 

  152. E.Ā Lutz. Sur lā€™equation y 2ā€‰=ā€‰x 3 āˆ’ ax āˆ’ b dans les corps p-adic. J. Reine Angew. Math., 177:237ā€“247, 1937.

    MathSciNetĀ  Google ScholarĀ 

  153. K.Ā Mahler. On the lattice points on curves of genus 1. Proc. London Math. Soc. (3), 39:431ā€“466, 1935.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  154. J.Ā I. Manin. The Hasse-Witt matrix of an algebraic curve. Izv. Akad. Nauk SSSR Ser. Mat., 25:153ā€“172, 1961.

    MathSciNetĀ  Google ScholarĀ 

  155. J.Ā I. Manin. The p-torsion of elliptic curves is uniformly bounded. Izv. Akad. Nauk SSSR Ser. Mat., 33:459ā€“465, 1969.

    MathSciNetĀ  Google ScholarĀ 

  156. J.Ā I. Manin. Cyclotomic fields and modular curves. Uspehi Mat. Nauk, 26(6(162)):7ā€“71, 1971. English translation: Russian Math. Surveys 26 (1971), no. 6, 7ā€“78.

    Google ScholarĀ 

  157. R.Ā C. Mason. The hyperelliptic equation over function fields. Math. Proc. Cambridge Philos. Soc., 93(2):219ā€“230, 1983.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  158. R.Ā C. Mason. Norm form equations. I. J. Number Theory, 22(2):190ā€“207, 1986.

    Google ScholarĀ 

  159. D.Ā Masser. Elliptic functions and transcendence. Springer-Verlag, Berlin, 1975. Lecture Notes in Mathematics, Vol. 437.

    Google ScholarĀ 

  160. D.Ā Masser and G.Ā WĆ¼stholz. Isogeny estimates for abelian varieties, and finiteness theorems. Ann. of Math. (2), 137(3):459ā€“472, 1993.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  161. D.Ā W. Masser. Specializations of finitely generated subgroups of abelian varieties. Trans. Amer. Math. Soc., 311(1):413ā€“424, 1989.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  162. D.Ā W. Masser and G.Ā WĆ¼stholz. Fields of large transcendence degree generated by values of elliptic functions. Invent. Math., 72(3):407ā€“464, 1983.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  163. D.Ā W. Masser and G.Ā WĆ¼stholz. Estimating isogenies on elliptic curves. Invent. Math., 100(1):1ā€“24, 1990.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  164. H.Ā Matsumura. Commutative algebra, volumeĀ 56 of Mathematics Lecture Note Series. Benjamin/Cummings Publishing Co., Inc., Reading, Mass., second edition, 1980.

    Google ScholarĀ 

  165. B.Ā Mazur. Modular curves and the Eisenstein ideal. Inst. Hautes Ɖtudes Sci. Publ. Math., (47):33ā€“186 (1978), 1977.

    Google ScholarĀ 

  166. B.Ā Mazur. Rational isogenies of prime degree (with an appendix by D. Goldfeld). Invent. Math., 44(2):129ā€“162, 1978.

    Google ScholarĀ 

  167. H.Ā McKean and V.Ā Moll. Elliptic curves. Cambridge University Press, Cambridge, 1997. Function theory, geometry, arithmetic.

    Google ScholarĀ 

  168. A.Ā J. Menezes, T.Ā Okamoto, and S.Ā A. Vanstone. Reducing elliptic curve logarithms to logarithms in a finite field. IEEE Trans. Inform. Theory, 39(5):1639ā€“1646, 1993.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  169. A.Ā J. Menezes, P.Ā C. van Oorschot, and S.Ā A. Vanstone. Handbook of Applied Cryptography. CRC Press Series on Discrete Mathematics and Its Applications. CRC Press, Boca Raton, FL, 1997.

    MATHĀ  Google ScholarĀ 

  170. L.Ā Merel. Bornes pour la torsion des courbes elliptiques sur les corps de nombres. Invent. Math., 124(1-3):437ā€“449, 1996.

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  171. J.-F. Mestre. Construction dā€™une courbe elliptique de rangā€‰ā‰„ā€‰12. C. R. Acad. Sci. Paris SĆ©r. I Math., 295(12):643ā€“644, 1982.

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  172. J.-F. Mestre. Courbes elliptiques et formules explicites. In Seminar on number theory, Paris 1981ā€“82 (Paris, 1981/1982), volumeĀ 38 of Progr. Math., pages 179ā€“187. BirkhƤuser Boston, Boston, MA, 1983.

    Google ScholarĀ 

  173. M.Ā Mignotte. Quelques remarques sur lā€™approximation rationnelle des nombres algĆ©briques. J. Reine Angew. Math., 268/269:341ā€“347, 1974. Collection of articles dedicated to Helmut Hasse on his seventy-fifth birthday, II.

    Google ScholarĀ 

  174. S.Ā D. Miller and R.Ā Venkatesan. Spectral analysis of Pollard rho collisions. In Algorithmic number theory, volume 4076 of Lecture Notes in Comput. Sci., pages 573ā€“581. Springer, Berlin, 2006.

    Google ScholarĀ 

  175. S.Ā D. Miller and R.Ā Venkatesan. Non-degeneracy of Pollard rho collisions, 2008. arXiv:0808.0469.

    Google ScholarĀ 

  176. V.Ā S. Miller. Use of elliptic curves in cryptography. In Advances in Cryptologyā€”CRYPTO ā€™85 (Santa Barbara, Calif., 1985), volume 218 of Lecture Notes in Comput. Sci., pages 417ā€“426. Springer, Berlin, 1986.

    Google ScholarĀ 

  177. J.Ā S. Milne. Arithmetic duality theorems, volumeĀ 1 of Perspectives in Mathematics. Academic Press Inc., Boston, MA, 1986.

    Google ScholarĀ 

  178. J.Ā S. Milne. Elliptic curves. BookSurge Publishers, Charleston, SC, 2006.

    MATHĀ  Google ScholarĀ 

  179. J.Ā Milnor. On LattĆØs maps. ArXiv:math.DS/0402147, Stony Brook IMS Preprint #2004/01.

    Google ScholarĀ 

  180. R.Ā Miranda. Algebraic curves and Riemann surfaces, volumeĀ 5 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1995.

    MATHĀ  Google ScholarĀ 

  181. A.Ā Miyaji, M.Ā Nakabayashi, and S.Ā Takano. Characterization of elliptic curve traces under FR-reduction. In Information security and cryptologyā€”ICISC 2000 (Seoul), volume 2015 of Lecture Notes in Comput. Sci., pages 90ā€“108. Springer, Berlin, 2001.

    Google ScholarĀ 

  182. P.Ā Monsky. Three constructions of rational points on Y 2ā€‰=ā€‰X 3 Ā± NX. Math. Z., 209(3):445ā€“462, 1992.

    Google ScholarĀ 

  183. F.Ā Morain. Building cyclic elliptic curves modulo large primes. In Advances in cryptologyā€”EUROCRYPT ā€™91 (Brighton, 1991), volume 547 of Lecture Notes in Comput. Sci., pages 328ā€“336. Springer, Berlin, 1991.

    Google ScholarĀ 

  184. L.Ā J. Mordell. The diophantine equation x 4 + my 4ā€‰=ā€‰z 2.ā€‰. Quart. J. Math. Oxford Ser. (2), 18:1ā€“6, 1967.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  185. L.Ā J. Mordell. Diophantine equations. Pure and Applied Mathematics, Vol. 30. Academic Press, London, 1969.

    Google ScholarĀ 

  186. D.Ā Mumford. Abelian varieties. Tata Institute of Fundamental Research Studies in Mathematics, No. 5. Published for the Tata Institute of Fundamental Research, Bombay, 1970.

    Google ScholarĀ 

  187. D.Ā Mumford, J.Ā Fogarty, and F.Ā Kirwan. Geometric invariant theory, volumeĀ 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)]. Springer-Verlag, Berlin, third edition, 1994.

    Google ScholarĀ 

  188. K.-I. Nagao. Construction of high-rank elliptic curves. Kobe J. Math., 11(2):211ā€“219, 1994.

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  189. K.-I. Nagao. \(\mathbb{Q}(T)\)-rank of elliptic curves and certain limit coming from the local points. Manuscripta Math., 92(1):13ā€“32, 1997. With an appendix by Nobuhiko Ishida, Tsuneo Ishikawa and the author.

    Google ScholarĀ 

  190. T.Ā Nagell. Solution de quelque problĆØmes dans la thĆ©orie arithmĆ©tique des cubiques planes du premier genre. Wid. Akad. Skrifter Oslo I, 1935. Nr. 1.

    Google ScholarĀ 

  191. NBSā€“DSS. Digital Signature Standard (DSS). FIPS Publication 186-2, National Bureau of Standards, 2000. http://csrc.nist.gov/publications/ PubsFIPS.html .

  192. A.Ā NĆ©ron. ProblĆØmes arithmĆ©tiques et gĆ©omĆ©triques rattachĆ©s Ć  la notion de rang dā€™une courbe algĆ©brique dans un corps. Bull. Soc. Math. France, 80:101ā€“166, 1952.

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  193. A.Ā NĆ©ron. ModĆØles minimaux des variĆ©tĆ©s abĆ©liennes sur les corps locaux et globaux. Inst. Hautes Ɖtudes Sci. Publ.Math. No., 21:128, 1964.

    MATHĀ  Google ScholarĀ 

  194. A.Ā NĆ©ron. Quasi-fonctions et hauteurs sur les variĆ©tĆ©s abĆ©liennes. Ann. of Math. (2), 82:249ā€“331, 1965.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  195. O.Ā Neumann. Elliptische Kurven mit vorgeschriebenem Reduktionsverhalten. I. Math. Nachr., 49:107ā€“123, 1971.

    Google ScholarĀ 

  196. J.Ā OesterlĆ©. Nouvelles approches du ā€œthĆ©orĆØmeā€ de Fermat. AstĆ©risque, (161-162):Exp.Ā No.Ā 694, 4, 165ā€“186 (1989), 1988. SĆ©minaire Bourbaki, Vol.Ā 1987/88.

    Google ScholarĀ 

  197. A.Ā Ogg. Modular forms and Dirichlet series. W. A. Benjamin, Inc., New York-Amsterdam, 1969.

    MATHĀ  Google ScholarĀ 

  198. A.Ā P. Ogg. Abelian curves of 2-power conductor. Proc. Cambridge Philos. Soc., 62:143ā€“148, 1966.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  199. A.Ā P. Ogg. Abelian curves of small conductor. J. Reine Angew. Math., 226:204ā€“215, 1967.

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  200. A.Ā P. Ogg. Elliptic curves and wild ramification. Amer. J. Math., 89:1ā€“21, 1967.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  201. L.Ā D. Olson. Torsion points on elliptic curves with given j-invariant. Manuscripta Math., 16(2):145ā€“150, 1975.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  202. PARI/GP, 2005. http://pari.math.u-bordeaux.fr/.

  203. A.Ā N. ParÅ”in. Algebraic curves over function fields. I. Izv. Akad. Nauk SSSR Ser. Mat., 32:1191ā€“1219, 1968.

    Google ScholarĀ 

  204. R.Ā G.Ā E. Pinch. Elliptic curves with good reduction away from 2. Math. Proc. Cambridge Philos. Soc., 96(1):25ā€“38, 1984.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  205. S.Ā C. Pohlig and M.Ā E. Hellman. An improved algorithm for computing logarithms over \(\mathop{\mathrm{GF}}\nolimits (p)\) and its cryptographic significance. IEEE Trans. Information Theory, IT-24(1):106ā€“110, 1978.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  206. J.Ā M. Pollard. Monte Carlo methods for index computation \((\mathop{\mathrm{mod}}\nolimits \ p)\). Math. Comp., 32(143):918ā€“924, 1978.

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  207. H.Ā Reichardt. Einige im Kleinen Ć¼berall lƶsbare, im Grossen unlƶsbare diophantische Gleichungen. J. Reine Angew. Math., 184:12ā€“18, 1942.

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  208. K.Ā A. Ribet. On modular representations of \(\mathop{\mathrm{Gal}}\nolimits (\overline{\mathbf{Q}}/\mathbf{Q})\) arising from modular forms. Invent. Math., 100(2):431ā€“476, 1990.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  209. R.Ā L. Rivest, A.Ā Shamir, and L.Ā Adleman. A method for obtaining digital signatures and public-key cryptosystems. Comm. ACM, 21(2):120ā€“126, 1978.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  210. A.Ā Robert. Elliptic curves. Springer-Verlag, Berlin, 1973. Notes from postgraduate lectures given in Lausanne 1971/72, Lecture Notes in Mathematics, Vol. 326.

    Google ScholarĀ 

  211. D.Ā E. Rohrlich. On L-functions of elliptic curves and anticyclotomic towers. Invent. Math., 75(3):383ā€“408, 1984.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  212. P.Ā Roquette. Analytic theory of elliptic functions over local fields. Hamburger Mathematische Einzelschriften (N.F.), Heft 1. Vandenhoeck & Ruprecht, Gƶttingen, 1970.

    Google ScholarĀ 

  213. M.Ā Rosen and J.Ā H. Silverman. On the rank of an elliptic surface. Invent. Math., 133(1):43ā€“67, 1998.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  214. K.Ā Rubin. Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer. Invent. Math., 64(3):455ā€“470, 1981.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  215. K.Ā Rubin. Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication. Invent. Math., 89(3):527ā€“559, 1987.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  216. K.Ā Rubin. The ā€œmain conjecturesā€ of Iwasawa theory for imaginary quadratic fields. Invent. Math., 103(1):25ā€“68, 1991.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  217. T.Ā Saito. Conductor, discriminant, and the Noether formula of arithmetic surfaces. Duke Math. J., 57(1):151ā€“173, 1988.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  218. T.Ā Satoh and K.Ā Araki. Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves. Comment. Math. Univ. St. Paul., 47(1):81ā€“92, 1998.

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  219. E.Ā F. Schaefer and M.Ā Stoll. How to do a p-descent on an elliptic curve. Trans. Amer. Math. Soc., 356(3):1209ā€“1231 (electronic), 2004.

    Google ScholarĀ 

  220. S.Ā H. Schanuel. Heights in number fields. Bull. Soc. Math. France, 107(4):433ā€“449, 1979.

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  221. W.Ā M. Schmidt. Diophantine approximation, volume 785 of Lecture Notes in Mathematics. Springer, Berlin, 1980.

    Google ScholarĀ 

  222. S.Ā Schmitt and H.Ā G. Zimmer. Elliptic curves, volumeĀ 31 of de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, 2003. A computational approach, With an appendix by Attila Pethő.

    Google ScholarĀ 

  223. R.Ā Schoof. Elliptic curves over finite fields and the computation of square roots mod p. Math. Comp., 44(170):483ā€“494, 1985.

    Google ScholarĀ 

  224. R.Ā Schoof. Counting points on elliptic curves over finite fields. J. ThĆ©or. Nombres Bordeaux, 7(1):219ā€“254, 1995. Les Dix-huitiĆØmes JournĆ©es ArithmĆ©tiques (Bordeaux, 1993).

    Google ScholarĀ 

  225. E.Ā S. Selmer. The Diophantine equation ax 3 + by 3 + cz 3ā€‰=ā€‰0. Acta Math., 85:203ā€“362 (1 plate), 1951.

    Google ScholarĀ 

  226. E.Ā S. Selmer. A conjecture concerning rational points on cubic curves. Math. Scand., 2:49ā€“54, 1954.

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  227. E.Ā S. Selmer. The diophantine equation ax 3 + by 3 + cz 3ā€‰=ā€‰0.ā€‰Completion of the tables. Acta Math., 92:191ā€“197, 1954.

    Google ScholarĀ 

  228. I.Ā A. Semaev. Evaluation of discrete logarithms in a group of p-torsion points of an elliptic curve in characteristic p. Math. Comp., 67(221):353ā€“356, 1998.

    Google ScholarĀ 

  229. J.-P. Serre. GĆ©omĆ©trie algĆ©brique et gĆ©omĆ©trie analytique. Ann. Inst. Fourier, Grenoble, 6:1ā€“42, 1955ā€“1956.

    Google ScholarĀ 

  230. J.-P. Serre. Complex multiplication. In Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), pages 292ā€“296. Thompson, Washington, D.C., 1967.

    Google ScholarĀ 

  231. J.-P. Serre. PropriĆ©tĆ©s galoisiennes des points dā€™ordre fini des courbes elliptiques. Invent. Math., 15(4):259ā€“331, 1972.

    Google ScholarĀ 

  232. J.-P. Serre. A course in arithmetic. Springer-Verlag, New York, 1973. Translated from the French, Graduate Texts in Mathematics, No. 7.

    Google ScholarĀ 

  233. J.-P. Serre. Local fields, volumeĀ 67 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1979. Translated from the French by Marvin Jay Greenberg.

    Google ScholarĀ 

  234. J.-P. Serre. Quelques applications du thĆ©orĆØme de densitĆ© de Chebotarev. Inst. Hautes Ɖtudes Sci. Publ. Math., (54):323ā€“401, 1981.

    ArticleĀ  MATHĀ  Google ScholarĀ 

  235. J.-P. Serre. Sur les reprĆ©sentations modulaires de degrĆ© 2 de \(\mathop{\mathrm{Gal}}\nolimits (\overline{\mathbf{Q}}/\mathbf{Q})\). Duke Math. J., 54(1):179ā€“230, 1987.

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  236. J.-P. Serre. Lectures on the Mordell-Weil theorem. Aspects of Mathematics. Friedr. Vieweg & Sohn, Braunschweig, third edition, 1997. Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt, With a foreword by Brown and Serre.

    Google ScholarĀ 

  237. J.-P. Serre. Abelian l-adic representations and elliptic curves, volumeĀ 7 of Research Notes in Mathematics. A K Peters Ltd., Wellesley, MA, 1998. With the collaboration of Willem Kuyk and John Labute, Revised reprint of the 1968 original.

    Google ScholarĀ 

  238. J.-P. Serre. Galois cohomology. Springer Monographs in Mathematics. Springer-Verlag, Berlin, english edition, 2002. Translated from the French by Patrick Ion and revised by the author.

    Google ScholarĀ 

  239. J.-P. Serre and J.Ā Tate. Good reduction of abelian varieties. Ann. of Math. (2), 88:492ā€“517, 1968.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  240. B.Ā Setzer. Elliptic curves of prime conductor. J. London Math. Soc. (2), 10:367ā€“378, 1975.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  241. B.Ā Setzer. Elliptic curves over complex quadratic fields. Pacific J. Math., 74(1):235ā€“250, 1978.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  242. I.Ā R. Shafarevich. Algebraic number fields. In Proc. Int. Cong. (Stockholm 1962), pages 25ā€“39. American Mathematical Society, Providence, R.I., 1963. Amer. Math. Soc. Transl., SeriesĀ 2, Vol.Ā 31.

    Google ScholarĀ 

  243. I.Ā R. Shafarevich. Basic algebraic geometry. Springer-Verlag, Berlin, study edition, 1977. Translated from the Russian by K. A. Hirsch, Revised printing of Grundlehren der mathematischen Wissenschaften, Vol. 213, 1974.

    Google ScholarĀ 

  244. I.Ā R. Shafarevich and J.Ā Tate. The rank of elliptic curves. In Amer. Math. Soc. Transl., volumeĀ 8, pages 917ā€“920. Amer. Math. Soc., 1967.

    Google ScholarĀ 

  245. A.Ā Shamir. Identity-based cryptosystems and signature schemes. In Advances in Cryptology (Santa Barbara, Calif., 1984), volume 196 of Lecture Notes in Comput. Sci., pages 47ā€“53. Springer, Berlin, 1985.

    Google ScholarĀ 

  246. G.Ā Shimura. Correspondances modulaires et les fonctions Ī¶ de courbes algĆ©briques. J. Math. Soc. Japan, 10:1ā€“28, 1958.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  247. G.Ā Shimura. On elliptic curves with complex multiplication as factors of the Jacobians of modular function fields. Nagoya Math. J., 43:199ā€“208, 1971.

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  248. G.Ā Shimura. On the zeta-function of an abelian variety with complex multiplication. Ann. of Math. (2), 94:504ā€“533, 1971.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  249. G.Ā Shimura. Introduction to the arithmetic theory of automorphic functions, volumeĀ 11 of Publications of the Mathematical Society of Japan. Princeton University Press, Princeton, NJ, 1994. Reprint of the 1971 original, Kano Memorial Lectures, 1.

    Google ScholarĀ 

  250. G.Ā Shimura and Y.Ā Taniyama. Complex multiplication of abelian varieties and its applications to number theory, volumeĀ 6 of Publications of the Mathematical Society of Japan. The Mathematical Society of Japan, Tokyo, 1961.

    Google ScholarĀ 

  251. T.Ā Shioda. An explicit algorithm for computing the Picard number of certain algebraic surfaces. Amer. J. Math., 108(2):415ā€“432, 1986.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  252. R.Ā Shipsey. Elliptic divisibility sequences. PhD thesis, Goldsmithā€™s College (University of London), 2000.

    Google ScholarĀ 

  253. V.Ā Shoup. Lower bounds for discrete logarithms and related problems. In Advances in cryptologyā€”EUROCRYPT ā€™97 (Konstanz), volume 1233 of Lecture Notes in Comput. Sci., pages 256ā€“266. Springer, Berlin, 1997. updated version at www.shoup.net/papers/dlbounds1.pdf.

  254. J.Ā H. Silverman. Lower bound for the canonical height on elliptic curves. Duke Math. J., 48(3):633ā€“648, 1981.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  255. J.Ā H. Silverman. The NĆ©ronā€“Tate height on elliptic curves. PhD thesis, Harvard University, 1981.

    Google ScholarĀ 

  256. J.Ā H. Silverman. Heights and the specialization map for families of abelian varieties. J. Reine Angew. Math., 342:197ā€“211, 1983.

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  257. J.Ā H. Silverman. Integer points on curves of genus 1. J. London Math. Soc. (2), 28(1):1ā€“7, 1983.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  258. J.Ā H. Silverman. The S-unit equation over function fields. Math. Proc. Cambridge Philos. Soc., 95(1):3ā€“4, 1984.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  259. J.Ā H. Silverman. Weierstrass equations and the minimal discriminant of an elliptic curve. Mathematika, 31(2):245ā€“251 (1985), 1984.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  260. J.Ā H. Silverman. Divisibility of the specialization map for families of elliptic curves. Amer. J. Math., 107(3):555ā€“565, 1985.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  261. J.Ā H. Silverman. Arithmetic distance functions and height functions in Diophantine geometry. Math. Ann., 279(2):193ā€“216, 1987.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  262. J.Ā H. Silverman. A quantitative version of Siegelā€™s theorem: integral points on elliptic curves and Catalan curves. J. Reine Angew. Math., 378:60ā€“100, 1987.

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  263. J.Ā H. Silverman. Computing heights on elliptic curves. Math. Comp., 51(183):339ā€“358, 1988.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  264. J.Ā H. Silverman. Wieferichā€™s criterion and the abc-conjecture. J. Number Theory, 30(2):226ā€“237, 1988.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  265. J.Ā H. Silverman. The difference between the Weil height and the canonical height on elliptic curves. Math. Comp., 55(192):723ā€“743, 1990.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  266. J.Ā H. Silverman. Advanced topics in the arithmetic of elliptic curves, volume 151 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1994.

    MATHĀ  Google ScholarĀ 

  267. J.Ā H. Silverman. The arithmetic of dynamical systems, volume 241 of Graduate Texts in Mathematics. Springer, New York, 2007.

    MATHĀ  Google ScholarĀ 

  268. N.Ā P. Smart. S-integral points on elliptic curves. Math. Proc. Cambridge Philos. Soc., 116(3):391ā€“399, 1994.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  269. N.Ā P. Smart. The discrete logarithm problem on elliptic curves of trace one. J. Cryptology, 12(3):193ā€“196, 1999.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  270. K.Ā Stange. The Tate pairing via elliptic nets. In Pairing Based Cryptography, Lecture Notes in Comput. Sci. Springer, 2007.

    Google ScholarĀ 

  271. K.Ā Stange. Elliptic Nets and Elliptic Curves. PhD thesis, Brown University, 2008.

    Google ScholarĀ 

  272. K.Ā Stange. Elliptic nets and elliptic curves, 2008. arXiv:0710.1316v2.

    Google ScholarĀ 

  273. H.Ā M. Stark. Effective estimates of solutions of some Diophantine equations. Acta Arith., 24:251ā€“259, 1973.

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  274. W.Ā Stein. The Modular Forms Database. http://modular.fas.harvard. edu/Tables .

  275. W.Ā Stein. Sage Mathematics Software, 2007. http://www.sagemath.org.

  276. N.Ā M. Stephens. The Diophantine equation X 3 + Y 3ā€‰=ā€‰DZ 3 and the conjectures of Birch and Swinnerton-Dyer. J. Reine Angew. Math., 231:121ā€“162, 1968.

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  277. D.Ā R. Stinson. Cryptography: Theory and Practice. CRC Press Series on Discrete Mathematics and Its Applications. Chapman & Hall/CRC, Boca Raton, FL, 2002.

    MATHĀ  Google ScholarĀ 

  278. W.Ā W. Stothers. Polynomial identities and Hauptmoduln. Quart. J. Math. Oxford Ser. (2), 32(127):349ā€“370, 1981.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  279. R.Ā J. Stroeker and N.Ā Tzanakis. Solving elliptic Diophantine equations by estimating linear forms in elliptic logarithms. Acta Arith., 67(2):177ā€“196, 1994.

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  280. J.Ā Tate. Letter to J.-P. Serre, 1968.

    Google ScholarĀ 

  281. J.Ā Tate. Duality theorems in Galois cohomology over number fields. In Proc. Internat. Congr. Mathematicians (Stockholm, 1962), pages 288ā€“295. Inst. Mittag-Leffler, Djursholm, 1963.

    Google ScholarĀ 

  282. J.Ā Tate. Endomorphisms of abelian varieties over finite fields. Invent. Math., 2:134ā€“144, 1966.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  283. J.Ā Tate. Algorithm for determining the type of a singular fiber in an elliptic pencil. In Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pages 33ā€“52. Lecture Notes in Math., Vol. 476. Springer, Berlin, 1975.

    Google ScholarĀ 

  284. J.Ā Tate. Variation of the canonical height of a point depending on a parameter. Amer. J. Math., 105(1):287ā€“294, 1983.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  285. J.Ā Tate. A review of non-Archimedean elliptic functions. In Elliptic curves, modular forms, & Fermatā€™s last theorem (Hong Kong, 1993), Ser. Number Theory, I, pages 162ā€“184. Int. Press, Cambridge, MA, 1995.

    Google ScholarĀ 

  286. J.Ā Tate. WC-groups over \(\mathfrak{p}\)-adic fields. In SĆ©minaire Bourbaki, Vol.Ā 4 (1957/58), pages Exp.Ā No.Ā 156, 265ā€“277. Soc. Math. France, Paris, 1995.

    Google ScholarĀ 

  287. J.Ā T. Tate. Algebraic cycles and poles of zeta functions. In Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), pages 93ā€“110. Harper & Row, New York, 1965.

    Google ScholarĀ 

  288. J.Ā T. Tate. Global class field theory. In Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), pages 162ā€“203. Thompson, Washington, D.C., 1967.

    Google ScholarĀ 

  289. J.Ā T. Tate. The arithmetic of elliptic curves. Invent. Math., 23:179ā€“206, 1974.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  290. R.Ā Taylor. Automorphy for some l-adic lifts of automorphic mod l representations. II. Inst. Hautes Ɖtudes Sci. Publ. Math. submitted.

    Google ScholarĀ 

  291. R.Ā Taylor and A.Ā Wiles. Ring-theoretic properties of certain Hecke algebras. Ann. of Math. (2), 141(3):553ā€“572, 1995.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  292. E.Ā Teske. A space efficient algorithm for group structure computation. Math. Comp., 67(224):1637ā€“1663, 1998.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  293. E.Ā Teske. Speeding up Pollardā€™s rho method for computing discrete logarithms. In Algorithmic Number Theory (Portland, OR, 1998), volume 1423 of Lecture Notes in Comput. Sci., pages 541ā€“554. Springer, Berlin, 1998.

    MATHĀ  Google ScholarĀ 

  294. E.Ā Teske. Square-root algorithms for the discrete logarithm problem (a survey). In Public-Key Cryptography and Computational Number Theory (Warsaw, 2000), pages 283ā€“301. de Gruyter, Berlin, 2001.

    Google ScholarĀ 

  295. D.Ā Ulmer. Elliptic curves with large rank over function fields. Ann. of Math. (2), 155(1):295ā€“315, 2002.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  296. B.Ā L. vanĀ der Waerden. Algebra. Vols. I and II. Springer-Verlag, New York, 1991. Based in part on lectures by E. Artin and E. Noether, Translated from the seventh German edition by Fred Blum and John R. Schulenberger.

    Google ScholarĀ 

  297. J.Ā VĆ©lu. IsogĆ©nies entre courbes elliptiques. C. R. Acad. Sci. Paris SĆ©r. A-B, 273:A238ā€“A241, 1971.

    Google ScholarĀ 

  298. P.Ā Vojta. A higher-dimensional Mordell conjecture. In Arithmetic geometry (Storrs, Conn., 1984), pages 341ā€“353. Springer, New York, 1986.

    Google ScholarĀ 

  299. P.Ā Vojta. Siegelā€™s theorem in the compact case. Ann. of Math. (2), 133(3):509ā€“548, 1991.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  300. J.Ā F. Voloch. Diagonal equations over function fields. Bol. Soc. Brasil. Mat., 16(2):29ā€“39, 1985.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  301. P.Ā M. Voutier. An upper bound for the size of integral solutions to Y mā€‰=ā€‰f(X). J. Number Theory, 53(2):247ā€“271, 1995.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  302. R.Ā J. Walker. Algebraic curves. Springer-Verlag, New York, 1978. Reprint of the 1950 edition.

    Google ScholarĀ 

  303. M.Ā Ward. Memoir on elliptic divisibility sequences. Amer. J. Math., 70:31ā€“74, 1948.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  304. L.Ā C. Washington. Elliptic curves. Discrete Mathematics and Its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, second edition, 2008. Number theory and cryptography.

    Google ScholarĀ 

  305. A.Ā Weil. Numbers of solutions of equations in finite fields. Bull. Amer. Math. Soc., 55:497ā€“508, 1949.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  306. A.Ā Weil. Jacobi sums as ā€œGrƶssencharaktere.ā€ Trans. Amer. Math. Soc., 73:487ā€“495, 1952.

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  307. A.Ā Weil. On algebraic groups and homogeneous spaces. Amer. J. Math., 77:493ā€“512, 1955.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  308. A.Ā Weil. Ɯber die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. Math. Ann., 168:149ā€“156, 1967.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  309. A.Ā Weil. Dirichlet Series and Automorphic Forms, volume 189 of Lecture Notes in Mathematics. Springer-Verlag, 1971.

    Google ScholarĀ 

  310. E.Ā T. Whittaker and G.Ā N. Watson. A course of modern analysis. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1996. Reprint of the fourth (1927) edition.

    Google ScholarĀ 

  311. A.Ā Wiles. Modular elliptic curves and Fermatā€™s last theorem. Ann. of Math. (2), 141(3):443ā€“551, 1995.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  312. A.Ā Wiles. The Birch and Swinnerton-Dyer conjecture. In The millennium prize problems, pages 31ā€“41. Clay Math. Inst., Cambridge, MA, 2006.

    Google ScholarĀ 

  313. G.Ā WĆ¼stholz. Recent progress in transcendence theory. In Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983), volume 1068 of Lecture Notes in Math., pages 280ā€“296. Springer, Berlin, 1984.

    MATHĀ  Google ScholarĀ 

  314. G.Ā WĆ¼stholz. Multiplicity estimates on group varieties. Ann. of Math. (2), 129(3):471ā€“500, 1989.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  315. D.Ā Zagier. Large integral points on elliptic curves. Math. Comp., 48(177):425ā€“436, 1987.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  316. H.Ā G. Zimmer. On the difference of the Weil height and the NĆ©ron-Tate height. Math. Z., 147(1):35ā€“51, 1976.

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  317. K.Ā Zsigmondy. Zur Theorie der Potenzreste. Monatsh. Math., 3:265ā€“284, 1892.

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  318. P.Ā Deligne. La conjecture de Weil. I. Inst. Hautes Ɖtudes Sci. Publ. Math., (43):273ā€“307, 1977.

    Google ScholarĀ 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Joseph H. Silverman .

Rights and permissions

Reprints and permissions

Copyright information

Ā© 2009 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Silverman, J.H. (2009). Elliptic Curves over Finite Fields. In: The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, vol 106. Springer, New York, NY. https://doi.org/10.1007/978-0-387-09494-6_5

Download citation

Publish with us

Policies and ethics