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.
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References
M.Ā Abdalla, M.Ā Bellare, and P.Ā Rogaway. The oracle Diffie-Hellman assumptions and an analysis of DHIES. In Topics in cryptologyāCT-RSA 2001 (San Francisco, CA), volume 2020 of Lecture Notes in Comput. Sci., pages 143ā158. Springer, Berlin, 2001.
D.Ā Abramovich. Formal finiteness and the torsion conjecture on elliptic curves. A footnote to a paper: āRational torsion of prime order in elliptic curves over number fieldsā [AstĆ©risque No.Ā 228 (1995), 3, 81ā100] by S. Kamienny and B. Mazur. AstĆ©risque, (228):3, 5ā17, 1995. Columbia University Number Theory Seminar (New York, 1992).
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.
T.Ā M. Apostol. Introduction to analytic number theory. Springer-Verlag, New York, 1976. Undergraduate Texts in Mathematics.
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.
N.Ā Arthaud. On Birch and Swinnerton-Dyerās conjecture for elliptic curves with complex multiplication. I. Compositio Math., 37(2):209ā232, 1978.
E.Ā Artin. Galois theory. Dover Publications Inc., Mineola, NY, second edition, 1998. Edited and with a supplemental chapter by Arthur N. Milgram.
M.Ā F. Atiyah and I.Ā G. Macdonald. Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, Mass.āLondonāDon Mills, Ont., 1969.
M.Ā F. Atiyah and C.Ā T.Ā C. Wall. Cohomology of groups. In Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), pages 94ā115. Thompson, Washington, D.C., 1967.
A.Ā O.Ā L. Atkin and F.Ā Morain. Elliptic curves and primality proving. Math. Comp., 61(203):29ā68, 1993.
A.Ā Baker. Transcendental number theory. Cambridge Mathematical Library. Cambridge University Press, Cambridge, second edition, 1990.
A.Ā Baker and J.Ā Coates. Integer points on curves of genus 1. Proc. Cambridge Philos. Soc., 67:595ā602, 1970.
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):141ā145, 1998.
A.Ā F. Beardon. Iteration of Rational Functions, volume 132 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1991. Complex analytic dynamical systems.
E.Ā Bekyel. The density of elliptic curves having a global minimal Weierstrass equation. J. Number Theory, 109(1):41ā58, 2004.
D.Ā Bernstein and T.Ā Lange. Faster addition and doubling on elliptic curves. In Advances in cryptologyāASIACRYPT 2007, volume 4833 of Lecture Notes in Comput. Sci., pages 29ā50. Springer, Berlin, 2007.
B.Ā J. Birch. Cyclotomic fields and Kummer extensions. In Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), pages 85ā93. Thompson, Washington, D.C., 1967.
B.Ā J. Birch. How the number of points of an elliptic curve over a fixed prime field varies. J. London Math. Soc., 43:57ā60, 1968.
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.
B.Ā J. Birch and H.Ā P.Ā F. Swinnerton-Dyer. Notes on elliptic curves. I. J. Reine Angew. Math., 212:7ā25, 1963.
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 2ā32. Lecture Notes in Math., Vol. 476. Springer, Berlin, 1975.
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.
D.Ā Boneh and M.Ā Franklin. Identity-based encryption from the Weil pairing. In Advances in CryptologyāCRYPTO 2001 (Santa Barbara, CA), volume 2139 of Lecture Notes in Comput. Sci., pages 213ā229. Springer, Berlin, 2001.
D.Ā Boneh, B.Ā Lynn, and H.Ā Shacham. Short signatures from the Weil pairing. In Advances in cryptologyāASIACRYPT 2001 (Gold Coast), volume 2248 of Lecture Notes in Comput. Sci., pages 514ā532. Springer, Berlin, 2001.
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.
A.Ā Bremner. On the equation Y 2ā=āX(X 2 + p). In Number theory and applications (Banff, AB, 1988), volume 265 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., pages 3ā22. Kluwer Acad. Publ., Dordrecht, 1989.
A.Ā Bremner and J.Ā W.Ā S. Cassels. On the equation Y 2ā=āX(X 2 + p). Math. Comp., 42(165):257ā264, 1984.
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):843ā939 (electronic), 2001.
F.Ā Brezing and A.Ā Weng. Elliptic curves suitable for pairing based cryptography. Des. Codes Cryptogr., 37(1):133ā141, 2005.
M.Ā L. Brown. Note on supersingular primes of elliptic curves over Q. Bull. London Math. Soc., 20(4):293ā296, 1988.
W.Ā D. Brownawell and D.Ā W. Masser. Vanishing sums in function fields. Math. Proc. Cambridge Philos. Soc., 100(3):427ā434, 1986.
Y.Ā Bugeaud. Bounds for the solutions of superelliptic equations. Compositio Math., 107(2):187ā219, 1997.
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):473ā481, 1985.
E.Ā R. Canfield, P.Ā ErdÅs, and C.Ā Pomerance. On a problem of Oppenheim concerning āfactorisatio numerorum.ā J. Number Theory, 17(1):1ā28, 1983.
H.Ā Carayol. Sur les reprĆ©sentations galoisiennes modulo l attachĆ©es aux formes modulaires. Duke Math. J., 59(3):785ā801, 1989.
J.Ā W.Ā S. Cassels. A note on the division values of ā(u). Proc. Cambridge Philos. Soc., 45:167ā172, 1949.
J.Ā W.Ā S. Cassels. Arithmetic on curves of genus 1. III. The Tate-Å afareviÄ and Selmer groups. Proc. London Math. Soc. (3), 12:259ā296, 1962.
J.Ā W.Ā S. Cassels. Arithmetic on curves of genus 1. IV. Proof of the Hauptvermutung. J. Reine Angew. Math., 211:95ā112, 1962.
J.Ā W.Ā S. Cassels. Arithmetic on curves of genus 1. V. Two counterexamples. J. London Math. Soc., 38:244ā248, 1963.
J.Ā W.Ā S. Cassels. Arithmetic on curves of genus 1. VIII. On conjectures of Birch and Swinnerton-Dyer. J. Reine Angew. Math., 217:180ā199, 1965.
J.Ā W.Ā S. Cassels. Diophantine equations with special reference to elliptic curves. J. London Math. Soc., 41:193ā291, 1966.
J.Ā W.Ā S. Cassels. Global fields. In Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), pages 42ā84. Thompson, Washington, D.C., 1967.
J.Ā W.Ā S. Cassels. Lectures on elliptic curves, volumeĀ 24 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1991.
T.Ā Chinburg. An introduction to Arakelov intersection theory. In Arithmetic geometry (Storrs, Conn., 1984), pages 289ā307. Springer, New York, 1986.
D.Ā V. Chudnovsky and G.Ā V. Chudnovsky. PadĆ© approximations and Diophantine geometry. Proc. Nat. Acad. Sci. U.S.A., 82(8):2212ā2216, 1985.
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.
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.
J.Ā Coates. Construction of rational functions on a curve. Proc. Cambridge Philos. Soc., 68:105ā123, 1970.
J.Ā Coates and A.Ā Wiles. On the conjecture of Birch and Swinnerton-Dyer. Invent. Math., 39(3):223ā251, 1977.
H.Ā Cohen. A Course in Computational Algebraic Number Theory, volume 138 of Graduate Texts in Mathematics. Springer-Verlag, Berlin, 1993.
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.
D.Ā A. Cox. The arithmetic-geometric mean of Gauss. Enseign. Math. (2), 30(3-4):275ā330, 1984.
J.Ā Cremona. Elliptic Curve Data. http://sage.math.washington. edu/cremona/index.html , http://www.math.utexas.edu/users/ tornaria/cnt/cremona.html .
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 .
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):42ā68, 2006.
L.Ā V. Danilov. The Diophantine equation x 3 ā y 2ā=āk and a conjecture of M. Hall. Mat. Zametki, 32(3):273ā275, 425, 1982. English translation: Math. Notes Acad. Sci. USSR 32 (1982), no. 3ā4, 617ā618 (1983).
H.Ā Davenport. On f 3ā(t) ā g 2ā(t). Norske Vid. Selsk. Forh. (Trondheim), 38:86ā87, 1965.
S.Ā David. Minorations de formes linĆ©aires de logarithmes elliptiques. MĆ©m. Soc. Math. France (N.S.), (62):iv+143, 1995.
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.
M.Ā Deuring. Die Typen der Multiplikatorenringe elliptischer Funktionenkƶrper. Abh. Math. Sem. Hansischen Univ., 14:197ā272, 1941.
M.Ā Deuring. Die Zetafunktion einer algebraischen Kurve vom Geschlechte Eins. Nachr. Akad. Wiss. Gƶttingen. Math.-Phys. Kl. Math.-Phys.-Chem. Abt., 1953:85ā94, 1953.
M.Ā Deuring. Die Zetafunktion einer algebraischen Kurve vom Geschlechte Eins. II. Nachr. Akad. Wiss. Gƶttingen. Math.-Phys. Kl. IIa., 1955:13ā42, 1955.
M.Ā Deuring. Die Zetafunktion einer algebraischen Kurve vom Geschlechte Eins. III. Nachr. Akad. Wiss. Gƶttingen. Math.-Phys. Kl. IIa., 1956:37ā76, 1956.
M.Ā Deuring. Die Zetafunktion einer algebraischen Kurve vom Geschlechte Eins. IV. Nachr. Akad. Wiss. Gƶttingen. Math.-Phys. Kl. IIa., 1957:55ā80, 1957.
W.Ā Diffie and M.Ā E. Hellman. New directions in cryptography. IEEE Trans. Information Theory, IT-22(6):644ā654, 1976.
L.Ā Dirichlet. Ćber den biquadratischen Charakter der Zahl āZwei.ā J. Reine Angew. Math., 57:187ā188, 1860.
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):5583ā5597, 2000.
D.Ā S. Dummit and R.Ā M. Foote. Abstract algebra. John Wiley & Sons Inc., Hoboken, NJ, third edition, 2004.
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):79ā89, 2005.
B.Ā Dwork. On the rationality of the zeta function of an algebraic variety. Amer. J. Math., 82:631ā648, 1960.
H.Ā M. Edwards. A normal form for elliptic curves. Bull. Amer. Math. Soc. (N.S.), 44(3):393ā422 (electronic), 2007.
M.Ā Eichler. QuaternƤre quadratische Formen und die Riemannsche Vermutung fĆ¼r die Kongruenzzetafunktion. Arch. Math., 5:355ā366, 1954.
D.Ā Eisenbud. Commutative algebra, volume 150 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995. With a view toward algebraic geometry.
T.Ā ElGamal. A public key cryptosystem and a signature scheme based on discrete logarithms. IEEE Trans. Inform. Theory, 31(4):469ā472, 1985.
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.
N.Ā Elkies. \(\mathbb{Z}^{28}\) in \(E(\mathbb{Q})\). Number Theory Listserver, May 2006.
N.Ā D. Elkies. The existence of infinitely many supersingular primes for every elliptic curve over \(\mathbb{Q}\). Invent. Math., 89(3):561ā567, 1987.
N.Ā D. Elkies. Distribution of supersingular primes. AstĆ©risque, (198-200):127ā132 (1992), 1991. JournĆ©es ArithmĆ©tiques, 1989 (Luminy, 1989).
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 21ā76. Amer. Math. Soc., Providence, RI, 1998.
J.-H. Evertse. On equations in S-units and the Thue-Mahler equation. Invent. Math., 75(3):561ā584, 1984.
J.-H. Evertse and J.Ā H. Silverman. Uniform bounds for the number of solutions to Y nā=āf(X). Math. Proc. Cambridge Philos. Soc., 100(2):237ā248, 1986.
G.Ā Faltings. EndlichkeitssƤtze fĆ¼r abelsche VarietƤten Ć¼ber Zahlkƶrpern. Invent. Math., 73(3):349ā366, 1983.
G.Ā Faltings. Calculus on arithmetic surfaces. Ann. of Math. (2), 119(2):387ā424, 1984.
G.Ā Faltings. Finiteness theorems for abelian varieties over number fields. In Arithmetic geometry (Storrs, Conn., 1984), pages 9ā27. Springer, New York, 1986. Translated from the German original [Invent.Ā Math.Ā 73 (1983), no.Ā 3, 349ā366; ibid.Ā 75 (1984), no.Ā 2, 381; MR 85g:11026ab] by Edward Shipz.
S.Ā Fermigier. Une courbe elliptique dĆ©finie sur Q de rangāā„ā22. Acta Arith., 82(4):359ā363, 1997.
E.Ā V. Flynn and C.Ā Grattoni. Descent via isogeny on elliptic curves with large rational torsion subgroups. J. Symbolic Comput., 43(4):293ā303, 2008.
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 452ā465. Springer, Berlin, 2006.
G.Ā Frey. Links between stable elliptic curves and certain Diophantine equations. Ann. Univ. Sarav. Ser. Math., 1(1):iv+40, 1986.
G.Ā Frey. Elliptic curves and solutions of A ā Bā=āC. In SĆ©minaire de ThĆ©orie des Nombres, Paris 1985ā86, volumeĀ 71 of Progr. Math., pages 39ā51. BirkhƤuser Boston, Boston, MA, 1987.
G.Ā Frey. Links between solutions of A ā Bā=āC and elliptic curves. In Number theory (Ulm, 1987), volume 1380 of Lecture Notes in Math., pages 31ā62. Springer, New York, 1989.
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., 62:865ā874, 1994.
A.Ā Frƶhlich. Local fields. In Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), pages 1ā41. Thompson, Washington, D.C., 1967.
A.Ā Frƶhlich. Formal groups. Lecture Notes in Mathematics, No. 74. Springer-Verlag, Berlin, 1968.
R.Ā Fueter. Ueber kubische diophantische Gleichungen. Comment. Math. Helv., 2(1):69ā89, 1930.
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.
J.Ā Gebel, A.Ā PethÅ, and H.Ā G. Zimmer. Computing integral points on elliptic curves. Acta Arith., 68(2):171ā192, 1994.
S.Ā Goldwasser and J.Ā Kilian. Almost all primes can be quickly certified. In STOC ā86: Proceedings of the eighteenth annual ACM symposium on Theory of computing, pages 316ā329, New York, 1986. ACM.
R.Ā Greenberg. On the Birch and Swinnerton-Dyer conjecture. Invent. Math., 72(2):241ā265, 1983.
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.
B.Ā Gross, W.Ā Kohnen, and D.Ā Zagier. Heegner points and derivatives of L-series. II. Math. Ann., 278(1-4):497ā562, 1987.
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):85ā87, 1983.
B.Ā H. Gross and D.Ā B. Zagier. Heegner points and derivatives of L-series. Invent. Math., 84(2):225ā320, 1986.
R.Ā Gross. A note on Rothās theorem. J. Number Theory, 36:127ā132, 1990.
R.Ā Gross and J.Ā Silverman. S-integer points on elliptic curves. Pacific J. Math., 167(2):263ā288, 1995.
K.Ā Gruenberg. Profinite groups. In Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), pages 116ā127. Thompson, Washington, D.C., 1967.
M.Ā Hall, Jr. The Diophantine equation x 3 ā y 2ā=āk. In Computers in number theory (Proc. Sci. Res. Council Atlas Sympos. No. 2, Oxford, 1969), pages 173ā198. Academic Press, London, 1971.
D.Ā Hankerson, A.Ā Menezes, and S.Ā Vanstone. Guide to elliptic curve cryptography. Springer Professional Computing. Springer-Verlag, New York, 2004.
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.
J.Ā Harris. Algebraic geometry, volume 133 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1992. A first course.
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.
R.Ā Hartshorne. Algebraic geometry. Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, No. 52.
M.Ā Hazewinkel. Formal groups and applications, volumeĀ 78 of Pure and Applied Mathematics. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1978.
M.Ā Hindry and J.Ā H. Silverman. The canonical height and integral points on elliptic curves. Invent. Math., 93(2):419ā450, 1988.
M.Ā Hindry and J.Ā H. Silverman. Diophantine geometry, volume 201 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2000. An introduction.
G.Ā Hochschild and J.-P. Serre. Cohomology of group extensions. Trans. Amer. Math. Soc., 74:110ā134, 1953.
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.
A.Ā Hurwitz. Ćber ternƤre diophantische Gleichungen dritten Grades. Vierteljahrschrift d. Naturf. Ges. ZĆ¼rich, 62:207ā229, 1917.
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.
J.-I. Igusa. Class number of a definite quaternion with prime discriminant. Proc. Nat. Acad. Sci. U.S.A., 44:312ā314, 1958.
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 385ā393. Springer, Berlin, 2000.
S.Ā Kamienny. Torsion points on elliptic curves and q-coefficients of modular forms. Invent. Math., 109(2):221ā229, 1992.
S.Ā Kamienny and B.Ā Mazur. Rational torsion of prime order in elliptic curves over number fields. AstĆ©risque, (228):3, 81ā100, 1995. With an appendix by A. Granville, Columbia University Number Theory Seminar (New York, 1992).
N.Ā M. Katz. An overview of Deligneās 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 275ā305. Amer. Math. Soc., Providence, R.I., 1976.
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.
M.Ā A. Kenku. On the number of Q-isomorphism classes of elliptic curves in each Q-isogeny class. J. Number Theory, 15(2):199ā202, 1982.
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 402ā415. Springer, Berlin, 2008.
A.Ā W. Knapp. Elliptic curves, volumeĀ 40 of Mathematical Notes. Princeton University Press, Princeton, NJ, 1992.
N.Ā Koblitz. Elliptic curve cryptosystems. Math. Comp., 48(177):203ā209, 1987.
N.Ā Koblitz. Introduction to elliptic curves and modular forms, volumeĀ 97 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1993.
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):522ā540, 670ā671, 1988.
S.Ā V. Kotov and L.Ā A. Trelina. S-ganze Punkte auf elliptischen Kurven. J. Reine Angew. Math., 306:28ā41, 1979.
D.Ā S. Kubert. Universal bounds on the torsion of elliptic curves. Proc. London Math. Soc. (3), 33(2):193ā237, 1976.
E.Ā Kunz. Introduction to plane algebraic curves. BirkhƤuser Boston Inc., Boston, MA, 2005. Translated from the 1991 German edition by Richard G. Belshoff.
M.Ā Lal, M.Ā F. Jones, and W.Ā J. Blundon. Numerical solutions of the Diophantine equation y 3 ā x 2ā=āk. Math. Comp., 20:322ā325, 1966.
S.Ā Lang. Elliptic curves: Diophantine analysis, volume 231 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1978.
S.Ā Lang. Introduction to algebraic and abelian functions, volumeĀ 89 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1982.
S.Ā Lang. Complex multiplication, volume 255 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York, 1983.
S.Ā Lang. Conjectured Diophantine estimates on elliptic curves. In Arithmetic and geometry, Vol. I, volumeĀ 35 of Progr. Math., pages 155ā171. BirkhƤuser Boston, Boston, MA, 1983.
S.Ā Lang. Fundamentals of Diophantine geometry. Springer-Verlag, New York, 1983.
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.
S.Ā Lang. Number theory III, volumeĀ 60 of Encyclopedia of Mathematical Sciences. Springer-Verlag, Berlin, 1991.
S.Ā Lang. Algebraic number theory, volume 110 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1994.
S.Ā Lang. Algebra, volume 211 of Graduate Texts in Mathematics. Springer-Verlag, New York, third edition, 2002.
S.Ā Lang and J.Ā Tate. Principal homogeneous spaces over abelian varieties. Amer. J. Math., 80:659ā684, 1958.
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.
M.Ā Laska. An algorithm for finding a minimal Weierstrass equation for an elliptic curve. Math. Comp., 38(157):257ā260, 1982.
M.Ā Laska. Elliptic curves over number fields with prescribed reduction type. Aspects of Mathematics, E4. Friedr. Vieweg & Sohn, Braunschweig, 1983.
D.Ā J. Lewis and K.Ā Mahler. On the representation of integers by binary forms. Acta Arith., 6:333ā363, 1960/1961.
S.Ā Lichtenbaum. The period-index problem for elliptic curves. Amer. J. Math., 90:1209ā1223, 1968.
C.-E. Lind. Untersuchungen Ć¼ber die rationalen Punkte der ebenen kubischen Kurven vom Geschlecht Eins. Thesis, University of Uppsala,, 1940:97, 1940.
J.Ā Liouville. Sur des classes trĆØs-Ć©tendues de quantitĆ©s dont la irrationalles algĆ©briques. C. R. Acad. Paris, 18:883ā885 and 910ā911, 1844.
E.Ā Lutz. Sur lāequation y 2ā=āx 3 ā ax ā b dans les corps p-adic. J. Reine Angew. Math., 177:237ā247, 1937.
K.Ā Mahler. On the lattice points on curves of genus 1. Proc. London Math. Soc. (3), 39:431ā466, 1935.
J.Ā I. Manin. The Hasse-Witt matrix of an algebraic curve. Izv. Akad. Nauk SSSR Ser. Mat., 25:153ā172, 1961.
J.Ā I. Manin. The p-torsion of elliptic curves is uniformly bounded. Izv. Akad. Nauk SSSR Ser. Mat., 33:459ā465, 1969.
J.Ā I. Manin. Cyclotomic fields and modular curves. Uspehi Mat. Nauk, 26(6(162)):7ā71, 1971. English translation: Russian Math. Surveys 26 (1971), no. 6, 7ā78.
R.Ā C. Mason. The hyperelliptic equation over function fields. Math. Proc. Cambridge Philos. Soc., 93(2):219ā230, 1983.
R.Ā C. Mason. Norm form equations. I. J. Number Theory, 22(2):190ā207, 1986.
D.Ā Masser. Elliptic functions and transcendence. Springer-Verlag, Berlin, 1975. Lecture Notes in Mathematics, Vol. 437.
D.Ā Masser and G.Ā WĆ¼stholz. Isogeny estimates for abelian varieties, and finiteness theorems. Ann. of Math. (2), 137(3):459ā472, 1993.
D.Ā W. Masser. Specializations of finitely generated subgroups of abelian varieties. Trans. Amer. Math. Soc., 311(1):413ā424, 1989.
D.Ā W. Masser and G.Ā WĆ¼stholz. Fields of large transcendence degree generated by values of elliptic functions. Invent. Math., 72(3):407ā464, 1983.
D.Ā W. Masser and G.Ā WĆ¼stholz. Estimating isogenies on elliptic curves. Invent. Math., 100(1):1ā24, 1990.
H.Ā Matsumura. Commutative algebra, volumeĀ 56 of Mathematics Lecture Note Series. Benjamin/Cummings Publishing Co., Inc., Reading, Mass., second edition, 1980.
B.Ā Mazur. Modular curves and the Eisenstein ideal. Inst. Hautes Ćtudes Sci. Publ. Math., (47):33ā186 (1978), 1977.
B.Ā Mazur. Rational isogenies of prime degree (with an appendix by D. Goldfeld). Invent. Math., 44(2):129ā162, 1978.
H.Ā McKean and V.Ā Moll. Elliptic curves. Cambridge University Press, Cambridge, 1997. Function theory, geometry, arithmetic.
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):1639ā1646, 1993.
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.
L.Ā Merel. Bornes pour la torsion des courbes elliptiques sur les corps de nombres. Invent. Math., 124(1-3):437ā449, 1996.
J.-F. Mestre. Construction dāune courbe elliptique de rangāā„ā12. C. R. Acad. Sci. Paris SĆ©r. I Math., 295(12):643ā644, 1982.
J.-F. Mestre. Courbes elliptiques et formules explicites. In Seminar on number theory, Paris 1981ā82 (Paris, 1981/1982), volumeĀ 38 of Progr. Math., pages 179ā187. BirkhƤuser Boston, Boston, MA, 1983.
M.Ā Mignotte. Quelques remarques sur lāapproximation rationnelle des nombres algĆ©briques. J. Reine Angew. Math., 268/269:341ā347, 1974. Collection of articles dedicated to Helmut Hasse on his seventy-fifth birthday, II.
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 573ā581. Springer, Berlin, 2006.
S.Ā D. Miller and R.Ā Venkatesan. Non-degeneracy of Pollard rho collisions, 2008. arXiv:0808.0469.
V.Ā S. Miller. Use of elliptic curves in cryptography. In Advances in CryptologyāCRYPTO ā85 (Santa Barbara, Calif., 1985), volume 218 of Lecture Notes in Comput. Sci., pages 417ā426. Springer, Berlin, 1986.
J.Ā S. Milne. Arithmetic duality theorems, volumeĀ 1 of Perspectives in Mathematics. Academic Press Inc., Boston, MA, 1986.
J.Ā S. Milne. Elliptic curves. BookSurge Publishers, Charleston, SC, 2006.
J.Ā Milnor. On LattĆØs maps. ArXiv:math.DS/0402147, Stony Brook IMS Preprint #2004/01.
R.Ā Miranda. Algebraic curves and Riemann surfaces, volumeĀ 5 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1995.
A.Ā Miyaji, M.Ā Nakabayashi, and S.Ā Takano. Characterization of elliptic curve traces under FR-reduction. In Information security and cryptologyāICISC 2000 (Seoul), volume 2015 of Lecture Notes in Comput. Sci., pages 90ā108. Springer, Berlin, 2001.
P.Ā Monsky. Three constructions of rational points on Y 2ā=āX 3 Ā± NX. Math. Z., 209(3):445ā462, 1992.
F.Ā Morain. Building cyclic elliptic curves modulo large primes. In Advances in cryptologyāEUROCRYPT ā91 (Brighton, 1991), volume 547 of Lecture Notes in Comput. Sci., pages 328ā336. Springer, Berlin, 1991.
L.Ā J. Mordell. The diophantine equation x 4 + my 4ā=āz 2.ā. Quart. J. Math. Oxford Ser. (2), 18:1ā6, 1967.
L.Ā J. Mordell. Diophantine equations. Pure and Applied Mathematics, Vol. 30. Academic Press, London, 1969.
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.
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.
K.-I. Nagao. Construction of high-rank elliptic curves. Kobe J. Math., 11(2):211ā219, 1994.
K.-I. Nagao. \(\mathbb{Q}(T)\)-rank of elliptic curves and certain limit coming from the local points. Manuscripta Math., 92(1):13ā32, 1997. With an appendix by Nobuhiko Ishida, Tsuneo Ishikawa and the author.
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.
NBSāDSS. Digital Signature Standard (DSS). FIPS Publication 186-2, National Bureau of Standards, 2000. http://csrc.nist.gov/publications/ PubsFIPS.html .
A.Ā NĆ©ron. ProblĆØmes arithmĆ©tiques et gĆ©omĆ©triques rattachĆ©s Ć la notion de rang dāune courbe algĆ©brique dans un corps. Bull. Soc. Math. France, 80:101ā166, 1952.
A.Ā NĆ©ron. ModĆØles minimaux des variĆ©tĆ©s abĆ©liennes sur les corps locaux et globaux. Inst. Hautes Ćtudes Sci. Publ.Math. No., 21:128, 1964.
A.Ā NĆ©ron. Quasi-fonctions et hauteurs sur les variĆ©tĆ©s abĆ©liennes. Ann. of Math. (2), 82:249ā331, 1965.
O.Ā Neumann. Elliptische Kurven mit vorgeschriebenem Reduktionsverhalten. I. Math. Nachr., 49:107ā123, 1971.
J.Ā OesterlĆ©. Nouvelles approches du āthĆ©orĆØmeā de Fermat. AstĆ©risque, (161-162):Exp.Ā No.Ā 694, 4, 165ā186 (1989), 1988. SĆ©minaire Bourbaki, Vol.Ā 1987/88.
A.Ā Ogg. Modular forms and Dirichlet series. W. A. Benjamin, Inc., New York-Amsterdam, 1969.
A.Ā P. Ogg. Abelian curves of 2-power conductor. Proc. Cambridge Philos. Soc., 62:143ā148, 1966.
A.Ā P. Ogg. Abelian curves of small conductor. J. Reine Angew. Math., 226:204ā215, 1967.
A.Ā P. Ogg. Elliptic curves and wild ramification. Amer. J. Math., 89:1ā21, 1967.
L.Ā D. Olson. Torsion points on elliptic curves with given j-invariant. Manuscripta Math., 16(2):145ā150, 1975.
PARI/GP, 2005. http://pari.math.u-bordeaux.fr/.
A.Ā N. ParÅ”in. Algebraic curves over function fields. I. Izv. Akad. Nauk SSSR Ser. Mat., 32:1191ā1219, 1968.
R.Ā G.Ā E. Pinch. Elliptic curves with good reduction away from 2. Math. Proc. Cambridge Philos. Soc., 96(1):25ā38, 1984.
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):106ā110, 1978.
J.Ā M. Pollard. Monte Carlo methods for index computation \((\mathop{\mathrm{mod}}\nolimits \ p)\). Math. Comp., 32(143):918ā924, 1978.
H.Ā Reichardt. Einige im Kleinen Ć¼berall lƶsbare, im Grossen unlƶsbare diophantische Gleichungen. J. Reine Angew. Math., 184:12ā18, 1942.
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):431ā476, 1990.
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):120ā126, 1978.
A.Ā Robert. Elliptic curves. Springer-Verlag, Berlin, 1973. Notes from postgraduate lectures given in Lausanne 1971/72, Lecture Notes in Mathematics, Vol. 326.
D.Ā E. Rohrlich. On L-functions of elliptic curves and anticyclotomic towers. Invent. Math., 75(3):383ā408, 1984.
P.Ā Roquette. Analytic theory of elliptic functions over local fields. Hamburger Mathematische Einzelschriften (N.F.), Heft 1. Vandenhoeck & Ruprecht, Gƶttingen, 1970.
M.Ā Rosen and J.Ā H. Silverman. On the rank of an elliptic surface. Invent. Math., 133(1):43ā67, 1998.
K.Ā Rubin. Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer. Invent. Math., 64(3):455ā470, 1981.
K.Ā Rubin. Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication. Invent. Math., 89(3):527ā559, 1987.
K.Ā Rubin. The āmain conjecturesā of Iwasawa theory for imaginary quadratic fields. Invent. Math., 103(1):25ā68, 1991.
T.Ā Saito. Conductor, discriminant, and the Noether formula of arithmetic surfaces. Duke Math. J., 57(1):151ā173, 1988.
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):81ā92, 1998.
E.Ā F. Schaefer and M.Ā Stoll. How to do a p-descent on an elliptic curve. Trans. Amer. Math. Soc., 356(3):1209ā1231 (electronic), 2004.
S.Ā H. Schanuel. Heights in number fields. Bull. Soc. Math. France, 107(4):433ā449, 1979.
W.Ā M. Schmidt. Diophantine approximation, volume 785 of Lecture Notes in Mathematics. Springer, Berlin, 1980.
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 PethÅ.
R.Ā Schoof. Elliptic curves over finite fields and the computation of square roots mod p. Math. Comp., 44(170):483ā494, 1985.
R.Ā Schoof. Counting points on elliptic curves over finite fields. J. ThĆ©or. Nombres Bordeaux, 7(1):219ā254, 1995. Les Dix-huitiĆØmes JournĆ©es ArithmĆ©tiques (Bordeaux, 1993).
E.Ā S. Selmer. The Diophantine equation ax 3 + by 3 + cz 3ā=ā0. Acta Math., 85:203ā362 (1 plate), 1951.
E.Ā S. Selmer. A conjecture concerning rational points on cubic curves. Math. Scand., 2:49ā54, 1954.
E.Ā S. Selmer. The diophantine equation ax 3 + by 3 + cz 3ā=ā0.āCompletion of the tables. Acta Math., 92:191ā197, 1954.
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):353ā356, 1998.
J.-P. Serre. GĆ©omĆ©trie algĆ©brique et gĆ©omĆ©trie analytique. Ann. Inst. Fourier, Grenoble, 6:1ā42, 1955ā1956.
J.-P. Serre. Complex multiplication. In Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), pages 292ā296. Thompson, Washington, D.C., 1967.
J.-P. Serre. PropriĆ©tĆ©s galoisiennes des points dāordre fini des courbes elliptiques. Invent. Math., 15(4):259ā331, 1972.
J.-P. Serre. A course in arithmetic. Springer-Verlag, New York, 1973. Translated from the French, Graduate Texts in Mathematics, No. 7.
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.
J.-P. Serre. Quelques applications du thĆ©orĆØme de densitĆ© de Chebotarev. Inst. Hautes Ćtudes Sci. Publ. Math., (54):323ā401, 1981.
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):179ā230, 1987.
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.
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.
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.
J.-P. Serre and J.Ā Tate. Good reduction of abelian varieties. Ann. of Math. (2), 88:492ā517, 1968.
B.Ā Setzer. Elliptic curves of prime conductor. J. London Math. Soc. (2), 10:367ā378, 1975.
B.Ā Setzer. Elliptic curves over complex quadratic fields. Pacific J. Math., 74(1):235ā250, 1978.
I.Ā R. Shafarevich. Algebraic number fields. In Proc. Int. Cong. (Stockholm 1962), pages 25ā39. American Mathematical Society, Providence, R.I., 1963. Amer. Math. Soc. Transl., SeriesĀ 2, Vol.Ā 31.
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.
I.Ā R. Shafarevich and J.Ā Tate. The rank of elliptic curves. In Amer. Math. Soc. Transl., volumeĀ 8, pages 917ā920. Amer. Math. Soc., 1967.
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 47ā53. Springer, Berlin, 1985.
G.Ā Shimura. Correspondances modulaires et les fonctions Ī¶ de courbes algĆ©briques. J. Math. Soc. Japan, 10:1ā28, 1958.
G.Ā Shimura. On elliptic curves with complex multiplication as factors of the Jacobians of modular function fields. Nagoya Math. J., 43:199ā208, 1971.
G.Ā Shimura. On the zeta-function of an abelian variety with complex multiplication. Ann. of Math. (2), 94:504ā533, 1971.
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.
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.
T.Ā Shioda. An explicit algorithm for computing the Picard number of certain algebraic surfaces. Amer. J. Math., 108(2):415ā432, 1986.
R.Ā Shipsey. Elliptic divisibility sequences. PhD thesis, Goldsmithās College (University of London), 2000.
V.Ā Shoup. Lower bounds for discrete logarithms and related problems. In Advances in cryptologyāEUROCRYPT ā97 (Konstanz), volume 1233 of Lecture Notes in Comput. Sci., pages 256ā266. Springer, Berlin, 1997. updated version at www.shoup.net/papers/dlbounds1.pdf.
J.Ā H. Silverman. Lower bound for the canonical height on elliptic curves. Duke Math. J., 48(3):633ā648, 1981.
J.Ā H. Silverman. The NĆ©ronāTate height on elliptic curves. PhD thesis, Harvard University, 1981.
J.Ā H. Silverman. Heights and the specialization map for families of abelian varieties. J. Reine Angew. Math., 342:197ā211, 1983.
J.Ā H. Silverman. Integer points on curves of genus 1. J. London Math. Soc. (2), 28(1):1ā7, 1983.
J.Ā H. Silverman. The S-unit equation over function fields. Math. Proc. Cambridge Philos. Soc., 95(1):3ā4, 1984.
J.Ā H. Silverman. Weierstrass equations and the minimal discriminant of an elliptic curve. Mathematika, 31(2):245ā251 (1985), 1984.
J.Ā H. Silverman. Divisibility of the specialization map for families of elliptic curves. Amer. J. Math., 107(3):555ā565, 1985.
J.Ā H. Silverman. Arithmetic distance functions and height functions in Diophantine geometry. Math. Ann., 279(2):193ā216, 1987.
J.Ā H. Silverman. A quantitative version of Siegelās theorem: integral points on elliptic curves and Catalan curves. J. Reine Angew. Math., 378:60ā100, 1987.
J.Ā H. Silverman. Computing heights on elliptic curves. Math. Comp., 51(183):339ā358, 1988.
J.Ā H. Silverman. Wieferichās criterion and the abc-conjecture. J. Number Theory, 30(2):226ā237, 1988.
J.Ā H. Silverman. The difference between the Weil height and the canonical height on elliptic curves. Math. Comp., 55(192):723ā743, 1990.
J.Ā H. Silverman. Advanced topics in the arithmetic of elliptic curves, volume 151 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1994.
J.Ā H. Silverman. The arithmetic of dynamical systems, volume 241 of Graduate Texts in Mathematics. Springer, New York, 2007.
N.Ā P. Smart. S-integral points on elliptic curves. Math. Proc. Cambridge Philos. Soc., 116(3):391ā399, 1994.
N.Ā P. Smart. The discrete logarithm problem on elliptic curves of trace one. J. Cryptology, 12(3):193ā196, 1999.
K.Ā Stange. The Tate pairing via elliptic nets. In Pairing Based Cryptography, Lecture Notes in Comput. Sci. Springer, 2007.
K.Ā Stange. Elliptic Nets and Elliptic Curves. PhD thesis, Brown University, 2008.
K.Ā Stange. Elliptic nets and elliptic curves, 2008. arXiv:0710.1316v2.
H.Ā M. Stark. Effective estimates of solutions of some Diophantine equations. Acta Arith., 24:251ā259, 1973.
W.Ā Stein. The Modular Forms Database. http://modular.fas.harvard. edu/Tables .
W.Ā Stein. Sage Mathematics Software, 2007. http://www.sagemath.org.
N.Ā M. Stephens. The Diophantine equation X 3 + Y 3ā=āDZ 3 and the conjectures of Birch and Swinnerton-Dyer. J. Reine Angew. Math., 231:121ā162, 1968.
D.Ā R. Stinson. Cryptography: Theory and Practice. CRC Press Series on Discrete Mathematics and Its Applications. Chapman & Hall/CRC, Boca Raton, FL, 2002.
W.Ā W. Stothers. Polynomial identities and Hauptmoduln. Quart. J. Math. Oxford Ser. (2), 32(127):349ā370, 1981.
R.Ā J. Stroeker and N.Ā Tzanakis. Solving elliptic Diophantine equations by estimating linear forms in elliptic logarithms. Acta Arith., 67(2):177ā196, 1994.
J.Ā Tate. Letter to J.-P. Serre, 1968.
J.Ā Tate. Duality theorems in Galois cohomology over number fields. In Proc. Internat. Congr. Mathematicians (Stockholm, 1962), pages 288ā295. Inst. Mittag-Leffler, Djursholm, 1963.
J.Ā Tate. Endomorphisms of abelian varieties over finite fields. Invent. Math., 2:134ā144, 1966.
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 33ā52. Lecture Notes in Math., Vol. 476. Springer, Berlin, 1975.
J.Ā Tate. Variation of the canonical height of a point depending on a parameter. Amer. J. Math., 105(1):287ā294, 1983.
J.Ā Tate. A review of non-Archimedean elliptic functions. In Elliptic curves, modular forms, & Fermatās last theorem (Hong Kong, 1993), Ser. Number Theory, I, pages 162ā184. Int. Press, Cambridge, MA, 1995.
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, 265ā277. Soc. Math. France, Paris, 1995.
J.Ā T. Tate. Algebraic cycles and poles of zeta functions. In Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), pages 93ā110. Harper & Row, New York, 1965.
J.Ā T. Tate. Global class field theory. In Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), pages 162ā203. Thompson, Washington, D.C., 1967.
J.Ā T. Tate. The arithmetic of elliptic curves. Invent. Math., 23:179ā206, 1974.
R.Ā Taylor. Automorphy for some l-adic lifts of automorphic mod l representations. II. Inst. Hautes Ćtudes Sci. Publ. Math. submitted.
R.Ā Taylor and A.Ā Wiles. Ring-theoretic properties of certain Hecke algebras. Ann. of Math. (2), 141(3):553ā572, 1995.
E.Ā Teske. A space efficient algorithm for group structure computation. Math. Comp., 67(224):1637ā1663, 1998.
E.Ā Teske. Speeding up Pollardās rho method for computing discrete logarithms. In Algorithmic Number Theory (Portland, OR, 1998), volume 1423 of Lecture Notes in Comput. Sci., pages 541ā554. Springer, Berlin, 1998.
E.Ā Teske. Square-root algorithms for the discrete logarithm problem (a survey). In Public-Key Cryptography and Computational Number Theory (Warsaw, 2000), pages 283ā301. de Gruyter, Berlin, 2001.
D.Ā Ulmer. Elliptic curves with large rank over function fields. Ann. of Math. (2), 155(1):295ā315, 2002.
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.
J.Ā VĆ©lu. IsogĆ©nies entre courbes elliptiques. C. R. Acad. Sci. Paris SĆ©r. A-B, 273:A238āA241, 1971.
P.Ā Vojta. A higher-dimensional Mordell conjecture. In Arithmetic geometry (Storrs, Conn., 1984), pages 341ā353. Springer, New York, 1986.
P.Ā Vojta. Siegelās theorem in the compact case. Ann. of Math. (2), 133(3):509ā548, 1991.
J.Ā F. Voloch. Diagonal equations over function fields. Bol. Soc. Brasil. Mat., 16(2):29ā39, 1985.
P.Ā M. Voutier. An upper bound for the size of integral solutions to Y mā=āf(X). J. Number Theory, 53(2):247ā271, 1995.
R.Ā J. Walker. Algebraic curves. Springer-Verlag, New York, 1978. Reprint of the 1950 edition.
M.Ā Ward. Memoir on elliptic divisibility sequences. Amer. J. Math., 70:31ā74, 1948.
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.
A.Ā Weil. Numbers of solutions of equations in finite fields. Bull. Amer. Math. Soc., 55:497ā508, 1949.
A.Ā Weil. Jacobi sums as āGrƶssencharaktere.ā Trans. Amer. Math. Soc., 73:487ā495, 1952.
A.Ā Weil. On algebraic groups and homogeneous spaces. Amer. J. Math., 77:493ā512, 1955.
A.Ā Weil. Ćber die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. Math. Ann., 168:149ā156, 1967.
A.Ā Weil. Dirichlet Series and Automorphic Forms, volume 189 of Lecture Notes in Mathematics. Springer-Verlag, 1971.
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.
A.Ā Wiles. Modular elliptic curves and Fermatās last theorem. Ann. of Math. (2), 141(3):443ā551, 1995.
A.Ā Wiles. The Birch and Swinnerton-Dyer conjecture. In The millennium prize problems, pages 31ā41. Clay Math. Inst., Cambridge, MA, 2006.
G.Ā WĆ¼stholz. Recent progress in transcendence theory. In Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983), volume 1068 of Lecture Notes in Math., pages 280ā296. Springer, Berlin, 1984.
G.Ā WĆ¼stholz. Multiplicity estimates on group varieties. Ann. of Math. (2), 129(3):471ā500, 1989.
D.Ā Zagier. Large integral points on elliptic curves. Math. Comp., 48(177):425ā436, 1987.
H.Ā G. Zimmer. On the difference of the Weil height and the NĆ©ron-Tate height. Math. Z., 147(1):35ā51, 1976.
K.Ā Zsigmondy. Zur Theorie der Potenzreste. Monatsh. Math., 3:265ā284, 1892.
P.Ā Deligne. La conjecture de Weil. I. Inst. Hautes Ćtudes Sci. Publ. Math., (43):273ā307, 1977.
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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
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