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Algebraic Varieties Kapitel lesen Erstes Kapitel lesen

2009 | OriginalPaper | Buchkapitel

Elliptic Curves over \(\mathbb{C}\)

verfasst von : Joseph H. Silverman

Erschienen in: The Arithmetic of Elliptic Curves

Verlag: Springer New York

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Abstract

Evaluation of the integral giving arc length on a circle, namely \(\int dx/\sqrt{1 - x^{2}}\), leads to an inverse trigonometric function. The analogous problem for the arc length of an ellipse yields an integral that is not computable in terms of so-called elementary functions.

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Vorheriges Kapitel Elliptic Curves over Finite Fields
Nächstes Kapitel Elliptic Curves over Local Fields
Literatur
1.
M. Abdalla, M. Bellare, and P. Rogaway. The oracle Diffie-Hellman assumptions and an analysis of DHIES. In Topics in cryptology—CT-RSA 2001 (San Francisco, CA), volume 2020 of Lecture Notes in Comput. Sci., pages 143–158. Springer, Berlin, 2001.
2.
D. Abramovich. Formal finiteness and the torsion conjecture on elliptic curves. A footnote to a paper: “Rational torsion of prime order in elliptic curves over number fields” [Astérisque No. 228 (1995), 3, 81–100] by S. Kamienny and B. Mazur. Astérisque, (228):3, 5–17, 1995. Columbia University Number Theory Seminar (New York, 1992).
3.
L. 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.
4.
T. M. Apostol. Introduction to analytic number theory. Springer-Verlag, New York, 1976. Undergraduate Texts in Mathematics.
5.
T. M. Apostol. Modular functions and Dirichlet series in number theory, volume 41 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1990.
6.
N. Arthaud. On Birch and Swinnerton-Dyer’s conjecture for elliptic curves with complex multiplication. I. Compositio Math., 37(2):209–232, 1978.
7.
E. Artin. Galois theory. Dover Publications Inc., Mineola, NY, second edition, 1998. Edited and with a supplemental chapter by Arthur N. Milgram.
8.
M. F. Atiyah and I. G. Macdonald. Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, Mass.–London–Don Mills, Ont., 1969.
9.
M. F. Atiyah and C. T. C. Wall. Cohomology of groups. In Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), pages 94–115. Thompson, Washington, D.C., 1967.
10.
11.
A. Baker. Transcendental number theory. Cambridge Mathematical Library. Cambridge University Press, Cambridge, second edition, 1990.MATH
12.
13.
R. Balasubramanian and N. Koblitz. The improbability that an elliptic curve has subexponential discrete log problem under the Menezes-Okamoto-Vanstone algorithm. J. Cryptology, 11(2):141–145, 1998.MathSciNetMATHCrossRef
14.
A. F. Beardon. Iteration of Rational Functions, volume 132 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1991. Complex analytic dynamical systems.
15.
E. Bekyel. The density of elliptic curves having a global minimal Weierstrass equation. J. Number Theory, 109(1):41–58, 2004.MathSciNetMATHCrossRef
16.
D. Bernstein and T. Lange. Faster addition and doubling on elliptic curves. In Advances in cryptology—ASIACRYPT 2007, volume 4833 of Lecture Notes in Comput. Sci., pages 29–50. Springer, Berlin, 2007.
17.
B. J. Birch. Cyclotomic fields and Kummer extensions. In Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), pages 85–93. Thompson, Washington, D.C., 1967.
18.
B. J. Birch. How the number of points of an elliptic curve over a fixed prime field varies. J. London Math. Soc., 43:57–60, 1968.MathSciNetMATHCrossRef
19.
B. J. Birch and W. Kuyk, editors. Modular functions of one variable. IV. Springer-Verlag, Berlin, 1975. Lecture Notes in Mathematics, Vol. 476.
20.
B. J. Birch and H. P. F. Swinnerton-Dyer. Notes on elliptic curves. I. J. Reine Angew. Math., 212:7–25, 1963.
21.
B. J. Birch and H. P. F. Swinnerton-Dyer. Elliptic curves and modular functions. In Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pages 2–32. Lecture Notes in Math., Vol. 476. Springer, Berlin, 1975.
22.
I. F. Blake, G. Seroussi, and N. 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.
23.
D. Boneh and M. Franklin. Identity-based encryption from the Weil pairing. In Advances in Cryptology—CRYPTO 2001 (Santa Barbara, CA), volume 2139 of Lecture Notes in Comput. Sci., pages 213–229. Springer, Berlin, 2001.
24.
D. Boneh, B. Lynn, and H. Shacham. Short signatures from the Weil pairing. In Advances in cryptology—ASIACRYPT 2001 (Gold Coast), volume 2248 of Lecture Notes in Comput. Sci., pages 514–532. Springer, Berlin, 2001.
25.
A. I. Borevich and I. R. Shafarevich. Number theory. Translated from the Russian by Newcomb Greenleaf. Pure and Applied Mathematics, Vol. 20. Academic Press, New York, 1966.
26.
A. Bremner. On the equation Y 2 = 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 3–22. Kluwer Acad. Publ., Dordrecht, 1989.
27.
A. Bremner and J. W. S. Cassels. On the equation Y 2 = X(X 2 + p). Math. Comp., 42(165):257–264, 1984.MathSciNetMATH
28.
C. Breuil, B. Conrad, F. Diamond, and R. Taylor. On the modularity of elliptic curves over Q: wild 3-adic exercises. J. Amer. Math. Soc., 14(4):843–939 (electronic), 2001.
29.
F. Brezing and A. Weng. Elliptic curves suitable for pairing based cryptography. Des. Codes Cryptogr., 37(1):133–141, 2005.MathSciNetMATHCrossRef
30.
31.
W. D. Brownawell and D. W. Masser. Vanishing sums in function fields. Math. Proc. Cambridge Philos. Soc., 100(3):427–434, 1986.MathSciNetMATHCrossRef
32.
33.
J. P. Buhler, B. H. Gross, and D. B. Zagier. On the conjecture of Birch and Swinnerton-Dyer for an elliptic curve of rank 3. Math. Comp., 44(170):473–481, 1985.MathSciNetMATH
34.
E. R. Canfield, P. Erdős, and C. Pomerance. On a problem of Oppenheim concerning “factorisatio numerorum.” J. Number Theory, 17(1):1–28, 1983.MathSciNetMATHCrossRef
35.
H. Carayol. Sur les représentations galoisiennes modulo l attachées aux formes modulaires. Duke Math. J., 59(3):785–801, 1989.MathSciNetMATHCrossRef
36.
37.
J. W. S. Cassels. Arithmetic on curves of genus 1. III. The Tate-Šafarevič and Selmer groups. Proc. London Math. Soc. (3), 12:259–296, 1962.
38.
J. W. S. Cassels. Arithmetic on curves of genus 1. IV. Proof of the Hauptvermutung. J. Reine Angew. Math., 211:95–112, 1962.
39.
J. W. S. Cassels. Arithmetic on curves of genus 1. V. Two counterexamples. J. London Math. Soc., 38:244–248, 1963.
40.
J. W. S. Cassels. Arithmetic on curves of genus 1. VIII. On conjectures of Birch and Swinnerton-Dyer. J. Reine Angew. Math., 217:180–199, 1965.
41.
J. W. S. Cassels. Diophantine equations with special reference to elliptic curves. J. London Math. Soc., 41:193–291, 1966.MathSciNetMATHCrossRef
42.
J. W. S. Cassels. Global fields. In Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), pages 42–84. Thompson, Washington, D.C., 1967.
43.
J. W. S. Cassels. Lectures on elliptic curves, volume 24 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1991.
44.
T. Chinburg. An introduction to Arakelov intersection theory. In Arithmetic geometry (Storrs, Conn., 1984), pages 289–307. Springer, New York, 1986.
45.
D. V. Chudnovsky and G. V. Chudnovsky. Padé approximations and Diophantine geometry. Proc. Nat. Acad. Sci. U.S.A., 82(8):2212–2216, 1985.MathSciNetMATHCrossRef
46.
C. H. Clemens. A scrapbook of complex curve theory, volume 55 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, 2003.
47.
L. Clozel, M. Harris, and R. Taylor. Automorphy for some l-adic lifts of automorphic mod l representations. 2007. IHES Publ. Math., submitted.
48.
49.
50.
H. Cohen. A Course in Computational Algebraic Number Theory, volume 138 of Graduate Texts in Mathematics. Springer-Verlag, Berlin, 1993.
51.
H. Cohen, G. Frey, R. Avanzi, C. Doche, T. Lange, K. Nguyen, and F. Vercauteren, editors. Handbook of Elliptic and Hyperelliptic Curve Cryptography. Discrete Mathematics and Its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2006.
52.
D. A. Cox. The arithmetic-geometric mean of Gauss. Enseign. Math. (2), 30(3-4):275–330, 1984.MathSciNetMATH
53.
54.
55.
J. E. Cremona, M. Prickett, and S. Siksek. Height difference bounds for elliptic curves over number fields. J. Number Theory, 116(1):42–68, 2006.MathSciNetMATHCrossRef
56.
L. V. Danilov. The Diophantine equation x 3 − y 2 = k and a conjecture of M. Hall. Mat. Zametki, 32(3):273–275, 425, 1982. English translation: Math. Notes Acad. Sci. USSR 32 (1982), no. 3–4, 617–618 (1983).
57.
H. Davenport. On f 3 (t) − g 2 (t). Norske Vid. Selsk. Forh. (Trondheim), 38:86–87, 1965.MathSciNetMATH
58.
S. David. Minorations de formes linéaires de logarithmes elliptiques. Mém. Soc. Math. France (N.S.), (62):iv+143, 1995.
59.
B. M. M. de Weger. Algorithms for Diophantine equations, volume 65 of CWI Tract. Stichting Mathematisch Centrum Centrum voor Wiskunde en Informatica, Amsterdam, 1989.
60.
M. Deuring. Die Typen der Multiplikatorenringe elliptischer Funktionenkörper. Abh. Math. Sem. Hansischen Univ., 14:197–272, 1941.MathSciNetMATHCrossRef
61.
M. Deuring. Die Zetafunktion einer algebraischen Kurve vom Geschlechte Eins. Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. Math.-Phys.-Chem. Abt., 1953:85–94, 1953.MathSciNetMATH
62.
M. Deuring. Die Zetafunktion einer algebraischen Kurve vom Geschlechte Eins. II. Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. IIa., 1955:13–42, 1955.
63.
M. Deuring. Die Zetafunktion einer algebraischen Kurve vom Geschlechte Eins. III. Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. IIa., 1956:37–76, 1956.
64.
M. Deuring. Die Zetafunktion einer algebraischen Kurve vom Geschlechte Eins. IV. Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. IIa., 1957:55–80, 1957.
65.
66.
L. Dirichlet. Über den biquadratischen Charakter der Zahl “Zwei.” J. Reine Angew. Math., 57:187–188, 1860.MathSciNetCrossRef
67.
Z. Djabri, E. F. Schaefer, and N. P. Smart. Computing the p-Selmer group of an elliptic curve. Trans. Amer. Math. Soc., 352(12):5583–5597, 2000.MathSciNetMATHCrossRef
68.
D. S. Dummit and R. M. Foote. Abstract algebra. John Wiley & Sons Inc., Hoboken, NJ, third edition, 2004.MATH
69.
R. Dupont, A. Enge, and F. Morain. Building curves with arbitrary small MOV degree over finite prime fields. J. Cryptology, 18(2):79–89, 2005.MathSciNetMATHCrossRef
70.
71.
H. M. Edwards. A normal form for elliptic curves. Bull. Amer. Math. Soc. (N.S.), 44(3):393–422 (electronic), 2007.
72.
M. Eichler. Quaternäre quadratische Formen und die Riemannsche Vermutung für die Kongruenzzetafunktion. Arch. Math., 5:355–366, 1954.MathSciNetMATHCrossRef
73.
D. Eisenbud. Commutative algebra, volume 150 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995. With a view toward algebraic geometry.
74.
T. ElGamal. A public key cryptosystem and a signature scheme based on discrete logarithms. IEEE Trans. Inform. Theory, 31(4):469–472, 1985.MathSciNetMATHCrossRef
75.
76.
N. Elkies. \(\mathbb{Z}^{28}\) in \(E(\mathbb{Q})\). Number Theory Listserver, May 2006.
77.
N. D. Elkies. The existence of infinitely many supersingular primes for every elliptic curve over \(\mathbb{Q}\). Invent. Math., 89(3):561–567, 1987.MathSciNetMATHCrossRef
78.
N. D. Elkies. Distribution of supersingular primes. Astérisque, (198-200):127–132 (1992), 1991. Journées Arithmétiques, 1989 (Luminy, 1989).
79.
N. D. Elkies. Elliptic and modular curves over finite fields and related computational issues. In Computational perspectives on number theory (Chicago, IL, 1995), volume 7 of AMS/IP Stud. Adv. Math., pages 21–76. Amer. Math. Soc., Providence, RI, 1998.
80.
81.
J.-H. Evertse and J. H. Silverman. Uniform bounds for the number of solutions to Y n  = f(X). Math. Proc. Cambridge Philos. Soc., 100(2):237–248, 1986.MathSciNetMATHCrossRef
82.
G. Faltings. Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math., 73(3):349–366, 1983.MathSciNetCrossRef
83.
84.
G. Faltings. Finiteness theorems for abelian varieties over number fields. In Arithmetic geometry (Storrs, Conn., 1984), pages 9–27. Springer, New York, 1986. Translated from the German original [Invent. Math. 73 (1983), no. 3, 349–366; ibid. 75 (1984), no. 2, 381; MR 85g:11026ab] by Edward Shipz.
85.
S. Fermigier. Une courbe elliptique définie sur Q de rang ≥ 22. Acta Arith., 82(4):359–363, 1997.MathSciNet
86.
E. V. Flynn and C. Grattoni. Descent via isogeny on elliptic curves with large rational torsion subgroups. J. Symbolic Comput., 43(4):293–303, 2008.MathSciNetMATHCrossRef
87.
D. Freeman. Constructing pairing-friendly elliptic curves with embedding degree 10. In Algorithmic number theory, volume 4076 of Lecture Notes in Comput. Sci., pages 452–465. Springer, Berlin, 2006.
88.
G. Frey. Links between stable elliptic curves and certain Diophantine equations. Ann. Univ. Sarav. Ser. Math., 1(1):iv+40, 1986.
89.
G. Frey. Elliptic curves and solutions of A − B = C. In Séminaire de Théorie des Nombres, Paris 1985–86, volume 71 of Progr. Math., pages 39–51. Birkhäuser Boston, Boston, MA, 1987.
90.
G. Frey. Links between solutions of A − B = C and elliptic curves. In Number theory (Ulm, 1987), volume 1380 of Lecture Notes in Math., pages 31–62. Springer, New York, 1989.
91.
G. Frey and H.-G. Rück. A remark concerning m-divisibility and the discrete logarithm problem in the divisor class group of curves. Math. Comp., 62:865–874, 1994.MathSciNetMATH
92.
A. Fröhlich. Local fields. In Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), pages 1–41. Thompson, Washington, D.C., 1967.
93.
A. Fröhlich. Formal groups. Lecture Notes in Mathematics, No. 74. Springer-Verlag, Berlin, 1968.
94.
95.
W. 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.
96.
J. Gebel, A. Pethő, and H. G. Zimmer. Computing integral points on elliptic curves. Acta Arith., 68(2):171–192, 1994.MathSciNetMATH
97.
S. Goldwasser and J. Kilian. Almost all primes can be quickly certified. In STOC ’86: Proceedings of the eighteenth annual ACM symposium on Theory of computing, pages 316–329, New York, 1986. ACM.
98.
99.
P. Griffiths and J. Harris. Principles of algebraic geometry. Wiley Classics Library. John Wiley & Sons Inc., New York, 1994. Reprint of the 1978 original.
100.
B. Gross, W. Kohnen, and D. Zagier. Heegner points and derivatives of L-series. II. Math. Ann., 278(1-4):497–562, 1987.
101.
B. Gross and D. Zagier. Points de Heegner et dérivées de fonctions L. C. R. Acad. Sci. Paris Sér. I Math., 297(2):85–87, 1983.
102.
103.
104.
105.
K. Gruenberg. Profinite groups. In Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), pages 116–127. Thompson, Washington, D.C., 1967.
106.
M. Hall, Jr. The Diophantine equation x 3 − y 2 = k. In Computers in number theory (Proc. Sci. Res. Council Atlas Sympos. No. 2, Oxford, 1969), pages 173–198. Academic Press, London, 1971.
107.
D. Hankerson, A. Menezes, and S. Vanstone. Guide to elliptic curve cryptography. Springer Professional Computing. Springer-Verlag, New York, 2004.MATH
108.
G. H. Hardy and E. M. Wright. An introduction to the theory of numbers. The Clarendon Press Oxford University Press, New York, fifth edition, 1979.MATH
109.
J. Harris. Algebraic geometry, volume 133 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1992. A first course.
110.
M. Harris, N. Shepherd-Barron, and R. Taylor. A family of Calabi-Yau varieties and potential automorphy. Ann. of Math. (2). to appear.
111.
R. Hartshorne. Algebraic geometry. Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, No. 52.
112.
M. Hazewinkel. Formal groups and applications, volume 78 of Pure and Applied Mathematics. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1978.
113.
M. Hindry and J. H. Silverman. The canonical height and integral points on elliptic curves. Invent. Math., 93(2):419–450, 1988.MathSciNetMATHCrossRef
114.
M. Hindry and J. H. Silverman. Diophantine geometry, volume 201 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2000. An introduction.
115.
116.
J. Hoffstein, J. Pipher, and J. H. Silverman. An introduction to mathematical cryptography. Undergraduate Texts in Mathematics. Springer-Verlag, New York, 2008.MATH
117.
A. Hurwitz. Über ternäre diophantische Gleichungen dritten Grades. Vierteljahrschrift d. Naturf. Ges. Zürich, 62:207–229, 1917.MATH
118.
D. Husemöller. 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.
119.
J.-I. Igusa. Class number of a definite quaternion with prime discriminant. Proc. Nat. Acad. Sci. U.S.A., 44:312–314, 1958.MathSciNetMATHCrossRef
120.
A. Joux. A one round protocol for tripartite Diffie-Hellman. In Algorithmic number theory (Leiden, 2000), volume 1838 of Lecture Notes in Comput. Sci., pages 385–393. Springer, Berlin, 2000.
121.
S. Kamienny. Torsion points on elliptic curves and q-coefficients of modular forms. Invent. Math., 109(2):221–229, 1992.MathSciNetMATHCrossRef
122.
S. Kamienny and B. Mazur. Rational torsion of prime order in elliptic curves over number fields. Astérisque, (228):3, 81–100, 1995. With an appendix by A. Granville, Columbia University Number Theory Seminar (New York, 1992).
123.
N. M. Katz. An overview of Deligne’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 275–305. Amer. Math. Soc., Providence, R.I., 1976.
124.
N. M. Katz and B. Mazur. Arithmetic moduli of elliptic curves, volume 108 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1985.
125.
M. A. Kenku. On the number of Q-isomorphism classes of elliptic curves in each Q-isogeny class. J. Number Theory, 15(2):199–202, 1982.MathSciNetMATHCrossRef
126.
J.-H. Kim, R. Montenegro, Y. Peres, and P. 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 402–415. Springer, Berlin, 2008.MATH
127.
A. W. Knapp. Elliptic curves, volume 40 of Mathematical Notes. Princeton University Press, Princeton, NJ, 1992.
128.
129.
N. Koblitz. Introduction to elliptic curves and modular forms, volume 97 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1993.
130.
V. 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):522–540, 670–671, 1988.
131.
S. V. Kotov and L. A. Trelina. S-ganze Punkte auf elliptischen Kurven. J. Reine Angew. Math., 306:28–41, 1979.MathSciNetMATH
132.
133.
E. Kunz. Introduction to plane algebraic curves. Birkhäuser Boston Inc., Boston, MA, 2005. Translated from the 1991 German edition by Richard G. Belshoff.
134.
M. Lal, M. F. Jones, and W. J. Blundon. Numerical solutions of the Diophantine equation y 3 − x 2 = k. Math. Comp., 20:322–325, 1966.
135.
S. Lang. Elliptic curves: Diophantine analysis, volume 231 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1978.
136.
S. Lang. Introduction to algebraic and abelian functions, volume 89 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1982.
137.
S. Lang. Complex multiplication, volume 255 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York, 1983.
138.
S. Lang. Conjectured Diophantine estimates on elliptic curves. In Arithmetic and geometry, Vol. I, volume 35 of Progr. Math., pages 155–171. Birkhäuser Boston, Boston, MA, 1983.
139.
140.
S. Lang. Elliptic functions, volume 112 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1987. With an appendix by J. Tate.
141.
S. Lang. Number theory III, volume 60 of Encyclopedia of Mathematical Sciences. Springer-Verlag, Berlin, 1991.
142.
S. Lang. Algebraic number theory, volume 110 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1994.
143.
S. Lang. Algebra, volume 211 of Graduate Texts in Mathematics. Springer-Verlag, New York, third edition, 2002.
144.
145.
S. Lang and H. 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.
146.
M. Laska. An algorithm for finding a minimal Weierstrass equation for an elliptic curve. Math. Comp., 38(157):257–260, 1982.MathSciNetMATHCrossRef
147.
M. Laska. Elliptic curves over number fields with prescribed reduction type. Aspects of Mathematics, E4. Friedr. Vieweg & Sohn, Braunschweig, 1983.
148.
D. J. Lewis and K. Mahler. On the representation of integers by binary forms. Acta Arith., 6:333–363, 1960/1961.
149.
150.
C.-E. Lind. Untersuchungen über die rationalen Punkte der ebenen kubischen Kurven vom Geschlecht Eins. Thesis, University of Uppsala,, 1940:97, 1940.MathSciNetMATH
151.
J. Liouville. Sur des classes très-étendues de quantités dont la irrationalles algébriques. C. R. Acad. Paris, 18:883–885 and 910–911, 1844.
152.
E. Lutz. Sur l’equation y 2 = x 3 − ax − b dans les corps p-adic. J. Reine Angew. Math., 177:237–247, 1937.MathSciNet
153.
154.
J. I. Manin. The Hasse-Witt matrix of an algebraic curve. Izv. Akad. Nauk SSSR Ser. Mat., 25:153–172, 1961.MathSciNet
155.
J. I. Manin. The p-torsion of elliptic curves is uniformly bounded. Izv. Akad. Nauk SSSR Ser. Mat., 33:459–465, 1969.MathSciNet
156.
J. I. Manin. Cyclotomic fields and modular curves. Uspehi Mat. Nauk, 26(6(162)):7–71, 1971. English translation: Russian Math. Surveys 26 (1971), no. 6, 7–78.
157.
158.
R. C. Mason. Norm form equations. I. J. Number Theory, 22(2):190–207, 1986.
159.
D. Masser. Elliptic functions and transcendence. Springer-Verlag, Berlin, 1975. Lecture Notes in Mathematics, Vol. 437.
160.
D. Masser and G. Wüstholz. Isogeny estimates for abelian varieties, and finiteness theorems. Ann. of Math. (2), 137(3):459–472, 1993.MathSciNetMATHCrossRef
161.
D. W. Masser. Specializations of finitely generated subgroups of abelian varieties. Trans. Amer. Math. Soc., 311(1):413–424, 1989.MathSciNetMATHCrossRef
162.
D. W. Masser and G. Wüstholz. Fields of large transcendence degree generated by values of elliptic functions. Invent. Math., 72(3):407–464, 1983.MathSciNetMATHCrossRef
163.
164.
H. Matsumura. Commutative algebra, volume 56 of Mathematics Lecture Note Series. Benjamin/Cummings Publishing Co., Inc., Reading, Mass., second edition, 1980.
165.
B. Mazur. Modular curves and the Eisenstein ideal. Inst. Hautes Études Sci. Publ. Math., (47):33–186 (1978), 1977.
166.
B. Mazur. Rational isogenies of prime degree (with an appendix by D. Goldfeld). Invent. Math., 44(2):129–162, 1978.
167.
H. McKean and V. Moll. Elliptic curves. Cambridge University Press, Cambridge, 1997. Function theory, geometry, arithmetic.
168.
A. J. Menezes, T. Okamoto, and S. A. Vanstone. Reducing elliptic curve logarithms to logarithms in a finite field. IEEE Trans. Inform. Theory, 39(5):1639–1646, 1993.MathSciNetMATHCrossRef
169.
A. J. Menezes, P. C. van Oorschot, and S. A. Vanstone. Handbook of Applied Cryptography. CRC Press Series on Discrete Mathematics and Its Applications. CRC Press, Boca Raton, FL, 1997.MATH
170.
L. Merel. Bornes pour la torsion des courbes elliptiques sur les corps de nombres. Invent. Math., 124(1-3):437–449, 1996.MathSciNetCrossRef
171.
J.-F. Mestre. Construction d’une courbe elliptique de rang ≥ 12. C. R. Acad. Sci. Paris Sér. I Math., 295(12):643–644, 1982.MathSciNetMATH
172.
J.-F. Mestre. Courbes elliptiques et formules explicites. In Seminar on number theory, Paris 1981–82 (Paris, 1981/1982), volume 38 of Progr. Math., pages 179–187. Birkhäuser Boston, Boston, MA, 1983.
173.
M. Mignotte. Quelques remarques sur l’approximation rationnelle des nombres algébriques. J. Reine Angew. Math., 268/269:341–347, 1974. Collection of articles dedicated to Helmut Hasse on his seventy-fifth birthday, II.
174.
S. D. Miller and R. Venkatesan. Spectral analysis of Pollard rho collisions. In Algorithmic number theory, volume 4076 of Lecture Notes in Comput. Sci., pages 573–581. Springer, Berlin, 2006.
175.
S. D. Miller and R. Venkatesan. Non-degeneracy of Pollard rho collisions, 2008. arXiv:0808.0469.
176.
V. S. Miller. Use of elliptic curves in cryptography. In Advances in Cryptology—CRYPTO ’85 (Santa Barbara, Calif., 1985), volume 218 of Lecture Notes in Comput. Sci., pages 417–426. Springer, Berlin, 1986.
177.
J. S. Milne. Arithmetic duality theorems, volume 1 of Perspectives in Mathematics. Academic Press Inc., Boston, MA, 1986.
178.
J. S. Milne. Elliptic curves. BookSurge Publishers, Charleston, SC, 2006.MATH
179.
J. Milnor. On Lattès maps. ArXiv:math.DS/0402147, Stony Brook IMS Preprint #2004/01.
180.
R. Miranda. Algebraic curves and Riemann surfaces, volume 5 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1995.MATH
181.
A. Miyaji, M. Nakabayashi, and S. Takano. Characterization of elliptic curve traces under FR-reduction. In Information security and cryptology—ICISC 2000 (Seoul), volume 2015 of Lecture Notes in Comput. Sci., pages 90–108. Springer, Berlin, 2001.
182.
P. Monsky. Three constructions of rational points on Y 2 = X 3 ± NX. Math. Z., 209(3):445–462, 1992.
183.
F. Morain. Building cyclic elliptic curves modulo large primes. In Advances in cryptology—EUROCRYPT ’91 (Brighton, 1991), volume 547 of Lecture Notes in Comput. Sci., pages 328–336. Springer, Berlin, 1991.
184.
185.
L. J. Mordell. Diophantine equations. Pure and Applied Mathematics, Vol. 30. Academic Press, London, 1969.
186.
D. Mumford. Abelian varieties. Tata Institute of Fundamental Research Studies in Mathematics, No. 5. Published for the Tata Institute of Fundamental Research, Bombay, 1970.
187.
D. Mumford, J. Fogarty, and F. Kirwan. Geometric invariant theory, volume 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)]. Springer-Verlag, Berlin, third edition, 1994.
188.
K.-I. Nagao. Construction of high-rank elliptic curves. Kobe J. Math., 11(2):211–219, 1994.MathSciNetMATH
189.
K.-I. Nagao. \(\mathbb{Q}(T)\)-rank of elliptic curves and certain limit coming from the local points. Manuscripta Math., 92(1):13–32, 1997. With an appendix by Nobuhiko Ishida, Tsuneo Ishikawa and the author.
190.
T. Nagell. Solution de quelque problèmes dans la théorie arithmétique des cubiques planes du premier genre. Wid. Akad. Skrifter Oslo I, 1935. Nr. 1.
191.
192.
A. Néron. Problèmes arithmétiques et géométriques rattachés à la notion de rang d’une courbe algébrique dans un corps. Bull. Soc. Math. France, 80:101–166, 1952.MathSciNetMATH
193.
A. Néron. Modèles minimaux des variétés abéliennes sur les corps locaux et globaux. Inst. Hautes Études Sci. Publ.Math. No., 21:128, 1964.MATH
194.
195.
O. Neumann. Elliptische Kurven mit vorgeschriebenem Reduktionsverhalten. I. Math. Nachr., 49:107–123, 1971.
196.
J. Oesterlé. Nouvelles approches du “théorème” de Fermat. Astérisque, (161-162):Exp. No. 694, 4, 165–186 (1989), 1988. Séminaire Bourbaki, Vol. 1987/88.
197.
A. Ogg. Modular forms and Dirichlet series. W. A. Benjamin, Inc., New York-Amsterdam, 1969.MATH
198.
199.
200.
201.
202.
203.
A. N. Paršin. Algebraic curves over function fields. I. Izv. Akad. Nauk SSSR Ser. Mat., 32:1191–1219, 1968.
204.
R. G. E. Pinch. Elliptic curves with good reduction away from 2. Math. Proc. Cambridge Philos. Soc., 96(1):25–38, 1984.MathSciNetMATHCrossRef
205.
S. C. Pohlig and M. 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):106–110, 1978.MathSciNetMATHCrossRef
206.
J. M. Pollard. Monte Carlo methods for index computation \((\mathop{\mathrm{mod}}\nolimits \ p)\). Math. Comp., 32(143):918–924, 1978.MathSciNetMATH
207.
H. Reichardt. Einige im Kleinen überall lösbare, im Grossen unlösbare diophantische Gleichungen. J. Reine Angew. Math., 184:12–18, 1942.MathSciNetMATH
208.
K. A. Ribet. On modular representations of \(\mathop{\mathrm{Gal}}\nolimits (\overline{\mathbf{Q}}/\mathbf{Q})\) arising from modular forms. Invent. Math., 100(2):431–476, 1990.MathSciNetMATHCrossRef
209.
R. L. Rivest, A. Shamir, and L. Adleman. A method for obtaining digital signatures and public-key cryptosystems. Comm. ACM, 21(2):120–126, 1978.MathSciNetMATHCrossRef
210.
A. Robert. Elliptic curves. Springer-Verlag, Berlin, 1973. Notes from postgraduate lectures given in Lausanne 1971/72, Lecture Notes in Mathematics, Vol. 326.
211.
212.
P. Roquette. Analytic theory of elliptic functions over local fields. Hamburger Mathematische Einzelschriften (N.F.), Heft 1. Vandenhoeck & Ruprecht, Göttingen, 1970.
213.
214.
K. Rubin. Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer. Invent. Math., 64(3):455–470, 1981.MathSciNetMATHCrossRef
215.
K. Rubin. Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication. Invent. Math., 89(3):527–559, 1987.MathSciNetMATHCrossRef
216.
K. Rubin. The “main conjectures” of Iwasawa theory for imaginary quadratic fields. Invent. Math., 103(1):25–68, 1991.MathSciNetMATHCrossRef
217.
218.
T. Satoh and K. Araki. Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves. Comment. Math. Univ. St. Paul., 47(1):81–92, 1998.MathSciNetMATH
219.
E. F. Schaefer and M. Stoll. How to do a p-descent on an elliptic curve. Trans. Amer. Math. Soc., 356(3):1209–1231 (electronic), 2004.
220.
S. H. Schanuel. Heights in number fields. Bull. Soc. Math. France, 107(4):433–449, 1979.MathSciNetMATH
221.
W. M. Schmidt. Diophantine approximation, volume 785 of Lecture Notes in Mathematics. Springer, Berlin, 1980.
222.
S. Schmitt and H. G. Zimmer. Elliptic curves, volume 31 of de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, 2003. A computational approach, With an appendix by Attila Pethő.
223.
R. Schoof. Elliptic curves over finite fields and the computation of square roots mod p. Math. Comp., 44(170):483–494, 1985.
224.
R. Schoof. Counting points on elliptic curves over finite fields. J. Théor. Nombres Bordeaux, 7(1):219–254, 1995. Les Dix-huitièmes Journées Arithmétiques (Bordeaux, 1993).
225.
E. S. Selmer. The Diophantine equation ax 3 + by 3 + cz 3 = 0. Acta Math., 85:203–362 (1 plate), 1951.
226.
E. S. Selmer. A conjecture concerning rational points on cubic curves. Math. Scand., 2:49–54, 1954.MathSciNetMATH
227.
E. S. Selmer. The diophantine equation ax 3 + by 3 + cz 3 = 0. Completion of the tables. Acta Math., 92:191–197, 1954.
228.
I. A. Semaev. Evaluation of discrete logarithms in a group of p-torsion points of an elliptic curve in characteristic p. Math. Comp., 67(221):353–356, 1998.
229.
J.-P. Serre. Géométrie algébrique et géométrie analytique. Ann. Inst. Fourier, Grenoble, 6:1–42, 1955–1956.
230.
J.-P. Serre. Complex multiplication. In Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), pages 292–296. Thompson, Washington, D.C., 1967.
231.
J.-P. Serre. Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent. Math., 15(4):259–331, 1972.
232.
J.-P. Serre. A course in arithmetic. Springer-Verlag, New York, 1973. Translated from the French, Graduate Texts in Mathematics, No. 7.
233.
J.-P. Serre. Local fields, volume 67 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1979. Translated from the French by Marvin Jay Greenberg.
234.
J.-P. Serre. Quelques applications du théorème de densité de Chebotarev. Inst. Hautes Études Sci. Publ. Math., (54):323–401, 1981.MATHCrossRef
235.
J.-P. Serre. Sur les représentations modulaires de degré 2 de \(\mathop{\mathrm{Gal}}\nolimits (\overline{\mathbf{Q}}/\mathbf{Q})\). Duke Math. J., 54(1):179–230, 1987.MathSciNetCrossRef
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.
237.
J.-P. Serre. Abelian l-adic representations and elliptic curves, volume 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.
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.
239.
240.
241.
242.
I. R. Shafarevich. Algebraic number fields. In Proc. Int. Cong. (Stockholm 1962), pages 25–39. American Mathematical Society, Providence, R.I., 1963. Amer. Math. Soc. Transl., Series 2, Vol. 31.
243.
I. 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.
244.
I. R. Shafarevich and J. Tate. The rank of elliptic curves. In Amer. Math. Soc. Transl., volume 8, pages 917–920. Amer. Math. Soc., 1967.
245.
A. Shamir. Identity-based cryptosystems and signature schemes. In Advances in Cryptology (Santa Barbara, Calif., 1984), volume 196 of Lecture Notes in Comput. Sci., pages 47–53. Springer, Berlin, 1985.
246.
247.
G. Shimura. On elliptic curves with complex multiplication as factors of the Jacobians of modular function fields. Nagoya Math. J., 43:199–208, 1971.MathSciNetMATH
248.
G. Shimura. On the zeta-function of an abelian variety with complex multiplication. Ann. of Math. (2), 94:504–533, 1971.MathSciNetMATHCrossRef
249.
G. Shimura. Introduction to the arithmetic theory of automorphic functions, volume 11 of Publications of the Mathematical Society of Japan. Princeton University Press, Princeton, NJ, 1994. Reprint of the 1971 original, Kano Memorial Lectures, 1.
250.
G. Shimura and Y. Taniyama. Complex multiplication of abelian varieties and its applications to number theory, volume 6 of Publications of the Mathematical Society of Japan. The Mathematical Society of Japan, Tokyo, 1961.
251.
T. Shioda. An explicit algorithm for computing the Picard number of certain algebraic surfaces. Amer. J. Math., 108(2):415–432, 1986.MathSciNetMATHCrossRef
252.
R. Shipsey. Elliptic divisibility sequences. PhD thesis, Goldsmith’s College (University of London), 2000.
253.
V. Shoup. Lower bounds for discrete logarithms and related problems. In Advances in cryptology—EUROCRYPT ’97 (Konstanz), volume 1233 of Lecture Notes in Comput. Sci., pages 256–266. Springer, Berlin, 1997. updated version at www.​shoup.​net/​papers/​dlbounds1.​pdf.
254.
255.
J. H. Silverman. The Néron–Tate height on elliptic curves. PhD thesis, Harvard University, 1981.
256.
J. H. Silverman. Heights and the specialization map for families of abelian varieties. J. Reine Angew. Math., 342:197–211, 1983.MathSciNetMATH
257.
258.
259.
J. H. Silverman. Weierstrass equations and the minimal discriminant of an elliptic curve. Mathematika, 31(2):245–251 (1985), 1984.MathSciNetMATHCrossRef
260.
J. H. Silverman. Divisibility of the specialization map for families of elliptic curves. Amer. J. Math., 107(3):555–565, 1985.MathSciNetMATHCrossRef
261.
J. H. Silverman. Arithmetic distance functions and height functions in Diophantine geometry. Math. Ann., 279(2):193–216, 1987.MathSciNetMATHCrossRef
262.
J. H. Silverman. A quantitative version of Siegel’s theorem: integral points on elliptic curves and Catalan curves. J. Reine Angew. Math., 378:60–100, 1987.MathSciNetMATH
263.
264.
265.
J. H. Silverman. The difference between the Weil height and the canonical height on elliptic curves. Math. Comp., 55(192):723–743, 1990.MathSciNetMATHCrossRef
266.
J. H. Silverman. Advanced topics in the arithmetic of elliptic curves, volume 151 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1994.MATH
267.
J. H. Silverman. The arithmetic of dynamical systems, volume 241 of Graduate Texts in Mathematics. Springer, New York, 2007.MATH
268.
269.
270.
K. Stange. The Tate pairing via elliptic nets. In Pairing Based Cryptography, Lecture Notes in Comput. Sci. Springer, 2007.
271.
K. Stange. Elliptic Nets and Elliptic Curves. PhD thesis, Brown University, 2008.
272.
K. Stange. Elliptic nets and elliptic curves, 2008. arXiv:0710.1316v2.
273.
H. M. Stark. Effective estimates of solutions of some Diophantine equations. Acta Arith., 24:251–259, 1973.MathSciNetMATH
274.
275.
276.
N. M. Stephens. The Diophantine equation X 3 + Y 3 = DZ 3 and the conjectures of Birch and Swinnerton-Dyer. J. Reine Angew. Math., 231:121–162, 1968.MathSciNetMATH
277.
D. R. Stinson. Cryptography: Theory and Practice. CRC Press Series on Discrete Mathematics and Its Applications. Chapman & Hall/CRC, Boca Raton, FL, 2002.MATH
278.
279.
R. J. Stroeker and N. Tzanakis. Solving elliptic Diophantine equations by estimating linear forms in elliptic logarithms. Acta Arith., 67(2):177–196, 1994.MathSciNetMATH
280.
J. Tate. Letter to J.-P. Serre, 1968.
281.
J. Tate. Duality theorems in Galois cohomology over number fields. In Proc. Internat. Congr. Mathematicians (Stockholm, 1962), pages 288–295. Inst. Mittag-Leffler, Djursholm, 1963.
282.
283.
J. 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 33–52. Lecture Notes in Math., Vol. 476. Springer, Berlin, 1975.
284.
285.
J. Tate. A review of non-Archimedean elliptic functions. In Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993), Ser. Number Theory, I, pages 162–184. Int. Press, Cambridge, MA, 1995.
286.
J. Tate. WC-groups over \(\mathfrak{p}\)-adic fields. In Séminaire Bourbaki, Vol. 4 (1957/58), pages Exp. No. 156, 265–277. Soc. Math. France, Paris, 1995.
287.
J. T. Tate. Algebraic cycles and poles of zeta functions. In Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), pages 93–110. Harper & Row, New York, 1965.
288.
J. T. Tate. Global class field theory. In Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), pages 162–203. Thompson, Washington, D.C., 1967.
289.
290.
R. Taylor. Automorphy for some l-adic lifts of automorphic mod l representations. II. Inst. Hautes Études Sci. Publ. Math. submitted.
291.
R. Taylor and A. Wiles. Ring-theoretic properties of certain Hecke algebras. Ann. of Math. (2), 141(3):553–572, 1995.MathSciNetMATHCrossRef
292.
293.
E. Teske. Speeding up Pollard’s rho method for computing discrete logarithms. In Algorithmic Number Theory (Portland, OR, 1998), volume 1423 of Lecture Notes in Comput. Sci., pages 541–554. Springer, Berlin, 1998.MATH
294.
E. Teske. Square-root algorithms for the discrete logarithm problem (a survey). In Public-Key Cryptography and Computational Number Theory (Warsaw, 2000), pages 283–301. de Gruyter, Berlin, 2001.
295.
296.
B. L. van 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.
297.
J. Vélu. Isogénies entre courbes elliptiques. C. R. Acad. Sci. Paris Sér. A-B, 273:A238–A241, 1971.
298.
P. Vojta. A higher-dimensional Mordell conjecture. In Arithmetic geometry (Storrs, Conn., 1984), pages 341–353. Springer, New York, 1986.
299.
300.
301.
P. M. Voutier. An upper bound for the size of integral solutions to Y m  = f(X). J. Number Theory, 53(2):247–271, 1995.MathSciNetMATHCrossRef
302.
R. J. Walker. Algebraic curves. Springer-Verlag, New York, 1978. Reprint of the 1950 edition.
303.
304.
L. C. Washington. Elliptic curves. Discrete Mathematics and Its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, second edition, 2008. Number theory and cryptography.
305.
306.
A. Weil. Jacobi sums as “Grössencharaktere.” Trans. Amer. Math. Soc., 73:487–495, 1952.MathSciNetMATH
307.
308.
309.
A. Weil. Dirichlet Series and Automorphic Forms, volume 189 of Lecture Notes in Mathematics. Springer-Verlag, 1971.
310.
E. T. Whittaker and G. N. Watson. A course of modern analysis. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1996. Reprint of the fourth (1927) edition.
311.
312.
A. Wiles. The Birch and Swinnerton-Dyer conjecture. In The millennium prize problems, pages 31–41. Clay Math. Inst., Cambridge, MA, 2006.
313.
G. Wüstholz. Recent progress in transcendence theory. In Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983), volume 1068 of Lecture Notes in Math., pages 280–296. Springer, Berlin, 1984.MATH
314.
315.
316.
317.
318.
P. Deligne. La conjecture de Weil. I. Inst. Hautes Études Sci. Publ. Math., (43):273–307, 1977.
Metadaten
Titel
Elliptic Curves over
verfasst von
Joseph H. Silverman
Copyright-Jahr
2009
Verlag
Springer New York
Buch
The Arithmetic of Elliptic Curves

Print ISBN: 978-0-387-09493-9

Electronic ISBN: 978-0-387-09494-6

Copyright-Jahr: 2009

DOI
https://doi.org/10.1007/978-0-387-09494-6_6

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