Elliptic Curves

Books on Elliptic Curves

  1. A. Weil, Number theory. An approach through history. From Hammurapi to Legendre, 1984 (History)
  2. J. Silverman, J. Tate, Rational points on elliptic curves. Undergraduate Texts in Mathematics, 1992, (elementary introduction)
  3. A. van der Poorten, Notes on Fermat's Last Theorem; Wiley 1996, 222 pp. (a little background to Wiles' proof of FLT).
  4. Yves Hellegouarch, Invitation aux mathematiques de Fermat - Wiles; Paris 1997, 397b pp, (elementary introduction to Wiles' proof).
  5. N. Koblitz, Introduction to elliptic curves and modular forms. Graduate Texts in Mathematics 97, 1993 (good introduction)
  6. N. Koblitz, Algebraic Aspects of Cryptography, Springer 1998 (contains an elementary introduction to hyperelliptic curves and their applications in cryptography)
  7. R. Pinch, Computational Number Theory (ECM)
  8. H. McKean, V. Moll, Elliptic Curves. Function Theory, Geometry, Arithmetic (pretty presentation, partially without proofs)
  9. T. Ono, Variations on a Theme of Euler : Quadratic Forms, Elliptic Curves, and Hopf Maps; 1994 (nice and expensive)
  10. D. Husemoeller, Elliptic curves; Graduate Texts in Mathematics 111, 1987 (out of print - a pity).
  11. J. Silverman, The arithmetic of elliptic curves. Graduate Texts in Mathematics 106, 1986 (THE, introduction to elliptic curves)
  12. J.W.S. Cassels, Lectures on elliptic curves. London Mathematical Society Student Texts 24, 1991 (seemed a bit strange at first, but now I like it. Many misprints)
  13. A. Knapp, Elliptic curves Mathematical Notes 40, Princeton Univ. Press 1992, $ 40 (excellent introduction to elliptic curves and modular forms)
  14. G. Cornell (ed.) et al, Modular forms and Fermat's last theorem. Springer 1997 (Wiles' Proof)
  15. J.E. Cremona, Algorithms for Modular Elliptic Curves (Tables and background)
  16. A. Robert, Elliptic curves, Lecture Notes in Math. 326, Springer-Verlag 1973 (out of print; this one is much better than it seems at first sight).
  17. Elliptic Functions and Elliptic Integrals by Viktor Prasolov and Yuri Solovyev (nice introduction to elliptic curves, functions and integrals).
  18. Fermat's Dream by Kazuya Kato has just appeared and introduces to number theory and elliptic curves.
  19. Arithmétique des courbes elliptiques et théorie d'Iwasawa by B. Perrin-Riou (studies elliptic curves with CM using Iwasawa theory).

Surveys

  1. J. Silverman wrote "A survey of the arithmetic theory of elliptic curves" in the Boston Proceedings mentioned above, as well as "Recent (and not so recent) developments in the arithmetic theory of elliptic curves" in Nieuw Arch. Wiskd. 7 (1989), 53-70.
  2. Henri Cohen's Elliptic curves", in `From number theory to physics' Springer-Verlag, 212-237 (1992);
  3. D. Zagier's "Elliptische Kurven: Fortschritte und Anwendungen" can be found in the Jahresbericht der DMV 92 (1990), 58-76.
  4. Roel Stroeker's "Aspects of elliptic curves. An introduction" is from Nieuw Arch. Wiskunde, III. Ser. 26 (1978), 371-412.
  5. H.G. Zimmer wrote "Zur Arithmetik der elliptischen Kurven", a survey of about 100pp covering the most important results and conjectures in Ber. Math.-Stat. Sekt. Forschungsges. Joanneum 271, (1986).
  6. L. Washington's "Number fields and elliptic curves" can be found in `Number theory and applications', Banff/Can. 1988, 245-278 (1989).
  7. The booklet `Zur Geschichte der Bestimmung rationaler Punkte auf elliptischen Kurven: Das Problem von Beha-Eddin `Amuli', Ber. Sitz. Joachim Jungius-Ges. Wiss., Hamburg 1(1982/83), 52 S. (1984) by Ch. Scriba deals with the history of a diophantine problem and its solution using the theory of elliptic curves.
  8. In the booklet `Lebendige Zahlen' by W. Borho et al. there is an article "Algebraische Kurven und diophantische Gleichungen" by Hanspeter Kraft.
  9. N. Schappacher and R. Schoof discussed Beppo Levi's contributions to the theory of elliptic curves in this article (dvi.gz).
  10. Not exactly bedtime lecture: the survey "Diophantine equations with special reference to elliptic curves" by J.W.S. Cassels in J. Lond. Math. Soc. 41, 193-291 (1966).
  11. Somewhat hard to find are the seminar reports from 1982 containing the articles "Courbes elliptiques" by R. Lardon, "La theorie de Kummer" by A. Faisant, "Fonctions modulaires et invariant modulaire" by G. Philibert, and "Courbes elliptiques et multiplication complexe" by M. Waldschmidt. Only the last one is available in english translation.
  12. Finally, there's the `mother of all surveys' on elliptic curves, Tate's "The arithmetic of elliptic curves": you can find it in Invent. Math. 23, 179-206 (1974).

Weblications

Link collections

  • Fermigier
  • Joye
  • Elliptic Curves and Modular Forms

    Tables and FAQs

  • FAQ
  • Cremona's Tables
  • Tom Womack's Tables
  • Integral points on Mordell curves y2 = x3 - k.

    Online scripts

  • Here's one by Koblitz.
  • Charles Daney on The Mathematics of FLT

    Surveys and Introductions

  • José Ignacio Cogolludo on congruent numbers
  • Claire Gallagher on Ramanujan's tau function.
  • Jeff Achter On computing the rank of elliptic curves, also as ps
  • Gael Benabou und Eric Colin de Verdière on solving quintic equations with elliptic functions (F).
  • Alice Silverberg has an article on Open Questions in Arithmetic Algebraic Geometry.
  • Joye's elementary introduction (French)
  • Loic Merel discusses The arithmetic of elliptic curves and diophantine equations; see also J. Theor. Nombres Bordeaux 11 (1999).
  • Ed Schaefer has texed some lectures on how to compute the Mordell-Weil groups of Jacobians.
  • Surveys on Iwasawa theory of elliptic curves by Ralph Greenberg.
  • H.G. Zimmer on Basic Algorithms of Elliptic Curves (D), also as ps

    Lecture Notes

    English:

  • Milne's Lecture Notes on elliptic curves are excellent. He also has notes on modular forms and modular functions.
  • There are lecture notes on modular forms by Igor Dolgachev going up to Taniyama-Shimura.
  • Connell's Handbook of elliptic curves is an ambitious project and still uncomplete.
  • Miles Reid has given a course on elliptic curves that is currently being TeXed.
  • John Cremona gives infos on his lectures on Rational points on curves.
  • Lecture Notes by William Stein (Serre conjectures etc.)
  • Brian Osserman has lecture notes on Kolyvagin's Application of Euler Systems to Elliptic Curves
  • There are notes to a similar seminar by Tom Weston
  • Stevenhagen's Lecture Notes on elliptic curves

    German:

  • W. Ruppert has lecture notes on elliptic curves and cryptography.

    French:

    Theses

    English:

  • Matt DeLong on connections between Selmer groups of elliptic curves with rational 3-isogenies and 3-class groups of quadratic number fields;
  • Garikai Campbell on elliptic curves with large rank and nontrivial torsion;
  • Wetherell on hyperelliptic diophantine equations via Chabauty
  • Adam van Tuyl on "The Field of N-torsion Points of an Elliptic Curve over a Finite Field"
  • Matt Baker on torsion points on modular curves.
  • D. Ulmer on The arithmetic of universal elliptic modular curves

    German:

  • Susanne Schmitt on the Mordell-Weil group of elliptic curves over number fields;
  • Jochen Stein on Zeros of L-function of certain elliptic curves
  • Michael Adam on torsion points of elliptic curves using what he calls "elementary algebraic geometry"
  • Theses on "Equations for X0(15)" by Franziska Hennig and on "Addition formulas for Jacobi varieties" by Sönke Maseberg can be found here.

    French:

  • Klaus Rolshausen on Eléments explicites dans K2 d'une courbe elliptique. (dvi.gz)
  • Charles-Antoine Louet and Oliver Wittenberg on "The rationality of the Zeta function of an algebraic variety over a finite field: Dwork's proof
  • Stéphane Fischler on applications of the abc conjecture on elliptic curves; also as ps
    More theses on demand from
  • Christian Wittmann on elliptic curves and cryptography (D);
  • Heuisu Ryu on construction of nontrivial Tate-Shafarevich groups using Heegner points
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