Books on Elliptic Curves
A. Weil, Number theory. An approach through history. From Hammurapi
to Legendre, 1984 (History)
- J. Silverman, J. Tate, Rational points on elliptic curves.
Undergraduate Texts in Mathematics, 1992, (elementary introduction)
- A. van der
Poorten, Notes on Fermat's Last Theorem; Wiley 1996, 222 pp.
(a little background to Wiles' proof of FLT).
- Yves Hellegouarch, Invitation aux mathematiques de Fermat - Wiles; Paris 1997, 397b pp, (elementary introduction to Wiles' proof).
- N. Koblitz, Introduction to elliptic curves and modular forms.
Graduate Texts in Mathematics 97, 1993 (good introduction)
N. Koblitz, Algebraic Aspects of Cryptography, Springer 1998
(contains an elementary introduction to hyperelliptic curves and their
applications in cryptography)
- R. Pinch,
Computational Number Theory (ECM)
- H. McKean,
V. Moll, Elliptic Curves. Function Theory, Geometry, Arithmetic
(pretty presentation, partially without proofs)
- T. Ono, Variations on a Theme of Euler : Quadratic Forms,
Elliptic Curves, and Hopf Maps; 1994 (nice and expensive)
- D. Husemoeller, Elliptic curves;
Graduate Texts in Mathematics 111, 1987 (out of print - a pity).
- J. Silverman, The arithmetic of elliptic curves.
Graduate Texts in Mathematics 106, 1986 (THE, introduction
to elliptic curves)
- 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)
- A. Knapp, Elliptic curves Mathematical Notes 40, Princeton Univ. Press
1992, $ 40 (excellent introduction to elliptic curves and modular forms)
- G. Cornell (ed.) et al, Modular forms and Fermat's last theorem.
Springer 1997 (Wiles' Proof)
- J.E. Cremona,
Algorithms for Modular Elliptic Curves (Tables and background)
- 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).
- Elliptic Functions and Elliptic Integrals by Viktor Prasolov and Yuri Solovyev
(nice introduction to elliptic curves, functions and integrals).
- Fermat's Dream by Kazuya Kato has just appeared
and introduces to number theory and elliptic curves.
- Arithmétique des courbes elliptiques et
théorie d'Iwasawa by B. Perrin-Riou (studies elliptic
curves with CM using Iwasawa theory).
- 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.
- Henri Cohen's Elliptic curves", in `From number theory to physics'
Springer-Verlag, 212-237 (1992);
- D. Zagier's "Elliptische Kurven: Fortschritte und Anwendungen"
can be found in the Jahresbericht der DMV 92 (1990), 58-76.
- Roel Stroeker's "Aspects of elliptic curves. An introduction"
is from Nieuw Arch. Wiskunde, III. Ser. 26 (1978), 371-412.
- 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,
- L. Washington's "Number fields and elliptic curves" can be found
in `Number theory and applications', Banff/Can. 1988, 245-278 (1989).
- 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.
- In the booklet `Lebendige Zahlen' by W. Borho et al. there is
an article "Algebraische Kurven und diophantische Gleichungen"
by Hanspeter Kraft.
- N. Schappacher and R. Schoof discussed Beppo Levi's contributions
to the theory of elliptic curves in
this article (dvi.gz).
- 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).
- 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
- 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).
Elliptic Curves and Modular Forms
Tables and FAQs
Tom Womack's Tables
Integral points on
y2 = x3 - k.
Here's one by
Charles Daney on
The Mathematics of FLT
Surveys and Introductions
José Ignacio Cogolludo on
Claire Gallagher on
Ramanujan's tau function.
Jeff Achter On
computing the rank of elliptic curves, also as
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
arithmetic of elliptic curves and diophantine equations; see also
Theor. Nombres Bordeaux 11 (1999).
Ed Schaefer has texed some
on how to compute the Mordell-Weil groups of Jacobians.
Iwasawa theory of elliptic curves by Ralph Greenberg.
H.G. Zimmer on
Algorithms of Elliptic Curves (D), also as
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
Dolgachev going up to Taniyama-Shimura.
of elliptic curves is an ambitious project and still uncomplete.
has given a course on elliptic curves that is currently being TeXed.
John Cremona gives infos on his lectures on Rational points on curves.
Notes by William Stein (Serre conjectures etc.)
Brian Osserman has lecture notes on
Application of Euler Systems to Elliptic Curves
There are notes to a similar seminar by
Lecture Notes on elliptic curves
W. Ruppert has lecture notes on elliptic curves and cryptography.
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
Wetherell on hyperelliptic diophantine equations via Chabauty
Adam van Tuyl on "The Field of
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
on the Mordell-Weil group of elliptic curves over number fields;
Jochen Stein on
of L-function of certain elliptic curves
Michael Adam on
points of elliptic curves using what he calls "elementary
Theses on "Equations for X0(15)" by Franziska Hennig
and on "Addition formulas for Jacobi varieties" by Sönke
Maseberg can be found
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:
Stéphane Fischler on applications of the
abc conjecture on elliptic curves; also as
More theses on demand from
Wittmann on elliptic curves and cryptography (D);
on construction of nontrivial Tate-Shafarevich groups using
Here's my Guestbook, please
leave your comments.