This is the web page of the DIAMANT / Mastermath course Elliptic Curves.



Lectures: G.L.M. Cornelissen (Utrecht)
P. Stevenhagen (Leiden)
Problem session: Peter Bruin (Leiden)
Marco Streng (Leiden)
Location:W&N Building S205, Vrije Universiteit, Amsterdam
Time:Tuesdays, February 5 – May 13, 2008, 10:15 – 13:00
Each class there will be three 45 minutes time slots: two lectures and one exercise class.


Elliptic curves are fundamental objects in a large part of mathematics. Along various historical paths, their origins can be traced to calculus, complex analysis and algebraic geometry, and their arithmetic aspects have made them key objects in modern cryptography and in Wiles' proof of Fermat's last theorem. This course is an introduction to the algebraic, geometric, complex analytic and number theoretical aspects of the theory of elliptic curves.


February 5PS intro, course rules, history, elliptic integrals, analogy sin/cos [notes, § 1 and 2]
February 12PS elliptic functions, Weierstrass isomorphism, group structure, endomorphisms [notes, § 2 and 3]
February 19GC diophantine equations in two variables, conics, chord and tangent construction [Silverman-Tate, I.1 and I.2]
February 26GC basic projective geometry, bezout, nine point theorem, [Silverman-Tate, I.3, II.1,3–5, A1, A2; read I.4 and A.3]
March 4GC torsion group/Lutz–Nagell theorem, heights [Silverman-Tate]
March 11PS Mordell's theorem [notes and Cassels, § 14]
March 18PS Mordell–Weil theorem; computation of the group of rational points of an elliptic curve over Q [notes and Cassels, § 14]
March 25break
April 1GC elliptic curves over finite fields, Hasse theorem, zeta function [Milne, Section IV.9]
April 8PS the elliptic curve factorization method (ECM), elliptic curve primality proving (ECPP), elliptic curve cryptography (ECC) [Silverman-Tate, Section IV.4]
April 15GC computer class with pari and Mwrank (in room P4.47).
April 22break
April 29GC elliptic curves over local fields, local-global principle [Cassels, §2--5, 10, 18]
May 6PScomplex multiplication [notes, § 3 and Cohen-Stevenhagen]
May 13PS complex multiplication
May 20GCmodularity and Fermat's Last Theorem


due datehomework
February 12Hand in 4 exercises from the first 14 pages of the lecture notes [notes].
February 19Hand in 4 exercises from sections 2 and 3 of the lecture notes [notes].
February 26Do exercises 1–3 from the exercise sheet and hand in exercise 2.
March 4Do exercises 4, 5, 8, 9 and 10 from the exercise sheet and hand in exercise 5. (Note: in an older version of these exercises, there was a typo in exercise 5.) From [Silverman-Tate], do exercises 1.17, 2.1, 2.10, 2.11, 2.12, and if you don't know projective geometry, A.1,4,5,8,9.
March 11Do exercises 3.1, 3.2, 3.3 of [Silverman-Tate] and hand in exercise 3.3. (Here is a copy of the exercises in case you don't have the book. A solution of exercise 3.2(a,b) can be found in the proof of Theorem 6.2 in [Silverman1].)
March 18Hand in 4 exercises from the following list: exercise 3.17, exercise 3.18 and section 4 from the [notes], and section 14 of [Cassels].
April 1Hand in 4 exercises from section 4 (Weak Mordell-Weil) from the [notes] (not the section on Elliptic curves over fields from an older version) and section 14 of [Cassels]. Do not hand in exercise 5 from the notes or the first curve of exercise 1 from Cassels. In exercise 3 from the notes and exercise 1 from Cassels, each curve counts as a separate exercise.
April 8Do exercises 4.1, 4.2, 4.3 (a) and (b), 4.5, (4.6*), 4.8, 4.12 of [Silverman-Tate] and exercise IV 5.6* of [Silverman 1]. Hand in exercise 9.13 of [Milne, Chapter IV]. (* means hard exercise)
April 15Hand in 4 exercises from the following set of exercises in [Silverman-Tate]. 4.1 -- 4.12, 4.13 (without using a computer), 4.17, 4.20 (for part (a), use the method of 4.19), 4.21 (especially recommended).
April 29Complete the computer class and hand in the exercise mentioned in bold. For information on using the software at home or your own university, see this page.
May 6Do as many exercises as possible from §2 of [Cassels] and do exercise 1(i) of §10. Prove that Selmer's curve 3x3 + 4y3 + 5z3 = 0 has points in Qp for all primes p (including the ‘infinite’ case of R), and do the same for the curve x4 − 17 = 2y2 of Lind and Reichardt. Hand in your solution for Selmer's curve.
May 13See the exercise sheet. References: [notes, § 3] and [Cohen-Stevenhagen].
May 20See the new exercise sheet.


The topics treated include a general discussion of elliptic integrals and functions, elliptic curves and their group law; Diophantine equations in two variables; Mordell's theorem, with computational aspects; Computer class (pari, MWrank); Elliptic curves over finite fields with applications (factoring integers, elliptic discrete logarithms and cryptography); complex multiplication; a survey of modularity of elliptic curves.


The final grade will be based exclusively on homework.


Linear algebra, groups, rings, fields, complex variables.


[notes] Lecture notes for Peter Stevenhagen's lectures: P. Stevenhagen: Elliptic Curves. PDF, PS
[Cassels]J.W.S. Cassels: Lectures on Elliptic Curves §§2–5 for the local-global principle, and §14 for 2-descent. Here is a scanned copy of §§2–6, 10 and 18, and here is one of §14.
[Cohen-Stevenhagen]H. Cohen and P. Stevenhagen - Computational class field theory. Chapter 15 in the upcoming book on algorithmic number theory. See pages 518--519 for how to enumerate all lattices having CM by a given ring.
[Milne]J.S. Milne: Elliptic Curves is electronically available online and (according to the book's web page) the paperback version costs only $17. Section IV.9 is a good reference for the Zeta function of a curve. The book replaces Milne's lecture notes that we linked to earlier: chapter 19 of the notes corresponds to section IV.9 of the book, exercise 19.8(b) of the notes corresponds to exercise 9.13 of chapter IV of the book.
[Silverman-Tate] Newcomers to the subject are suggested to buy the book J.H. Silverman and J. Tate: Rational Points on Elliptic Curves. Undergraduate Texts in Mathematics, Springer-Verlag, Corr. 2nd printing, 1994, ISBN: 978-0-387-97825-3: it contains a lot of the material treated in the course.
[Silverman1] Advanced students with a good knowledge of algebraic geometry are recommended to (also) buy J.H. Silverman: The arithmetic of elliptic curves. Corrected reprint of the 1986 original. Graduate Texts in Mathematics, 106. Springer-Verlag, New York, 1992. ISBN: 0-387-96203-4.
[Silverman2]Further references: J.H. Silverman: Advanced topics in the arithmetic of elliptic curves. Graduate Texts in Mathematics 151, Springer-Verlag, 1994. ISBN: 0-387-94328-5.

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