Monday 11:40 - 12:30, SBZ-11 Wednesday 10:40 - 12:30, seminar roomYou will be graded on take home exams and homework.

You will also need a little bit of Galois theory; let me advertise my own seminar on Galois theory this semester: if more than one student is interested, we'll do it. It will also give you a short introduction to algebraic number theory. Email me if you're interested.

- J. Silverman, J. Tate: Rational points on elliptic curves

The most elementary introduction. - J.W.S. Cassels: Elliptic Curves.

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Actually it's quite a good book, even for men (Cassels' book, I mean). - L. Washington: Elliptic Curves

This one has just appeared. It is very elementary. - A. Knapp: Elliptic Curves

Gives a nice introduction to the analytic aspects - J. Silverman: Arithmetic of Elliptic Curves

The standard reference - Husemöller: Elliptic Curves

There will be a new edition in 2004

- Group law on conics and elliptic curves
- Review of p-adic numbers
- Nagell-Lutz: torsion points and formal group laws
- Heights
- Mordell-Weil: 2-descent
- Galois Cohomology
- Selmer and Tate-Shafarevich groups
- Function fields and divisors
- The group law via divisors
- Weil pairing
- Modular Forms
- Advanced Topics (Taniyama-Shimura, complex multiplication, Eichler-Shimura, Fermat's Last Theorem, abelian varieties, ...)

- macros required for the tex files.
- Due Feb 23: problems (tex file); solutions.
- Due March 03: problems (tex file); solutions.
- Due March 17: problems (tex file);
- Due at the end of March: Midterm 1 problems (tex file); solutions.
- Due April 19: problems (tex file); (hints).
- Midterm 2 problems (tex file);
- Final problems

John Voight gives a description of how to use pari (and other programs) for computing with elliptic curves.

- We 11.02.04 Historical Remarks;
Fermat's Last Theorem for exponent 4.
Lecture 1

Additional Material: here's a timeline for Fermat's Last Theorem. - Fr 13.02.04 Overview: Arithmetic of Conics and Cubics.
Lecture 2

Additional Material: I shortly mentioned p-adic numbers and Hasse's Local-Global Principle. Capi Corrales Rodriganez wrote a very gentle introduction to p-adic numbers, and Catherine Goldstein gave a similarly down-to-earth introduction to elliptic curves from the p-adic point of view. We'll later return to these topics and discuss them in more detail (and with proofs!). - Mo 16.02.04 Overview: L-series, Birch-Swinnerton-Dyer conjecture.
Lecture 3

Here's Wiles' ``Clay'' article on the Birch and Swinnerton-Dyer conjecture. - We 18.02.04 Projective Closure of affine curves. Lecture 4
- Fr 20.02.04 Lecture 5: Singular points
- Mo 23.02.04 Lecture 6: algebraic groups; group laws on lines and conics
- We 25.02.04 Lecture 7: Tate's formulas; Group Laws on Cubics. Here's Tate's Inventiones article
- Mo 01.03.04 no class; moved to 20.02.04
- We 03.03.04 (10:40 SAZ02) Lecture 8: conics over finite fields; Pollard's p-1 method. record factors found by p-1;
- Mo 08.03.04 (11:40 SAZ02) Lecture 9: factorization algorithms; at this page you can watch numbers getting factored using ECM.
- We 10.03.04 Lecture 10: The Hasse bound (Manin's Proof)
- Mo 15.03.04 Lecture 11: Manin's Proof Part II
- We 17.03.04 Lecture 12: p-adic numbers. For a couple of introductory essays, see this page.
- Mo 22.03.04 (10:40)
Lecture 13:
Hensel's Lemma; conics over Z
_{p} - We 24.03.04 No class; moved to 20.02.04 and 22.03.04
- Mo 29.03.04 Lecture 14: Reduction modulo p.
- We 31.03.04 Lecture 15: Elliptic curves over Local Fields
- Mo 05.04.04 Lecture 16: Theorem of Nagell-Lutz on torsion points;
- We 07.04.04 Lecture 17: Elliptic curves over the complex numbers. Here's an article by Schappacher and Schoof on the life and the work of Beppo Levi. Here's a nice little web page on torsion points
- Mo 12.04.04 Lecture 18: families of elliptic curves with cyclic torsion groups of order up to 10; nonexistence of torsion groups of order 11 over the rationals. Here's the article by Billing and Mahler; Ogg's article on the nonexistence of rational torsion points of order 17; and finally the article by Mazur and Tate on the nonexistence of rational torsion points of order 13, just to show you that even this special case is highly nontrivial.
- We 14.04.04 Lecture 19: Modular Curves, Isogenies, Mazur's Proof. Here's a nice article by Lenstra and de Smit on connections between one of Escher's drawings and a certain elliptic curve.
- Mo 19.04.04 (10:40) Lecture 20: Weak Mordell-Weil: the finiteness of E(Q)/2E(Q)
- We 21.04.04 No class; one hour moved to 19.04, the other one to some location t.b.a.
- Mo 26.04.04 Lecture 21: Heights
- We 28.04.04 Lecture 22: Theorem of Mordell-Weil
- Mo 03.05.04 Lecture 23: 2-descent via isogenies.
- We 05.05.04 Lecture 24: exact sequences, Tate's formulas; here's the intro to It's my turn
- Mo 10.05.04 Lecture 25: First steps in Galois cohomology
- We 12.05.04 Lecture 25:
Selmer and Tate-Shafarevich groups;
15.30: Proof of Fermat's Last Theorem

**payback time**: Mesut has the evaluation sheets; please fill them out if you find the time. - Mo 17.05.04. No class (I will be here); moved to 12.05.04
- Here I will list a few files containing material we could not
cover. More to follow.
- Here's an explanation of the general 2-descent on elliptic curves.
- A few incoherent remarks about Wiles' proof of FLT.