Franz Lemmermeyer

# Topics in Algebraic Geometry: Elliptic Curves John Baez, a well known mathematical physicist, writes an enlightening weekly column about what is happening in his area. Here's week 201.

### Schedule

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

### Content and Preliminaries

From the quiz I gather that all of you have some background in complex analysis, and that the majority is interested in abelian varieties. As for the other topics: I will review some of them briefly in class. Of course I will get nowhere if I do everything from scratch, so here's my suggestion: I will discuss affine and projective spaces, the snake lemma, p-adic numbers, and eventually a simple case of Riemann-Roch in Algebraic Geometry; you don't have to attend, but probably should read the notes. In this class, I will explain these concepts more briefly, but maybe also more informally.
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.

### Books

1. J. Silverman, J. Tate: Rational points on elliptic curves
The most elementary introduction.
2. J.W.S. Cassels: Elliptic Curves.
From Amazon.com:
```     Customers interested in LMSST: 24 Lectures on Elliptic Curves
may also be interested in:
Curves For Women
Franchise Opportunity for Ladies Only 30 Minute Workout Club.```

Actually it's quite a good book, even for men (Cassels' book, I mean).
3. L. Washington: Elliptic Curves
This one has just appeared. It is very elementary.
4. A. Knapp: Elliptic Curves
Gives a nice introduction to the analytic aspects
5. J. Silverman: Arithmetic of Elliptic Curves
The standard reference
6. Husemöller: Elliptic Curves
There will be a new edition in 2004

### Topics

• 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, ...)

### Software

Here's a windows executable of pari. If you type in ?, you'll get a list of chapters; ?4 lists e.g. the number theoretical functions, and ?gcd tells you what gcd does. You can find a more detailed manual at the pari homepage in Bordeaux.
John Voight gives a description of how to use pari (and other programs) for computing with elliptic curves.

### Schedule

• 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 Zp
• 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.
1. Here's an explanation of the general 2-descent on elliptic curves.
2. A few incoherent remarks about Wiles' proof of FLT.
Here are Nigel Boston's notes on Wiles' proof of Fermat's Last Theorem. And here is a page with several low-brow introductions to the proof, along with the original articles.