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.
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
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.
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).
- 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
- 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, ...)
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
John Voight gives a description of how to use pari (and other
computing with elliptic curves.
Here are Nigel Boston's
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.
- We 11.02.04 Historical Remarks;
Fermat's Last Theorem for exponent 4.
Additional Material: here's a
for Fermat's Last Theorem.
- Fr 13.02.04 Overview: Arithmetic of Conics and Cubics.
Additional Material: I shortly mentioned p-adic numbers and Hasse's
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.
Here's Wiles' ``Clay'' article
on the Birch and Swinnerton-Dyer conjecture.
- We 18.02.04 Projective Closure of affine curves.
- 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.
factors found by p-1;
- Mo 08.03.04 (11:40 SAZ02) Lecture 9:
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
- Mo 22.03.04 (10:40)
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
- 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
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
- 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
- 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.