Number Theory of Cubic Curves
(mastermath course 2007/2008)
Lecturers
Literature
- Joseph H. Silverman and John Tate, Rational Points on Elliptic Curves.
Springer-Verlag, Undergraduate Texts in Mathematics, 1992.
Final Grade:
Student are expected to hand in solutions to problems which will be given
during most of the classes (see the schedule below), and to do a final take home exam.
The take home will be available by December 14th, and should
be handed in by January 18th. To pass the course, you should have
a mark M at least 4 for the take home exam, and a final grade
(4M+H1+H2+H3+H4+H5+H6)/10 at least 5.5. Here the Hi are the 6 best grades
for your homework.
Take Home Exam:
The take home exam, including instructions how to hand in the problems,
can be found here.
(January 7th: I corrected an error in Problem 7... sorry!)
Schedule
Number Theory of Cubic Curves, Fall 2007:
- Friday, September 14th:
The affine and the projective plane, cubic curves
(Silverman-Tate Appendix, first two sections)
Classifying real plane cubic curves according to Newton and to
Moebius
(Here are slides
from a lecture on the latter topic)
- Friday, September 21th:
The chord and tangent methods, statement of Mordell's theorem
Group law on smooth projective cubics with a point
(Silverman-Tate, Chapter 1)
- Homework set 1 (due September 28th at the start of class):
two problems of your own choice from Chapter 1 Exc. 9, 12, 14, 18abc, 20.
- Friday, September 28th:
Torsion points, the Nagell-Lutz theorem, reduction modulo p
(Silverman-Tate, II.1, II.3, II.5 and statements in II.4 and IV.3)
- Homework set 2 (due October 5th at the start of class):
two problems of your choice from Chapter II Exc. 2, 11 and Chapter IV Exc. 9.
- Friday, October 5th:
Torsion points are integral
(Silverman-Tate II.4)
- Homework set 3 (due October 12th at the start of class):
1) Chapter II Exc. 10 and 12(d).
2) Take a a nonzero even integer and f(x)=x^3+ax-a. Show that
y^2=f(x) defines a smooth cubic curve and, with the point at infinity as
the zero element for the group law, (1,1) on this curve has infinite order.
- Friday, October 12th:
The congruent number problem, See, e.g., the nice and elementary
survey by V. Chandrasekar.
- Homework set 4 (due October 26th at the start of class):
Find (e.g., using the web) all congruent numbers below 20, and for each of them,
give at least two different rectangular triangles with rational sides and the given area.
- Friday, October 19th: no class
- Friday, October 26th:
Part (a) of the Proposition in III.5 (in a more general setting),
and applications of this to proving independence of points.
Mestre's "completing the square" trick.
- Homework set 5 (due November 2nd at the start of class):
1) Chapter III Exc. 3.5
2) Construct two cubic curves with three independent rational points:
one cubic with also a rational point of order two, and one without
any nontrivial rational points of finite order.
- Friday, November 2nd:
The "weak Mordell-Weil theorem"
(Silverman-Tate III.4 and III.5)
- Homework set 6 (due November 23rd(!!) at the start of class):
Two of the problems III.3.7, III.3.8, III.3.9be.
- Friday, November 9th:
Rank calculations (Silverman-Tate III.6)
In today's class, we looked for various websites containing data
on cubic curves with several independent points. Moreover, some of you
sent me internet addresses containing even more such examples.
Here is a list. In some cases, a cubic curve y^2+a1*x*y+a3*y=x^3+a2*x^2+a4*x+a6
is denotes as [a1, a2, a3, a4, a6].
a paper by Ezra Brown and
Bruce Myers
a history of rank records
a lecture by Alice Silverberg
Tom Womack's list
of `easiest' cubic curves with many independent points
- Friday, November 16th:
heights and completing the proof of Mordell's theorem.
(Silverman-Tate III.1, III.2, III.3)
- Friday, November 23rd:
The Hasse inequality.
(We discuss Manin's proof from 1956, which is explained
in J.S. Chahal's paper
in Nieuw Archief voor Wiskunde fourth series 13 (1995).)
- Homework set 7 (due November 30th at the start of class):
These three problems.
- Friday, November 30th:
Another application of cubic curves to elementary geometry.
See this text
- Homework set 8 (last set, due December 7th):
These two problems.
- Friday, December 7th: last week of classes!
Elliptic curves primality proving
(See this text)
and elliptic curves integer factorization (Silverman-Tate IV-4)