Math 532

Algebraic Geometry

Fall  2012


Instructor:   Kai Behrend

Office Hour:   Wednesdays 13:00-14:00, Math Annex 1213

Time:   MWF 12-13

Place:   MATH 102

Course Outline: pdf 

Homework: 1.4.1(ii), 1.4.3, 1.4.4, 1.4.5, 1.4.6, 1.4.8
from Gathmann's notes. Due date: Monday, Oct 1.

Homework: 2.6.1, 2.6.3, 2.6.6, 2.6.8, 2.6.9, 2.6.12
from Gathmann's notes. Due date: Monday, Oct 22.
For 2.6.9, prove also that for every P in X the local ring is a
subring of the function field, and that if X is a variety, then
different points lead to different subrings. Moreover, prove that for
every open subset of X, the ring of regular function is the
intersection of all local rings of all its points.

Homework: 3.5.1, 3.5.2, 3.5.3, 3.5.4, 3.5.5,
4.6.2, 4.6.3, 4.6.5, 4.6.6, 4.6.7, 5.6.12
from Gathmann's notes. Due date: Friday, Nov 23.

Projects: Here are 9 subjects on Elliptic Curves. Choose one
of them each, and prepare a presentation on it. You will present them
to each other towards the end of the term, possibly in the exam
period. You should meet with me to discuss the content of your
project and the relevant literature. Some of the later topics are
more advanced and suitable for students interested in number theory.
These later subjects would be somewhat expository.

Subject Name Date and Location
Definition of elliptic curves, Weierstrass normal form. K. ChenDec. 10, 3:00-4:00 Math 102
Classification: the j-invariant. L. HsuDec. 10, 4:00-5:00 Math 102
The group law. A. KeetDec. 11, 3:30-4:30 Math 102
Elliptic functions (Elliptic curves over C). K. LiDec. 11, 4:30-5:30 Math 102
The Picard group. M. BergeronDec. 12, 3:00-4:00 Math 102
Elliptic curves over finite fields (Chapter V of Silverman). K. BehrendDec. 12, 4:00-5:00 Math 102
Mordell-Weil theorem over Q. X. LiuDec. 13, 3:00-4:00 Math 102
Elliptic curves and congruence numbers (Chapter 1 of Koblitz). A. VlasevDec. 13, 4:00-5:00 Math 102
Hasse-Weil L-function of an elliptic curve. Birch-Swinnerton-Dyer conjecture. M. RupertDec. 14, 3:00-4:00 Math 102
Vector bundles over elliptic curves. E. Mohyedin KermaniDec. 14, 4:00-5:00 Math 102