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. Chen Dec. 10, 3:00-4:00 Math 102 Classification: the j-invariant. L. Hsu Dec. 10, 4:00-5:00 Math 102 The group law. A. Keet Dec. 11, 3:30-4:30 Math 102 Elliptic functions (Elliptic curves over C). K. Li Dec. 11, 4:30-5:30 Math 102 The Picard group. M. Bergeron Dec. 12, 3:00-4:00 Math 102 Elliptic curves over finite fields (Chapter V of Silverman). K. Behrend Dec. 12, 4:00-5:00 Math 102 Mordell-Weil theorem over Q. X. Liu Dec. 13, 3:00-4:00 Math 102 Elliptic curves and congruence numbers (Chapter 1 of Koblitz). A. Vlasev Dec. 13, 4:00-5:00 Math 102 Hasse-Weil L-function of an elliptic curve. Birch-Swinnerton-Dyer conjecture. M. Rupert Dec. 14, 3:00-4:00 Math 102 Vector bundles over elliptic curves. E. Mohyedin Kermani Dec. 14, 4:00-5:00 Math 102