CLASS HOURS AND LOCATION: Tue-Thu 11:00 - 12:30 RLM 9.166. The course will start on January 22nd.
UNIQUE NUMBER: 59240
OFFICE HOURS: Wed 9:00 -- 11:00 or by appointment.
TEXTBOOK: We will not use a textbook but there several good references such as Lang's Fundamentals of diophantine geometry, Lang's Number theory III : diophantine geometry, Serre's Lectures on the Mordell-Weil Theorem, Hindry and Silverman's Diophantine Geometry: an introduction and Bombieri and Gubler's Heights in Diophantine Geometry.
PREREQUISITES: Graduate Algebra. It will be useful, but not essential, to have some prior experience with number fields and heights (as in Vaaler's course last Spring) and some algebraic geometry. Contact me if you have any questions about prerequisites.
GRADE POLICY: Each student will be required to take notes for a week and TeX them.
COURSE DESCRIPTION: Diophantine equations are polynomial equations in several variables where the solutions are sought among the integers or rational numbers. A major insight of Twentieth Century Mathematics has been that the geometry of the algebraic variety described by these equations have a profound influence on the nature of the solutions in integers or rational numbers. This course will discuss these insights. We will give some proofs but sometimes we will content ourselves with an overview, going for insight rather than detail in this sometimes technical subject. We will try to discuss the many conjectures that are still open in the subject.
Topics to be covered might include:
Review of field valuations. Heights in projective spaces and varieties. Local distance functions.
Diophantine approximation and the Thue-Siegel-Roth-Schmidt theorem. Vojta's conjectures.
Integral points on curves, Siegel's theorem. Rational points on curves and abelian varieties, Mordell-Weil theorem and the Mordell conjecture. The Birch and Swinnerton-Dyer conjecture.
Local-Global principles. The Brauer-Manin obstruction and Scharaschkin's conjecture.
NOTES: Now available in one file here.