Diophantine geometry lies at the intersection of number theory and algebraic geometry. A classical problem in number theory asks for the solubility of polynomial equations over the integers or rationals. From the point of view of algebraic geometry, such systems of polynomials define algebraic varieties, and solutions correspond to rational points. Studying the geometry of this variety then provides one with information about the solubility of the equations.
The aim of this course is to introduce participants to the basic concepts in Diophantine geometry and broader arithmetic geometry. Participants will learn techniques to describe the set of rational points on a variety and apply them in exercise problems. They will become familiar with concepts and techniques relevant to other areas of number theory, arithmetic and algebraic geometry and they will be given an outlook to current research.
The course will cover the required background in algebraic geometry and number theory, including relevant properties of varieties, cohomology, p-adic numbers and Hensel’s lemma, and then continue with local-to-global principles, the Brauer-Manin obstruction, and descent. Depending on time and interests of the audience, we will continue with different aspects of Diophantine geometry.
Target group: PhD students and postdocs interested in arithmetic or algebraic geometry or number theory
Prerequisites: Some background in algebraic geometry. The course “An introduction to Algebraic Geometry” held in winter 19/20 would serve as an excellent background and complement this course, but is not required.
Evaluation: Performance in exercise problems and oral presentation
Teaching format: Lectures with exercise classes
ECTS: 6 Year: 2020
MAT-ALG Mathematics - Algebra
MAT-GEO Mathematics - Geometry and Topology
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