Arithmetic Geometry (SS 2020)
Lectures
- Tuesday, 02:15 pm - 3:45 pm, online
- Wednesday, 09:15 am - 10:45 am, online
- First Lecture: Tuesday, 21.04.2020, 02:15 pm
- The course takes place online via the Zoom platform; the respective links can be found on the respective Moodle homepage
Summary of the course
Arithmetic intersection theory has been introduced by S. Arakelov in order to provide a conceptual tool to measure the arithmetic complexity of rational points on algebraic varieties defined over number fields. Arakelov’s approach has been generalized by H. Gillet and C. Soulé to the higher dimensional setting. In this course we will provide a comprehensive introduction to arithmetic intersection theory and its latest developments.
We will start by reviewing classical intersection theory from algebraic geometry. Next we will give an introduction to the theory of Green’s currents, which can be seen as the differential geometric counterpart of algebraic geometric intersection theory. Combining these two theories will lead to arithmetic intersection theory, which will in particular provide the tool to measure the arithmetic complexity of closed subvarieties of algebraic varieties defined over number fields.
In the second part of the course we will talk about generalizations of the theory of arithmetic intersections, in particular to certain singular settings which enables us, among others, to apply the theory to Shimura varieties of non-compact type. In this context we will also discuss various examples.
The course will mainly be quite self-contained. As prerequisites we recommend a thorough knowledge of algebra and some basic knowledge in algebraic geometry and differential geometry. It is planned that we will design the exercise classes in the form of a student seminar which will be devoted to establish a thorough introduction to algebraic geometric intersection theory.
Selected literature
- William Fulton: Algebraic Curves, Addison-Wesley Publishing Company.
- William Fulton: Intersection Theory, Springer Verlag.
- Henri Gillet and Christophe Soulé: Arithmetic intersection theory, Publications mathématiques de l’I.H.É.S., tome 72 (1990), p. 93-174.
- Phillip Griffiths and Joseph Harris: Principles of Algebraic Geometry, John Wiley & Sons Inc.
- Christoph Soulé et al.: Lectures on Arakelov Geometry, Cambridge University Press.
Exercise classes / Seminar
- Wednesday, 11:15 am - 12:45 pm, online
- First Class: Wednesday, 22.04.2020, 11:15 am
- The seminar takes place online via the Zoom platform; the respective links can be found on the respective Moodle homepage
- There is a Moodle course for the event. You have to register in this to obtain course materials, participate in discussions at the forum and receive communications from fellow participants and course instructors. If you do not have a CMS account, you can easily create an external account on the Moodle homepage.
- The Moodle homepage can be found here: https://moodle.hu-berlin.de/course/view.php?id=93794
- Course number: 3314424
- Course name: Arithmetic Geometry (Lecture)
- To obtain the enrolment key for the Moodle, please send an email to Antareep Mandal.
Examination
- Type: Online oral examination
- Date: 06.10.2020
- Registration deadline: 22.09.2020
- Cancellation deadline: 29.09.2020