An introduction to height theory, geometric viewpoint
In this course, we will establish a general geometric process allowing us to construct heights on any projective algebraic variety V. We recover the Weil height, which has been studied in the first semester from an arithmetic viewpoint, when V is the projective space. When V is assumed to be abelian, the constructed height is then the well-known Néron-Tate height. Next, we will see that the Néron-Tate height has similar properties to that of Weil. Finally, we will prove that the Néron-Tate height on elliptic curves has a (explicit) decomposition into local height functions, which are defined arithmetically. Time permitting, we will give some applications of this decomposition.
Lecturer
Date
21st March ~ 20th June, 2024
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Thursday | 13:30 - 16:55 | A3-2-303 | ZOOM 02 | 518 868 7656 | BIMSA |
Prerequisite
Master in Number theory and/or Algebraic geometry. REMARK : It is not necessary to have followed the first semester course to understand this one.
Syllabus
Class 1 : I will recall the two definitions of the Weil height given in the first semester (one involving the Mahler measure and the other one using the places of a number field) explaining why they are not easy to generalize. We will then see a third way of defining it, which is the "good" approach to construct heights on projective algebraic varieties.
Class 2-4 : Construction of heights on projective algebraic varieties.
Class 5 : Statements of basic results on Néron-Tate heights.
Class 6-7 : Background on elliptic curves.
Class 8-10 : Decomposition of the Néron-Tate height on elliptic curves into local height functions.
Class 11-12 : Applications of this decomposition.
Class 2-4 : Construction of heights on projective algebraic varieties.
Class 5 : Statements of basic results on Néron-Tate heights.
Class 6-7 : Background on elliptic curves.
Class 8-10 : Decomposition of the Néron-Tate height on elliptic curves into local height functions.
Class 11-12 : Applications of this decomposition.
Reference
-For the construction of heights, see D. Mumford "Abelian varieties"
- Regarding the decomposition into local height functions, I will follow the book "Advanced topics in the Arithmetic of Elliptic curves" by J. Silverman.
- Regarding the decomposition into local height functions, I will follow the book "Advanced topics in the Arithmetic of Elliptic curves" by J. Silverman.
Audience
Graduate
, Postdoc
, Researcher
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Arnaud Plessis is an assistant professor at BIMSA from September 2023. His research is mainly focused on diophantine geometry. He obtained his Phd. thesis in 2019 at Université de Caen Normandie. Before joining BIMSA, he has been Attaché Temporaire d'Enseignement et de Recherche (a kind of postdoctoral with course duties) at Université Grenoble Alpes from September 2019 to August 2020. Then, he has been postdoctor at Morningside Center of Mathematics, Chinese Academy of Sciences, from September 2020 to August 2023.