Heights on projective varieties: an arithmetic construction
In my previous course, I gave a purely GEOMETRIC construction of canonical (a.k.a. Call-Silverman) heights on projective varieties (more precisely, on polarized dynamical systems). This construction allows us to solve GEOMETRIC problems (e.g. Mordell-Weil Theorem), but it is irrelevant for providing ARITHMETIC properties of canonical heights. Hence, it is of interest of giving a purely ARITHMETIC definition of these heights. After proving Mordell-Weil Theorem, we will establish that the canonical heights are decomposed into a sum of local heights, which involves arithmetic. Finally, we will see that this decomposition is explicit in the elliptic setting.
Lecturer
Date
18th September ~ 11th December, 2024
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Wednesday | 13:30 - 16:55 | A3-2-201 | ZOOM 08 | 787 662 9899 | BIMSA |
Prerequisite
All prerequisites/remainders will be made in Class 1 so that this course is self-contained
Syllabus
Class 1: Prerequisites/remainders.
Class 2: Call-Silverman height in the special case of abelian varieties.
Class 3-5 : Proof of Mordell-Weil Theorem
Class 6-10: Establishing the local decomposition of canonical heights.
Class 11-12: Explicitness of this decomposition for elliptic curves
Class 2: Call-Silverman height in the special case of abelian varieties.
Class 3-5 : Proof of Mordell-Weil Theorem
Class 6-10: Establishing the local decomposition of canonical heights.
Class 11-12: Explicitness of this decomposition for elliptic curves
Reference
I will prove the weak Mordell-Weil Theorem by following the exposition of the book "Heights in Diophantine geometry". Then, I refer to the Call and Silverman's paper "Canonical heights on varieties with morphisms", Section 2 for the local decomposition. Finally, the explicitness of this decomposition in the elliptic setting can be found in Silverman's book "Advanced Topics in the arithmetic of elliptic curves", Chapter VI.
Audience
Advanced Undergraduate
, 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.