Sommersemester 2013-2014: Neron models of smooth proper varieties

This is the Website for the Seminar Neron models of smooth proper varieties which was held at the University of Mainz during the Sommersemester 2013-2014.

André Néron was a French mathematician who lived from 1922 until 1985. His name is attached to many mathematical concepts such as the Néron-Severi group, the Néron-Tate height and the Néron model. Néron models were invented by Néron in the early sixties, and are fundamental to the study of arithmetic aspects of abelian varieties. In fact, Néron's motivation to work on models of abelian varieties over a local field K was motivated by the apparent lack of a "good" model  for such varieties over the valuation ring O_K. Surprisingly, he was not only able to prove the existence of a "good" model, but also used it to study rational points of abelian varieties over global fields. The theory of Néron models has had far reaching applications from Faltings' proof of the Tate, Shafarevich and Mordell conjectures in 1983 to Wiles' proof of the modularity of elliptic curves over Q in 1994. Although the classical motivation to study Néron models lies in the study of abelian varieties, in this seminar we will look beyond models of abelian varieties and also consider varieties with ample canonical bundle.


Practical information

Organizers: Ariyan Javanpeykar, Abolfazl Mohajer, Ronan Terpereau

Schedule: Tuesday 14h in Room 04-432

Special day in the weekend: Saturday 17th of May

On the "special day" we will discuss how to use SAGE to compute several examples of Néron models of elliptic curves.

Special talk by Bas Edixhoven: Tuesday 17th of June

Bas Edixhoven (Leiden University) will visit us and speak about his work on Weil's theorem.


  • We will follow closely the book on Néron models by Bosch, Lütkebohmert and Raynaud.
  • We will study the article by Edixhoven and Romagny: click here
  • We will study the recent article by Qing Liu and Jilong Tong: click here
  • We will study the article by Serre and Tate on good reduction of abelian varieties: click here


Program, Outline, Notes



  1. Introduction (Ariyan Javanpeykar), 22nd of April
  2. Weierstrass models (Sonia Samol), 29th of April
  3. Kodaira-Néron classification (Ariyan Javanpeykar), 6th of May
  4. Tate's algorithm (Ariyan Javanpeykar), 13th of May
  5. Special day, Saturday 17th of May
  6. Lipman's resolution of singularities (Ronan Terpereau), 20th of May
  7. Minimal regular models (Axel Stäbler), 27th of May
  8. Models with a birational group law: non-minimal components (Ben Anthes), 3rd of June
  9. Néron's smoothening process (Thomas Weißschuh), 10th of June
  10. Weil's extension theorem (Bas Edixhoven, Leiden University), 17th of June
  11. No talk,  24th of June
  12. Néron models of not-necessarily proper group schemes (Ronan Terpereau), 1st of July
  13. No talk, 8th of July
  14. Picard functors and relative Jacobians I (Abolfazl Mohajer), 15th of July
  15. Picard functors and relative Jacobians II (Abolfazl Mohajer), 22nd of July
  16. Néron models of algebraic curves (Ariyan Javanpeykar), 29th of July



  • Sonia Samol
  • Manfred Lehn
  • Lan Guitang
  • Susanne Mueller
  • Manuel Blickle
  • Ben Anthes
  • Axel Stäbler
  • Thomas Weißschuh
  • Ronan Terpereau
  • Abolfazl Mohajer
  • Ariyan Javanpeykar


Please let us know if you are interested in participating, and/or giving a talk.