In June 2013 I defended my PhD at Leiden University and the University of Paris-Sud 11. My thesis advisors were Jean-Benoît Bost, Bas Edixhoven and Robin de Jong. You can find a copy of my PhD thesis below.
E-mail: peykar (at) uni-mainz.de
Phone number: +49 6131-39-22436
Please do not hesitate to contact me for any questions, remarks, or comments.
My research focuses on applications of Arakelov theory to problems in arithmetic geometry, themes around Belyi's theorem and the arithmetic Shafarevich conjecture.
Keywords and themes are: Arakelov geometry, Belyi's theorem, Shafarevich conjecture, computational aspects of étale cohomology, Arakelov inequalities, special subvarieties.
- Polynomial bounds for Arakelov invariants of Belyi curves, with an appendix by Peter Bruin. (arXiv, journal link) Algebra and Number Theory, Vol. 8 (2014), No. 1, 89–140.
- Szpiro's small points conjecture for cyclic covers. (joint work with Rafael von Kaenel) (arxiv, journal link) Documenta Math., 19 (2014) 1085-1103.
- Néron models and the arithmetic Shafarevich conjecture. (pdf) Bull. London Math. Soc., to appear.
- Good reduction of algebraic groups and flag varieties. (joint work with Daniel Loughran) (pdf) Submitted.
- Vorlesung/Course "Elliptic curves, complex multiplication, modular curves", together with Stefan Müller-Stach Wintersemester 2014-2015.
- Übungsleiter "Lineaire Algebra und Geometrie II". Teacher: Theo de Jong. Wintersemester 2014-2015.
- Übungsleiter "Algebraische Kurven und Riemannsche Flächen". Teacher: Duco van Straten. Summersemester 2013-2014
- Mitarbeiter für "Zahlentheorie". Teacher: Manuel Blickle. Wintersemester 2013-2014
- Arakelov invariants of Belyi curves. Ph.D. thesis (Leiden University and University of Paris-Sud 11)
- Stellingen. These are the "Propositions".
In my thesis, I studied Arakelov invariants of curves. I proved explicit polynomial bounds for Arakelov invariants (such as the Faltings height) in terms of the Belyi degree. The first application (and the primary motivation to do all this) is related to the work of Couveignes and Edixhoven on computing coefficients of modular forms. The results of my thesis show that their algorithm runs in polynomial time under GRH. Also, in the direction of "computing" etale cohomology, these explicit bounds settle a conjecture of Edixhoven, de Jong and Schepers on the Faltings height of a cover of curves. You can find a published proof of these results in  above. In my thesis, the necessary work is done in Chapters 1, 2 and 3. In , joint with Rafael von Kaenel, we apply the results of  and Baker's theory of linear forms to prove Szpiro's small points conjecture for cyclic covers of the projective line. This article corresponds to Chapter 4 of my thesis. Note that the exposition in my thesis is very different from that of .
Seminars, Workgroups, Groupes de travail
- Abelian varieties: Old & New. University of Mainz. Wintersemester 2014-2015
In this seminar we study the moduli of abelian varieties. The aim of this seminar is to familiarize students with the moduli theory of abelian varieties (i.e., Shimura varieties).
- Neron models of smooth proper varieties (after Artin, Bosch, Lütkebohmert, Neron, Raynaud, and Liu & Tong). University of Mainz. Summersemester 2013-2014
In this seminar we study the book "Neron models" by Bosch, Lütkebohmert and Raynaud, the recent work of Romagny and Edixhoven on Weil's theorem, and the article by Qing Liu and Jilong Tong on Neron models for curves. The aim of this seminar is to familiarize students with techniques in algebraic geometry that are (only a bit) more difficult than those in the book by Hartshorne.
- The Shafarevich conjecture (after Arakelov, Campana, Kovacs, Kebekus, Parshin, Patakfalvi, Viehweg, Zuo,...). Leiden University, Summersemester 2012-2013
Notes, Master's thesis, Bachelor's thesis, etc.
Below you will find several notes written a very long time ago. In particular, they contain many inaccuracies (due to yours truly). I do feel like some of them could be helpful to some people. Therefore, I made them available here. Comments, questions and remarks are more than welcome.
- Master's thesis (June 17, 2010):Grothendieck-Riemann-Roch, with an application to covers of varieties. Supervisor: Professor Jaap Murre. Leiden University
- The Lefschetz trace formula: Notes Leiden, Notes Paris
- The Atiyah-Jänich theorem. Short "Mémoire" written in 2009 for a course given by Thierry Masson at Paris-Sud 11.
- The Atiyah-Singer index theorem, a course by Jean-Michel Bismut. Notes written in 2009 for personal use in Paris-Sud 11.
- Bachelor's thesis (July 11, 2008): Representation, homotopy and gauge theory for SU2(R). Supervisors: Franck Doray and Koenraad Schalm. Leiden University