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 can be currently divided into three parts: Arakelov geometry and Belyi's theorem, the arithmetic Shafarevich conjecture, and special subvarieties of Shimura varieties.
In my thesis I studied Arakelov theory on curves and found applications to arithmetic geometry by using Belyi maps. The main results of my thesis are in ,  and  below.
My interest in Belyi's theorem started during my thesis and has led to a preprint on a higher-dimensional analogue of Belyi's theorem for complete intersections of general type; see .
After my work with Rafael von Kaenel on Szpiro's small points conjecture , I started to investigate finiteness results for integral points on varieties and stacks over Z. Such finiteness statements conjecturally fit into a larger framework. The first conjecture, on the finiteness of the set of integral points on the stack of curves of smooth proper curves M_g of genus g (g > 1), was stated explicitly by Shafarevich at the 1962 ICM in Stockholm. His conjecture is commonly referred to as the arithmetic Shafarevich conjecture for curves and was proven by Faltings in his famous paper on the Mordell conjecture. It seems reasonable to suspect that Shafarevich's conjecture applies to a larger class of varieties, e.g., varieties of general type.
My research on this generalized version of the Shafarevich conjecture has led to three papers; see ,  and . Together with Daniel Loughran, we formulated a conjectural analogue of Shafarevich's conjecture for complete intersections; see .
Keywords and themes are: Arakelov geometry, Belyi's theorem, Shafarevich conjecture, computational aspects of étale cohomology, Arakelov inequalities, special subvarieties, Torelli theorems, good reduction, Lang's conjecture.
- 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 for certain canonically polarized varieties. (journal link) Bull. London Math. Soc., (2015) 47 (1).
- Good reduction of algebraic groups and flag varieties. (joint work with Daniel Loughran) (arxiv, journal link ) Archiv der Math., (2015) Vol. 104, Issue 2, 133-143.
- An effective Arakelov-theoretic version of the hyperbolic isogeny theorem. (pdf) Submitted.
- Complete intersections: Moduli, Torelli, and good reduction. (joint with Daniel Loughran) (arxiv) Submitted.
- Belyi's theorem for complete intersections of general type. (pdf) Submitted.
- Vorlesung "Algebraische Geometrie II", Wintersemester 2015-2016.
- Vorlesung/Course "Elliptic curves, complex multiplication, modular curves", together with Stefan Müller-Stach Wintersemester 2014-2015.
- Übungsleiter "Lineare 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
- Fano varieties. MPI Bonn and University of Mainz. Wintersemester 2015-2016.
We study Fano varieties and their moduli spaces. We meet precisely six times: three times in Bonn and three times in Mainz. The topics will vary each time.
- Topics in Hodge theory. University of Mainz. Wintersemester 2015-2016.
We discuss several topics in Hodge theory, e.g., mixed Hodge modules, period maps, and absolute Hodge cycles.
- Torelli theorems and Hodge theory. University of Mainz. Summersemester 2014-2015
In this seminar we familiarize students with period domains and period maps by studying as many examples as possible (such as curves, (polarized) abelian varieties, K3 surfaces, and hypersurfaces).
- 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
Summer school on Algebraic Stacks
- Algebraic stacks and related topics. University of Mainz. 31-08-2015 until 04-09-2015.
This summer school is organized by the SFB/TRR 45 Bonn-Essen-Mainz and takes place in Mainz from August 31st until September 4th, 2015.