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Workshop "Periods and heights of CM abelian varieties"

This workshop is organized by J. Ayoub, C. Fuchs, A. Kresch and G. Wüstholz. It takes place from July 6-11, 2014 at the Böglerhof in Alpbach/Tyrol, Austria.

This summer school will continue a many-year tradition of schools in Alpbach featuring high-profile research results in arithmetic and geometry. The topic builds on p-adic periods from the ProDoc Workshop 2010. PhD students/postdocs (mainly from Zurich) present the topics, and everyone gains a deep exposure.

Colmez' conjecture relates the Faltings height, measuring the complexity of a CM abelian variety, to L-functions, which are generalizations of the Riemann zeta-function. For a CM elliptic curve (CM, or complex multiplication, refers to extra symmetries), the conjecture reproduces the Chowla-Selberg formula. The goal of the workshop is to understand Colmez' proof, which treats the case of abelian CM fields. More detailes on the program will be published here in due time.

A pre-workshop took place at FIM from April 7-10, 2014. For further details see here.

Participants

Joseph Ayoub
Marco Besier (Mainz)
Peter Bruin (Warwick)
Simon Felten (Mainz)
Andrea Ferraguti
Javier Fresan (Bonn)
Clemens Fuchs (Salzburg)
Martin Gallauer
Sergey Gorchinskiy (Moscow)
Daniel Harrer (Freiburg)
Christoph Hutle (Salzburg)
Ariyan Javanpeykar (Mainz)
Peter Jossen (Lausanne)
Rafael von Känel (MPI Bonn)
Andrew Kresch
Lars Kühne (Pisa)
Vincent Maillot (Paris)
Matthias Nickel (Mainz)
Roland Paulin (Salzburg)
Simon Pepin-Lehalleur
Maximilian Preisinger (Mainz)
Michael Rottmaier (Freiburg)
Sonia Samol (Mainz)
Max Schmidtke (Freiburg)
Vaibhav Vaish
Francesco Veneziano (Graz)
Alberto Vezzani
Konrad Völkel (Freiburg)
Thomas Weißschuh (Mainz)
Gisbert Wüstholz

The participants consist of PhD students and postdocs in arithmetic and geometry from Zurich as well as colleagues from Bonn, Freiburg, Graz, Lausanne, Mainz, Moscow, Pisa, Salzburg and Warwick.

Program

The program will start on Sunday evening, the first talk is scheduled for 17:15-18:45. The detailed program of the planned talks will be published here: Program-2014

Sunday, 06.07.2014: Arrival day - Introduction and overview
Introductory talk (Fresan)
Monday: The periods mod bar{Q}^*
Deligne's theorem on absolute Hodge cycles on abelian varities (Pepin-Lehallleur); Andersons's period distribution, Shimura's monomial relations and the gamma distribution; Fermat curves (Vaish), proof of Anderson's theorem (Preisinger/Samol/Weißschuh)
Tuesday: Formal groups
Commutative formal groups, the Dieudonne and Tate modules of a commutative formal group, Tate's results (Vezzani); Lubin-Tate formal groups (Ferraguti); CM formal groups, the main structure theorem (Paulin/Hutle/Veneziano)
Wednesday: The p-adic periods of CM formal groups
p-adic periods for CM formal groups (Gallauer)
Possible afternoon program: Informal discussion, excursions, hiking tours
Thursday: Colmez's period distribution and formulation of the main theorem
Crystalline cohomology, the action of the crystalline Weil group (Harrer/Rottmaier/Schmidtke); p-adic absolute Hodge cycles (Jossen); the period distribution ht, the main result, link with the Faltings height (Javanpeykar)
Friday, 11.07.2014: Fermat curves, completion of the proof and outlook
Colemans's result on the action of the crystalline Weil group on the cohomology of Fermat curves, completion of the proof (Bruin); complements: the motivic setting (Gorchinskiy); complements: the logarithmic derivatives of Dirichlet L-functions at negative integers (Maillot)

(Tentative) Schedule:
09:00 - 10:30: First talk
10:30 - 11:00: Coffee break
11:00 - 12:30: Second talk
12:45: Lunch
13:45 - 15:15: Third talk
15:30 - 18:00: Time for informal discussion
19:00: Dinner

Lecture Notes and Literature

Program written by Damian Rössler (version from April 7, 2014): Program-2014
Suitable references can be found here.

 

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