Mathematical Sciences Research Institute - Cohomological Approaches to Rational Points

This workshop will explore the various ways in which cohomology gives information on the existence and density of rational points. Here are the main topics to be covered. Étale cohomology of varieties over number fields: the Brauer-Manin obstruction and beyond. Étale cohomology of varieties over finite fields: as an example, Esnault's result that smooth rationally connected varieties over a finite field have a rational point. Coherent cohomology, deformation methods: applications to varieties over function fields in one or two variables over the complex field, also to varieties over finite and local fields. Arakelov geometry, algebraic geometry over the algebraic closure of the rationals, connexions with hyperbolicity, distribution of algebraic points of small height.
Workshop Schedule (Abstracts of talks)
Monday, March 27
9:00-9:30	Welcome by David Eisenbud and Staff
9:30-10:30	Minhyong Kim (University of Arizona and Purdue University): Non-abelian cohomology varieties and Diophantine geometry
10:30-11:00	Tea break.
11:00-12:00	Gaël Rémond (Institut Fourier, Grenoble) : On a conjecture of Pink
12:00-2:15	Lunch break.
2:15-3:15	Shou-Wu Zhang (Columbia University): Distributions in arithmetic dynamics
3:15-3:45	Tea break.
3:45-4:45	Carlo Gasbarri (Università Tor Vegata, Roma): Dyson’s Theorem for the product of two curves

Tuesday, March 28
9:30-10:30	Hélène Esnault (Universität Duisburg-Essen): Rational points over finite fields and cohomological methods
10:30-11:00	Tea break.
11:00-12:00	János Kollár (Princeton University): A conjecture of Ax and degenerations of Fano varieties
12:00-2:15	Lunch break.
2:15-3:15	Jason Starr (Massachussetts Institute of Technology): Rational 1-connectedness and rational points
3:15-3:45	Tea break.
3:45-4:45	Max Lieblich (Princeton University): Period and index in the Brauer group of a relative curve

Wednesday, March 29
9:30-10:30	Shuji Saito (University of Tokyo): Weak Bloch-Beilinson conjecture for zero cycles over p-adic fields. (joint work with Kanetomo SATO)
10:30-11:00	Tea break.
11:00-12:00	Alexander Merkurjev (University of California at Los Angeles): Zero-cycles on homogeneous varieties
12:00-2:15	Lunch break.
2:15-3:15	Emmanuel Peyre (Université Joseph Fourier (Grenoble) and MSRI): Height zeta functions : an arithmetic-motivic dictionary
3:15-3:45	Tea break.
3:45-4:45	David Bourqui (IRMAR, Université Rennes I): Height zeta functions : the case of toric varieties
5:00-6:00	Reception.

Thursday, March 30
9:30-10:30	Michael Stoll (International University Bremen): Finite descent obstructions and rational points
10:30-11:00	Tea break
11:00-12:00	Olivier Wittenberg (Université Paris-Sud and MSRI): Rational points on pencils of curves of genus 1
12:00-2:15	Lunch break
2:15-3:15	Philippe Gille (CNRS, Université Paris-Sud): Rational points on homogeneous spaces : a survey
3:15-3:45	Tea break
3:45-4:45	David Harari (Université Paris-Sud): 1-motives and principal homogeneous spaces of algebraic groups

Friday, March 31
9:30-10:30	Alexei Skorobogatov (Imperial College, London and MSRI): The elementary obstruction for homogeneous spaces
10:30-11:00	Tea break
11:00-12:00	Karl Rubin (University of California at Irvine and MSRI): Growth of ranks of abelian varieties in dihedral extensions

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Currently Available Videos

Minhyong Kim , Non-abelian Cohomology Varieties and Diophantine Geometry March 27,2006, 09:30 AM to 10:30 AM
Gaël Rémond , On a Conjecture of Pink March 27,2006, 11:00 AM to 12:00 PM
Shou-Wu Zhang , Distributions in Arithmetic Dynamics March 27,2006, 02:15 PM to 03:15 PM
Carlo Gasbarri , Dyson’s Theorem for the Product of Two Curves March 27,2006, 03:45 PM to 04:45 PM
Helene Esnault , Rational Points over Finite Fields and Cohomological Methods March 28,2006, 09:30 AM to 10:30 AM
János Kollár , A Conjecture of Ax and Degenerations of Fano Varieties March 28,2006, 11:00 AM to 12:00 PM
Jason Starr , Rational 1-connectedness and Rational Points March 28,2006, 02:15 PM to 03:15 PM
Max Lieblich , Period and Index in the Brauer Group of a Relative Curve March 28,2006, 03:45 PM to 04:45 PM
Shuji Saito , Weak Bloch-Beilinson Conjecture for Zero Cycles over p-adic Fields. (joint work with Kanetomo March 29,2006, 09:30 AM to 10:30 AM
Alexander Merkurjev , Zero-cycles on Homogeneous Varieties March 29,2006, 11:00 AM to 12:00 PM
Emmanuel Peyre , Height Zeta Functions : An Arithmetic-motivic Dictionary March 29,2006, 02:15 PM to 03:15 PM
David Bourqui , Height Zeta Functions : The Case of Toric Varieties March 29,2006, 03:45 PM to 04:45 PM
Michael Stoll , Finite Descent Obstructions and Rational Points March 30,2006, 09:30 AM to 10:30 AM
Olivier Wittenberg , Rational Points on Pencils of Curves of Genus 1 March 30,2006, 11:00 AM to 12:00 PM
Philippe Gille , Rational Points on Homogeneous Spaces : A Survey March 30,2006, 02:15 PM to 03:15 PM
David Harari , 1-Motives and Principal Homogeneous Spaces of Algebraic Groups March 30,2006, 03:45 PM to 04:45 PM
Alexei Skorobogatov , The Elementary Obstruction for Homogeneous Spaces March 31,2006, 09:30 AM to 10:30 AM
Karl Rubin , Growth of Ranks of Abelian Varieties in Dihedral Extensions March 31,2006, 11:00 AM to 12:00 PM

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For more information:
Questions about this workshop should be sent either by email to

302@msri.org

or by regular mail to:

Cohomological Approaches to Rational Points Mathematical Sciences Research Institute 17 Gauss Way, Berkeley, CA 94720-5070. USA

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