[Turkmath:7031] Motivic Themes from Algebraic Geometry, June 8 - 13, 2010

[Kursat Aker]

  Motivic Themes from Algebraic Geometry


      June 8 - 13, 2010

      TÜBI.TAK - FEZA GÜRSEY INSTITUTE

*Lectures:*

    * *Hélène Esnault*, Universität Duisburg-Essen

      /Rational points on rationally connected varieties: motivic
      aspects/ (/four/ 90-minute lectures)

         1. Theorem of Chevalley-Warning
         2. Etale cohomology, trace formula
         3. Chow groups, some of the motivic conjectures
         4. Rational points on Fanos over finite fields.

      *References:*

          o Antoine Chambert-Loir, Points rationnels et groupes
            fondamentaux : applications de la cohomologie p-adique,
            math.AG/0303052
            http://front.math.ucdavis.edu/math.AG/0303052

    * *Johannes Nicaise*, Katholieke Universiteit Leuven

      /Construction of Néron models/ (/four/ 90-minute lectures)

      Let K be a complete discretely valued field with algebraically
      closed residue field k, and X a smooth and proper K-variety. In
      general, one cannot hope to find a smooth and proper model for X
      over the valuation ring R of K, but we can replace properness by a
      weaker notion that only involves the rational points on X. A weak
      Néron model for X is a smooth model Y for X over the valuation
      ring R of K such that every K-rational point on X extends to an
      R-section on Y. Such a weak Néron model always exists. Starting
      with an arbitrary proper R-model for X, we can construct a weak
      Néron model Y by a very elegant canonical smoothing process.

      Since Y is smooth over R, its special fiber is a good measure for
      the set of K-points on X. Using motivic integration, Loeser and
      Sebag have shown that the class of this special fiber in an
      appropriate Grothendieck ring is independent of the weak Néron
      model. This class is called the motivic Serre invariant of X.
      Under certain conditions, it admits a cohomological interpretation
      in terms of the Galois action on the étale cohomology of X, which
      is analogous to the Grothendieck-Lefschetz-Verdier trace formula
      for varieties over a finite field.

      In the lectures, we will discuss the following topics:

          o The construction of weak Néron models, and their relation
            with Néron models of abelian varieties.
          o The definition of the motivic Serre invariant via motivic
            integration.
          o The trace formula for the motivic Serre invariant.

      *References:*

          o The main reference for the theory of weak Néron models is
            S. Bosch, W. Lütkebohmert, M. Raynaud: "Néron models"
            (especially Chapter 3).
          o For the part about the motivic Serre invariant and the trace
            formula, one can look at
            F. Loeser and J. Sebag: "Motivic integration on smooth rigid
            varieties and invariants of degenerations" (Duke Math. J,
            2003).
          o J. Nicaise: "A trace formula for varieties over a discretely
            valued field" (to appear in J. Reine Angew. Math., arxiv:
            arXiv:0805.1323v2)

      *Requirements:* /No special background will be assumed except for
      a basic knowledge of the theory of schemes./

Accommodation (including breakfast, lunch and dinner) will be provided 
by TÜBI.TAK - Feza Gürsey Enstitüsü for participants from outside 
I.stanbul if needed.

Program is also featured on the Istanbul Mathematical Agenda: 
http://www.google.com/calendar/embed?src=jdf754c331751cbt6q9vc281es%40group.calendar.google.com&ctz=Europe/Istanbul 
<http://www.google.com/calendar/embed?src=jdf754c331751cbt6q9vc281es%40group.calendar.google.com&ctz=Europe/Istanbul> 


All participants are encouraged to fill in the following application 
form. Filling in the form is essential for the TÜBI.TAK - FEZA GÜRSEY 
INSTITUTE to provide the best service for all participants.

*Number of participants is limited to _25_ people.*

*Deadline:* May 14, 2010

*To Apply:* http://www.gursey.gov.tr/apps/app-frm-gen.php?id=motivic1006

*Organizers:*
Sinan Ünver (Koç Üniversitesi),
Kürs,at Aker (TÜBI.TAK - Feza Gürsey Enstitüsü).

*Contact:* motivic1006[]gursey.gov.tr

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