[Turkmath:7103] [FGE-Duyuru] FINAL DAY/CHANGE OF PROGRAMME: Motivic Themes from Algebraic Geometry, June 7 - 13, 2010, TÜBÝTAK - FEZA GÜRSEY INSTITUTE

[Kursat Aker]

*

IMPORTANT: Change of programme for the Final Day of Motivic Themes from 
Algebraic Geometry is attached.*


  Motivic Themes from Algebraic Geometry


      June 7 - 13, 2010

      TÃœBÄ°TAK - FEZA GÃœRSEY 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./

    * *Fabien Trihan**, University of Nottingham
         1. /On the Birch and Swinnerton-Dyer conjecture over function
            fields/ [Kato-T]

            We prove that if for some prime l, the l-primary part of the
            Tate-Shafarevich group is finite then the conjecture of BSD
            over function fields holds.

         2. /The parity conjecture over function fields/ [T-Wuthrich]

            We prove that for any elliptic curve over a function field
            of characteristic p>2, the p-corank of the Selmer group of E
            and the analytic rank of E have the same parity.

         3. /The (non-commutative) Iwasawa Main Conjecture/

            We give an analogue of the non-commutative Iwasawa Main
            conjecture of [Coates-Fukaya-Kato-Sujatha-Venjakob] for
            semi-stable abelian varieties over unramified towers.

      *References:*

          o Coates, John ; Fukaya, Takako ; Kato, Kazuya ; Sujatha,
            Ramdorai ; Venjakob, Otmar . The GL2 main conjecture for
            elliptic curves without complex multiplication. Publ. Math.
            Inst. Hautes Études Sci. No. 101 (2005), 163--208.
          o Kato, Kazuya ; Trihan, Fabien . On the conjectures of Birch
            and Swinnerton-Dyer in characteristic p0. Invent. Math. 153
            (2003), no. 3, 537--592.

    * *Sinan Ünver*, Koç Üniversitesi

      /Introduction to Rational points on rationally connected
      varieties/ (/three/ 60-minute lectures)

      *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

    * *Şafak Özden*, Mimar Sinan Güzel Sanatlar Üniversitesi

      /Introduction to Rigid Analytic Geometry/ (/two/ 60-minute lectures)

      *References:*

          o S. Bosch, U. Güntzer , R. Remmert: Non-Archimedean Analysis:
            A Systematic Approach to Rigid Analytic Geometry

*(*)* Supported by TÜBİTAK Grant No. 107T897 Matematik İşbirliği Ağı: 
Cebir ve Uygulamaları.

Accommodation (including breakfast, lunch and dinner) will be provided 
by TÜBİTAK - Feza Gürsey Enstitüsü for participants from outside 
Ä°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 TÃœBÄ°TAK - FEZA GÃœRSEY 
INSTITUTE to provide the best service for all participants.

*Number of participants is limited to _25_ people.*

*Deadline:* May 21, 2010

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

*Web site:* http://www.gursey.gov.tr/new/motivic1006/

*Program: 
*http://www.gursey.gov.tr/new/motivic1006/motivic1006program.pdf?m=prog

*Organizers:*
Sinan Ünver (Koç Üniversitesi),
Kürşat Aker (TÜBİTAK - Feza Gürsey Enstitüsü).

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



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