[Turkmath:7079] Motivic Themes from Algebraic Geometry, June 7 - 13, 2010, TÜBİTAK - FEZA GÜRSEY INSTITUTE

1 Haz 2010 Sal 14:35:45 EEST

  Motivic Themes from Algebraic Geometry

      June 7 - 13, 2010

      TÃœBÄ°TAK - FEZA GÃœRSEY INSTITUTE

*Lectures:*

    * *HÃ©lÃ¨ne Esnault*, UniversitÃ¤t 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 NÃ©ron 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
      NÃ©ron 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 NÃ©ron model always exists. Starting
      with an arbitrary proper R-model for X, we can construct a weak
      NÃ©ron 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 NÃ©ron
      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 Ã©tale 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 NÃ©ron models, and their relation
            with NÃ©ron 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 NÃ©ron models is
            S. Bosch, W. LÃ¼tkebohmert, M. Raynaud: "NÃ©ron 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 Ã‰tudes 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 Ãœnver*, KoÃ§ Ãœniversitesi

      /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

    * *Åafak Ã–zden*, Mimar Sinan GÃ¼zel Sanatlar Ãœniversitesi

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

      *References:*

          o S. Bosch, U. GÃ¼ntzer , R. Remmert: Non-Archimedean Analysis:
            A Systematic Approach to Rigid Analytic Geometry

*(*)* Supported by TÃœBÄ°TAK Grant No. 107T897 Matematik Ä°ÅŸbirliÄŸi AÄŸÄ±: 
Cebir ve UygulamalarÄ±.

Accommodation (including breakfast, lunch and dinner) will be provided 
by TÃœBÄ°TAK - Feza GÃ¼rsey EnstitÃ¼sÃ¼ 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 Ãœnver (KoÃ§ Ãœniversitesi),
KÃ¼rÅŸat Aker (TÃœBÄ°TAK - Feza GÃ¼rsey EnstitÃ¼sÃ¼).

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

-------------- sonraki bölüm --------------
Bir HTML eklentisi temizlendi...
URL: http://yunus.listweb.bilkent.edu.tr/pipermail/turkmath/attachments/20100601/57884b96/attachment.htm 
-------------- sonraki bölüm --------------
YazÄ± olmayan bir eklenti temizlendi...
Ä°sim: char3E.png
TÃ¼r: image/png
Boyut: 175 bayt
TanÄ±m: kullanÄ±lamÄ±yor
Url: http://yunus.listweb.bilkent.edu.tr/pipermail/turkmath/attachments/20100601/57884b96/attachment.png 