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<h1>Motivic Themes from Algebraic Geometry</h1>
<h3>June 7 - 13, 2010<br>
<br>
TÜBİTAK - FEZA GÜRSEY INSTITUTE</h3>
<p><b>Lectures:</b>
</p>
<ul>
<li><b>Hélène Esnault</b>, Universität Duisburg-Essen
<p><i>Rational points on rationally connected varieties: motivic
aspects</i> (<i>four</i> 90-minute lectures)
</p>
<ol>
<li>Theorem of Chevalley-Warning
</li>
<li>Etale cohomology, trace formula
</li>
<li>Chow groups, some of the motivic conjectures
</li>
<li>Rational points on Fanos over finite fields.</li>
</ol>
<p><b>References:</b>
</p>
<ul>
<li>Antoine Chambert-Loir, Points rationnels et groupes
fondamentaux : applications de la cohomologie <span class="typeset"><nobr><span
class="scale"><span class="icmmi10">p</span></span></nobr></span>-adique,
math.AG/0303052<br>
<a href="http://front.math.ucdavis.edu/math.AG/0303052">http://front.math.ucdavis.edu/math.AG/0303052</a>
</li>
</ul>
<br>
</li>
<li><b>Johannes Nicaise</b>, Katholieke Universiteit Leuven
<p><i>Construction of Néron models</i> (<i>four</i> 90-minute
lectures)
</p>
<p>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.
</p>
<p>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.
</p>
<p>In the lectures, we will discuss the following topics:
</p>
<ul>
<li>The construction of weak Néron models, and their relation
with Néron
models of abelian varieties.
</li>
<li>The definition of the motivic Serre invariant via motivic
integration.
</li>
<li>The trace formula for the motivic Serre invariant.</li>
</ul>
<p><b>References:</b>
</p>
<ul>
<li>The main reference for the theory of weak Néron models is
<br>
S. Bosch, W. Lütkebohmert, M. Raynaud: "Néron models" (especially
Chapter 3). </li>
<li>For the part about the motivic Serre invariant and the trace
formula, one can look at
<br>
F. Loeser and J. Sebag: "Motivic integration on smooth rigid varieties
and invariants of degenerations<http: www.math.ens.fr="" %7eloeser=""
dmj11902_4.pdf="">" (Duke Math. J, 2003).
</http:></li>
<li>J. Nicaise: "A trace formula for varieties over a discretely
valued field"
(to appear in J. Reine Angew. Math., arxiv: arXiv:0805.1323v2<http:
arxiv.org="" abs="" 0805.1323v2="">)
</http:></li>
</ul>
<p><b>Requirements:</b> <i>No special background will be assumed
except for a basic knowledge of
the theory of schemes.</i>
</p>
</li>
<li><b>Fabien Trihan</b>*, University of Nottingham
<p></p>
<ol>
<li><i>On the Birch and Swinnerton-Dyer conjecture over function
fields</i> [Kato-T]
<p>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.
</p>
</li>
<li><i>The parity conjecture over function fields</i>
[T-Wuthrich]
<p>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.
</p>
</li>
<li><i>The (non-commutative) Iwasawa Main Conjecture</i>
<p>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.
</p>
</li>
</ol>
<p><b>References:</b>
</p>
<ul>
<li>Coates, John ; Fukaya, Takako ; Kato,
Kazuya ; Sujatha, Ramdorai ; Venjakob, Otmar . The <span
class="typeset"><nobr><span class="scale"><span class="icmmi10">G</span><span
class="icmmi10">L</span><span style="position: relative; top: 0.27em;"><span
class="size2"><span class="icmr10">2</span></span><span class="spacer"
style="margin-left: 0.05em;"></span></span></span></nobr></span>
main conjecture for elliptic curves without complex multiplication.
Publ. Math. Inst. Hautes Études Sci. No. 101 (2005), 163--208.
</li>
<li>Kato, Kazuya ; Trihan, Fabien . On the
conjectures of Birch and Swinnerton-Dyer in characteristic <span
class="typeset"><nobr><span class="scale"><span class="icmmi10">p</span><span
style="position: relative; margin-left: 0.277em;"><img
src="cid:part1.03040908.00080106@gursey.gov.tr"
style="height: 11px; width: 12px; vertical-align: -1px; margin-right: 0.059em;"></span><span
style="position: relative; margin-left: 0.277em;"><span class="icmr10">0</span></span></span></nobr></span>.
Invent. Math. 153 (2003), no. 3, 537--592.
</li>
</ul>
<br>
</li>
<li><b>Sinan Ünver</b>, Koç Üniversitesi
<p><i>Introduction to Rational points on rationally connected
varieties</i> (<i>three</i> 60-minute lectures) </p>
<p><b>References:</b>
</p>
<ul>
<li>Antoine Chambert-Loir, Points rationnels et groupes
fondamentaux : applications de la cohomologie <span class="typeset"><nobr><span
class="scale"><span class="icmmi10">p</span></span></nobr></span>-adique,
math.AG/0303052<br>
<a href="http://front.math.ucdavis.edu/math.AG/0303052">http://front.math.ucdavis.edu/math.AG/0303052</a>
</li>
</ul>
<br>
</li>
<li><b>Şafak Özden</b>, Mimar Sinan Güzel Sanatlar Üniversitesi
<p><i>Introduction to Rigid Analytic Geometry</i> (<i>two</i>
60-minute lectures)
</p>
<p><b>References:</b>
</p>
<ul>
<li>S. Bosch, U. Güntzer , R. Remmert: Non-Archimedean Analysis:
A Systematic Approach to Rigid Analytic Geometry </li>
</ul>
</li>
</ul>
<p><b>(*)</b> Supported by TÜBİTAK Grant No. 107T897 Matematik
İşbirliği Ağı: Cebir ve Uygulamaları.
<!-- <p><a href="algeo0310program.pdf" target="_blank">Daily programme as a PDF file.</a> --></p>
<p>Accommodation (including breakfast, lunch and dinner) will be
provided by TÜBİTAK - Feza Gürsey Enstitüsü for participants from
outside İstanbul if needed. </p>
<p>Program is also featured on the Istanbul Mathematical Agenda:
<a target="_blank"
href="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</a>
</p>
<p>All participants are encouraged to fill in the following application
form. Filling in the form is essential for the TÜBİTAK - FEZA GÜRSEY
INSTITUTE to provide the best service for all participants.<!-- <p>If you would like to be accommodated at FGI please apply to the sister programme:<br/> --><!-- <a href="http://www.gursey.gov.tr/new/mathphd1005/" target="_blank">Matematik Lisansüstü Öğrencileri Seminerleri</a> (Math Postgraduate Seminars) --> </p>
<p><b>Number of participants is limited to <u>25</u> people.</b>
</p>
<p><b>Deadline:</b> May 21, 2010
</p>
<p><b>To Apply:</b> <a
href="http://www.gursey.gov.tr/apps/app-frm-gen.php?id=motivic1006">http://www.gursey.gov.tr/apps/app-frm-gen.php?id=motivic1006</a><br>
</p>
<p><b>Web site:</b> <a class="moz-txt-link-freetext" href="http://www.gursey.gov.tr/new/motivic1006/">http://www.gursey.gov.tr/new/motivic1006/</a><br>
</p>
<p><b>Program: </b><a class="moz-txt-link-freetext" href="http://www.gursey.gov.tr/new/motivic1006/motivic1006program.pdf?m=prog">http://www.gursey.gov.tr/new/motivic1006/motivic1006program.pdf?m=prog</a><br>
</p>
<p>
</p>
<p><b>Organizers:</b> <br>
Sinan Ünver (Koç Üniversitesi),
<br>
Kürşat Aker (TÜBİTAK - Feza Gürsey Enstitüsü).
</p>
<p><b>Contact:</b> motivic1006[]gursey.gov.tr<br>
</p>
<br>
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