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<p class="MsoTitle"><span
style="font-size:18.0pt;font-family:"Times New
Roman","serif"">SEMINARIO
DI <span style="mso-no-proof:yes">MATEMATICA</span><o:p></o:p></span></p>
<p class="MsoTitle"><span
style="font-size:10.0pt;font-family:"Times New
Roman","serif""><o:p><span
style="text-decoration:none"> </span></o:p></span></p>
<h2><span style="font-family:"Times New
Roman","serif";mso-no-proof:yes">Mercoledì
10 ottobre 2012</span><span style="font-family:"Times New
Roman","serif""><o:p></o:p></span></h2>
<p class="MsoNormal" style="margin-left:9.0pt;text-align:center;
line-height:18.0pt;tab-stops:324.0pt" align="center"><span
style="font-size:16.0pt;font-family:
"Times New Roman","serif"">Ore <span
style="mso-no-proof:yes">15.00</span><o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:9.0pt;text-align:center;
line-height:18.0pt;tab-stops:324.0pt" align="center"><span
style="font-size:16.0pt;font-family:
"Times New Roman","serif""><o:p> </o:p></span></p>
<p class="MsoNormal" style="margin-left:9.0pt;text-align:center;
line-height:18.0pt;tab-stops:324.0pt" align="center"><u><span
style="font-size:16.0pt;
font-family:"Times New Roman","serif"">Scuola
Normale Superiore<o:p></o:p></span></u></p>
<p class="MsoNormal" style="margin-left:9.0pt;text-align:center;
line-height:18.0pt;tab-stops:324.0pt" align="center"><span
style="font-size:16.0pt;font-family:
"Times New Roman","serif"">Pisa<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:9.0pt;text-align:center;
line-height:18.0pt;tab-stops:324.0pt" align="center"><span
style="font-size:16.0pt;font-family:
"Times New Roman","serif"">Aula <span
style="mso-no-proof:yes">Mancini</span><o:p></o:p></span></p>
<p class="MsoNormal" style="tab-stops:324.0pt"><b><u><span
style="font-size:10.0pt;
font-family:"Times New Roman","serif""><o:p><span
style="text-decoration:none"> </span></o:p></span></u></b></p>
<p class="MsoNormal" style="text-align:center" align="center"><b
style="mso-bidi-font-weight:
normal"><span style="font-size:26.0pt;font-family:"Times
New Roman","serif"">Niels
Borne<i style="mso-bidi-font-style:normal"><span
style="mso-no-proof:yes"><o:p></o:p></span></i></span></b></p>
<p class="MsoNormal" style="text-align:center" align="center"><i
style="mso-bidi-font-style:
normal"><span style="font-family:"Times New
Roman","serif"">Université de Lille
1<o:p></o:p></span></i></p>
<p class="MsoNormal"><span style="font-family:"Times New
Roman","serif""><o:p> </o:p></span></p>
<p class="MsoNormal" style="text-align:center;tab-stops:324.0pt"
align="center"><span
style="font-size:16.0pt;font-family:"Times New
Roman","serif"">Terrà un
seminario dal titolo:<o:p></o:p></span></p>
<p class="MsoNormal" style="tab-stops:324.0pt"><span
style="font-family:"Times New
Roman","serif""><o:p> </o:p></span></p>
<pre style="text-align:center"><b style="mso-bidi-font-weight:normal"><i style="mso-bidi-font-style:normal"><span style="font-size:26.0pt;font-family:
"Times New Roman","serif"">“</span></i></b><b style="mso-bidi-font-weight:normal"><span style="font-size:26.0pt;font-family:"Times New Roman","serif";mso-bidi-font-family:
"Courier New"">Lifting sections along torsors</span></b><b style="mso-bidi-font-weight:
normal"><i style="mso-bidi-font-style:normal"><span style="font-size:26.0pt;
font-family:"Times New Roman","serif"">”</span></i></b><b style="mso-bidi-font-weight:
normal"><span style="font-size:26.0pt;font-family:"Times New Roman","serif""><o:p></o:p></span></b></pre>
<p class="MsoNormal" style="text-align:justify"><b
style="mso-bidi-font-weight:
normal"><i style="mso-bidi-font-style:normal"><span
style="font-size:10.0pt;
font-family:"Times New Roman","serif""><o:p> </o:p></span></i></b></p>
<p class="MsoNormal" style="text-align:justify"><b
style="mso-bidi-font-weight:
normal"><i style="mso-bidi-font-style:normal"><span
style="font-size:10.0pt;
font-family:"Times New Roman","serif""><o:p> </o:p></span></i></b></p>
<p class="MsoNormal" style="text-align:justify"><b
style="mso-bidi-font-weight:
normal"><i style="mso-bidi-font-style:normal"><span
style="font-size:10.0pt;
font-family:"Times New Roman","serif"">Abstract:<o:p></o:p></span></i></b></p>
<p class="MsoBodyText"><i style="mso-bidi-font-style:normal"><span
style="mso-bidi-font-size:12.0pt;font-family:"Times New
Roman","serif"" lang="EN-IE">Let k
be a number field and X/k a proper hyperbolic curve. Each
rational point gives
a section of the morphism of étale fundamental groups π</span></i><i
style="mso-bidi-font-style:normal"><span
style="mso-bidi-font-size:
12.0pt;font-family:"Times New
Roman","serif";position:relative;top:2.5pt;
mso-text-raise:-2.5pt;mso-font-kerning:7.0pt" lang="EN-IE">1</span></i><i
style="mso-bidi-font-style:
normal"><span
style="mso-bidi-font-size:12.0pt;font-family:"Times New
Roman","serif"" lang="EN-IE">(X)
Gal</span></i><i style="mso-bidi-font-style:normal"><span
style="mso-bidi-font-size:12.0pt;font-family:"Times New
Roman","serif";
position:relative;top:2.5pt;mso-text-raise:-2.5pt;mso-font-kerning:7.0pt"
lang="EN-IE">k</span></i><i style="mso-bidi-font-style:normal"><span
style="mso-bidi-font-size:
12.0pt;font-family:"Times New
Roman","serif"" lang="EN-IE">. Grothendieck’s
section
conjecture predicts that this gives a bijection between the
set of rational
points and the set of such sections, up to conjugacy. A
stunning consequence of
a refined version of this conjecture is that for each open
subset U of X, a
section of π</span></i><i style="mso-bidi-font-style:normal"><span
style="mso-bidi-font-size:12.0pt;font-family:"Times New
Roman","serif";
position:relative;top:2.5pt;mso-text-raise:-2.5pt;mso-font-kerning:7.0pt"
lang="EN-IE">1</span></i><i style="mso-bidi-font-style:normal"><span
style="mso-bidi-font-size:
12.0pt;font-family:"Times New
Roman","serif"" lang="EN-IE">(X) </span></i><!--[if gte msEquation 12]><m:oMath><i
style='mso-bidi-font-style:normal'><span lang=EN-IE style='mso-bidi-font-size:
12.0pt;font-family:"Cambria Math","serif"'><m:r>⟶</m:r></span></i></m:oMath><![endif]--><!--[if !msEquation]--><span
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src="cid:part1.08060005.00000905@sns.it"
v:shapes="_x0000_i1025" height="19" width="19"><!--[endif]--></span><!--[endif]--><i
style="mso-bidi-font-style:
normal"><span
style="mso-bidi-font-size:12.0pt;font-family:"Times New
Roman","serif"" lang="EN-IE"><span
style="mso-spacerun:yes"> </span>Gal</span></i><i
style="mso-bidi-font-style:
normal"><span
style="mso-bidi-font-size:12.0pt;font-family:"Times New
Roman","serif";
position:relative;top:2.5pt;mso-text-raise:-2.5pt;mso-font-kerning:7.0pt"
lang="EN-IE">k</span></i><i style="mso-bidi-font-style:normal"><span
style="mso-bidi-font-size:
12.0pt;font-family:"Times New
Roman","serif"" lang="EN-IE"><span
style="mso-spacerun:yes"> </span>lifts to a section of π</span></i><i
style="mso-bidi-font-style:normal"><span
style="mso-bidi-font-size:
12.0pt;font-family:"Times New
Roman","serif";position:relative;top:2.5pt;
mso-text-raise:-2.5pt;mso-font-kerning:7.0pt" lang="EN-IE">1</span></i><i
style="mso-bidi-font-style:
normal"><span
style="mso-bidi-font-size:12.0pt;font-family:"Times New
Roman","serif"" lang="EN-IE">(U)</span></i><!--[if gte msEquation 12]><m:oMath><i
style='mso-bidi-font-style:normal'><span lang=EN-IE style='mso-bidi-font-size:
12.0pt;font-family:"Cambria Math","serif"'><m:r> ⟶</m:r></span></i></m:oMath><![endif]--><!--[if !msEquation]--><span
style="font-size:11.0pt;line-height:115%;font-family:"Calibri","sans-serif";
mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times
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position:relative;top:3.0pt;mso-text-raise:-3.0pt;mso-ansi-language:IT;
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o:title="" chromakey="white"/>
</v:shape><![endif]--><!--[if !vml]--><img
src="cid:part2.01030102.01000000@sns.it"
v:shapes="_x0000_i1025" height="19" width="22"><!--[endif]--></span><!--[endif]--><i
style="mso-bidi-font-style:
normal"><span
style="mso-bidi-font-size:12.0pt;font-family:"Times New
Roman","serif"" lang="EN-IE"><span
style="mso-spacerun:yes"> </span>Gal</span></i><i
style="mso-bidi-font-style:
normal"><span
style="mso-bidi-font-size:12.0pt;font-family:"Times New
Roman","serif";
position:relative;top:2.5pt;mso-text-raise:-2.5pt;mso-font-kerning:7.0pt"
lang="EN-IE">k</span></i><i style="mso-bidi-font-style:normal"><span
style="mso-bidi-font-size:
12.0pt;font-family:"Times New
Roman","serif"" lang="EN-IE">. This
consequence, the
cuspidalization conjecture, is also part of various strategies
to attack the
section conjecture. As a first approach, one can study the
lifting of a section
of π</span></i><i style="mso-bidi-font-style:normal"><span
style="mso-bidi-font-size:12.0pt;font-family:"Times New
Roman","serif";
position:relative;top:2.5pt;mso-text-raise:-2.5pt;mso-font-kerning:7.0pt"
lang="EN-IE">1</span></i><i style="mso-bidi-font-style:normal"><span
style="mso-bidi-font-size:
12.0pt;font-family:"Times New
Roman","serif"" lang="EN-IE">(X) </span></i><!--[if gte msEquation 12]><m:oMath><i
style='mso-bidi-font-style:normal'><span lang=EN-IE style='mso-bidi-font-size:
12.0pt;font-family:"Cambria Math","serif"'><m:r>⟶</m:r></span></i></m:oMath><![endif]--><!--[if !msEquation]--><span
style="font-size:11.0pt;line-height:115%;font-family:"Calibri","sans-serif";
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mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin;
mso-bidi-font-family:"Times
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position:relative;top:3.0pt;mso-text-raise:-3.0pt;mso-ansi-language:IT;
mso-fareast-language:IT;mso-bidi-language:AR-SA"><!--[if gte vml 1]><v:shape
id="_x0000_i1025" type="#_x0000_t75" style='width:14.25pt;height:14.25pt'>
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</v:shape><![endif]--><!--[if !vml]--><img
src="cid:part1.08060005.00000905@sns.it"
v:shapes="_x0000_i1025" height="19" width="19"><!--[endif]--></span><!--[endif]--><i
style="mso-bidi-font-style:
normal"><span
style="mso-bidi-font-size:12.0pt;font-family:"Times New
Roman","serif"" lang="EN-IE"><span
style="mso-spacerun:yes"> </span>Gal</span></i><i
style="mso-bidi-font-style:
normal"><span
style="mso-bidi-font-size:12.0pt;font-family:"Times New
Roman","serif";
position:relative;top:2.5pt;mso-text-raise:-2.5pt;mso-font-kerning:7.0pt"
lang="EN-IE">k</span></i><i style="mso-bidi-font-style:normal"><span
style="mso-bidi-font-size:
12.0pt;font-family:"Times New
Roman","serif"" lang="EN-IE"><span
style="mso-spacerun:yes"> </span>to a section of π</span></i><i
style="mso-bidi-font-style:normal"><span
style="mso-bidi-font-size:
12.0pt;font-family:"Times New
Roman","serif";position:relative;top:2.5pt;
mso-text-raise:-2.5pt;mso-font-kerning:7.0pt" lang="EN-IE">1</span></i><i
style="mso-bidi-font-style:
normal"><span
style="mso-bidi-font-size:12.0pt;font-family:"Times New
Roman","serif"" lang="EN-IE">(E)
</span></i><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:
normal'><span lang=EN-IE style='mso-bidi-font-size:12.0pt;font-family:"Cambria Math","serif"'><m:r>⟶</m:r></span></i></m:oMath><![endif]--><!--[if !msEquation]--><span
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position:relative;top:3.0pt;mso-text-raise:-3.0pt;mso-ansi-language:IT;
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</v:shape><![endif]--><!--[if !vml]--><img
src="cid:part1.08060005.00000905@sns.it"
v:shapes="_x0000_i1025" height="19" width="19"><!--[endif]--></span><!--[endif]--><i
style="mso-bidi-font-style:
normal"><span
style="mso-bidi-font-size:12.0pt;font-family:"Times New
Roman","serif"" lang="EN-IE"><span
style="mso-spacerun:yes"> </span>Gal</span></i><i
style="mso-bidi-font-style:
normal"><span
style="mso-bidi-font-size:12.0pt;font-family:"Times New
Roman","serif";
position:relative;top:2.5pt;mso-text-raise:-2.5pt;mso-font-kerning:7.0pt"
lang="EN-IE">k</span></i><i style="mso-bidi-font-style:normal"><span
style="mso-bidi-font-size:
12.0pt;font-family:"Times New
Roman","serif"" lang="EN-IE">, where E/X is a
natural torsor
under a torus, trivial over U. I will explain how to get
positive results in
this setting for very specific U and only when k = <b
style="mso-bidi-font-weight:
normal">Q</b>. This is joint work (in progress) with Michel
Emsalem and Jakob
Stix</span></i><span
style="font-size:10.0pt;font-family:"Times New
Roman","serif"" lang="EN-IE">.
</span><span lang="EN-IE"><o:p></o:p></span></p>
<p class="MsoNormal"><span style="mso-ansi-language:EN-US"
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style="font-size:10.0pt;font-family:"Times New
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EN-US" lang="EN-US"><o:p> </o:p></span></i></b></p>
<p class="MsoNormal" style="text-align:justify"><span
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gli interessati sono invitati a
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<pre class="moz-signature" cols="72">--
Valeria Giuliani
Scuola Normale Superiore
Area Didattica
Servizio Supporto attivita Didattiche
tel.050509260
Fax.050509045
<a class="moz-txt-link-abbreviated" href="mailto:v.giuliani@sns.it">v.giuliani@sns.it</a>
<a class="moz-txt-link-abbreviated" href="mailto:adi.sd@sns.it">adi.sd@sns.it</a></pre>
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