ON MANIN'S CONJECTURE FOR A FAMILY OF CHÂTELET SURFACES

Description

Niveau: Supérieur, Doctorat, Bac+8
ON MANIN'S CONJECTURE FOR A FAMILY OF CHÂTELET SURFACES Régis de la Bretèche, Tim Browning and Emmanuel Peyre Abstract. — The Manin conjecture is established for Châtelet surfaces over Q arising as minimal proper smooth models of the surface Y 2 + Z2 = f(X) in A3Q, where f ? Z[X] is a totally reducible polynomial of degree 3 without repeated roots. These surfaces do not satisfy weak approximation. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. A family of Châtelet surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3. Points of bounded height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4. Description of versal torsors . . . . . . . . . . . . . . . . . . . . . . . . . . .

• s2 ?


• a4 a2


• ?? ??1s ?


• thus gc


• oc ??


• ??1s


• ????


• ??

Sujets

• Châtelet
• Browning

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 16
• Langue: English

Extrait

FORAFOANMMILAYNIONF'SCCHOÂNTJEELCETTUSRUERFACES

RégisdelaBretèche,TimBrowningandEmmanuelPeyre

Abstract
.TheManinconjectureisestablishedforChâteletsurfacesover
Q
arisingas
minimalpropersmoothmodelsofthesurface
Y
2
+
Z
2
=
f
(
X
)
in
A
3
Q
,where
f
∈
Z
[
X
]
isatotallyreduciblepolynomialofdegree3withoutrepeatedroots.
Thesesurfacesdonotsatisfyweakapproximation.

Contents
1.Introduction..................................................1
2.AfamilyofChâteletsurfaces..................................2
3.Pointsofboundedheight......................................5
4.Descriptionofversaltorsors....................................6
5.Jumpingup..................................................12
6.Formulationofthecountingproblem............................25
7.Estimating
U
(
T
)
:anupperbound..............................27
8.Estimating
U
(
T
)
:anasymptoticformula........................29
9.Thedénouement..............................................34
10.Jumpingdown..............................................36
References......................................................40

1.Introduction
ThepurposeofthispaperistoproveManin'sconjectureaboutpointsofboundedheight
forafamilyofChâteletsurfacesover
Q
.ThesesurfaceshavebeenconsideredbyF.Châtelet
in[
Ch1
]and[
Ch2
],byV.A.Iskovskikh[
Is
],byD.CorayandM.A.Tsfasman[
CoTs
],and
2000
MathematicsSubjectClassication
.primary14E08;secondary11D45,12G05,14F43.
SupportedbyEPSRCgrantnumber
EP/E053262/1
andtheANRprojectPEPR.

2
RÉGISDELABRETÈCHE,TIMBROWNINGandEMMANUELPEYRE

byJ.-L.Colliot-Thélène,J.-J.Sansuc,andP.Swinnerton-Dyerin[
CTSSD1
]and[
CTSSD2
],
amongothers.
Thesurfacesconsideredherearesmoothpropermodelsoftheafnesurfacesgivenin
A
3
Q
byanequationoftheform
Y
2
+
Z
2
=
X
(
a
3
X
+
b
3
)(
a
4
X
+
b
4
)
,
forsuitable
a
3
,
b
3
,
a
4
,
b
4
∈
Z
.
Itisimportanttonotethatthesurfacesweconsiderdonotsatisfyweakapproximation,the
lackofwhichisexplainedbytheBrauer-Maninobstruction,asdescribedin[
CTSSD1
]and
[
CTSSD2
].Uptonow,theonlycasesforwhichManin'sprinciplewasprovendespiteweak
approximationnotholdingwereobtainedusingharmonicanalysisandrequiredtheaction
ofanalgebraicgrouponthevarietywithanopenorbit.Themethodusedinthispaperis
completelydifferent.FollowingideasofP.Salberger[
Sal
],weuseversaltorsorsintroduced
byColliot-ThélèneandSansucin[
CTS1
],[
CTS2
],and[
CTS3
]toestimatethenumberof
rationalpointsofboundedheightonthesurface.Suchacombinationofdescentmethods
withanalyticnumbertheorywasusedin[
HBS
]toprovethattheBrauer-Maninobstruction
toweakapproximationistheonlyoneforhypersurfacesrelatedtonormforms.Therefore
wecanreasonablyhopethatfurtherdevelopmentsofthesetechniquesmaybesuccessfulin
provingtherenedconjecturesofManinforothersuchvarieties.
Thispaperisorganisedasfollows:insection2,werecallsomefactsaboutthegeometry
ofthesurfaces.Insection3,wedenetheheightandstateourmainresult.Section4contains
thedescriptionoftheversaltorsorsweuse.Insection5,wedescribetheliftingofrational
pointstotheversaltorsors.Thisliftingreducestheinitialproblemtotheestimationofsome
arithmeticsumsdenotedby
U
(
T
)
.Thefollowingsectionscontainthekeyanalyticaltools
usedintheproof.Insection7wegiveauniformupperboundfor
U
(
T
)
andinsection8
anasymptoticformulaforit.Thelastsectionisdevotedtoaninterpretationoftheleading
constant.
Letusxsomenotationfortheremainderofthistext.
Notationandconventions
.If
k
isaeld,wedenoteby
k
analgebraicclosureof
k
.For
anyvariety
X
over
k
andany
k
-algebra
A
,wedenoteby
X
A
theproduct
X
×
Spec
(
k
)
Spec
(
A
)
andby
X
(
A
)
thesetHom
Spec
(
k
)
(
Spec
(
A
)
,X
)
.Wealsoput
X
=
X
k
.Thecohomological
Brauergroupof
X
isdenedasBr
(
X
)=
H
é2t
(
X,
G
m
)
,where
G
m
denotesthemultiplicative
group.Theprojectivespaceofdimension
n
over
A
isdenotedby
P
nA
andtheafnespaceby
A
nA
.Forany
(
x
0
,...,x
n
)
∈
k
n
+
1
{
0
}
wedenoteby
(
x
0
:

:
x
n
)
itsimagein
P
n
(
k
)
.
2.AfamilyofChâteletsurfaces
Letusx
a
1
,
a
2
,
a
3
,
a
4
,
b
1
,
b
2
,
b
3
,b
4
∈
Z
suchthat
aajiD
i,j
=

b
i
b
j

6
=
0
forany
i,j
∈{
1
,
2
,
3
,
4
}
with
i
6
=
j
.Wethenconsiderthelinearforms
L
i
denedby
L
i
(
U,V
)=
a
i
U
+
b
i
V
for
i
∈{
1
,
2
,
3
,
4
}
anddenethehypersurface
S
1
of
P
2
Q
×
A
1
Q
given

MANIN'SCONJECTUREFORCHÂTELETSURFACES

3

bytheequation
4X
2
+
Y
2
=
T
2
L
i
(
U,
1
)
Y1=iandthehypersurface
S
2
givenbytheequation
4X
′
2
+
Y
′
2
=
T
′
2
L
i
(
1
,V
)
.
Y1=iLet
U
1
betheopensubsetof
S
1
denedby
U
6
=
0and
U
2
betheopensubsetof
S
2
dened
by
V
6
=
0.Themap
F
:
U
1
→
U
2
whichmaps
((
X
:
Y
:
T
)
,U
)
onto
((
X
:
Y
:
U
2
T
)
,
1
/U
)
isanisomorphismandwedene
S
asthesurfaceobtainedbyglueing
S
1
to
S
2
usingthe
isomorphism
F
.Thesurface
S
isasmoothprojectivesurfaceandisaparticularcaseofa
Châteletsurface.ThegeometryofsuchsurfaceshasbeendescribedbyJ.-L.Colliot-Thélène,
J.-J.SansucandP.Swinnerton-Dyerin[
CTSSD2
,§7].Forthesakeofcompleteness,letus
recallpartofthisdescriptionwhichwillbeusefulforthedescriptionofversaltorsors.
Themaps
S
1
→
P
1
Q
(resp.
S
2
→
P
1
Q
)whichmaps
((
X
:
Y
:
T
)
,U
)
onto
(
U
:1
)
(resp.
((
X
′
:
Y
′
:
T
′
)
,V
)
onto
(
1:
V
)
)gluetogethertogiveaconicbration
π
:
S
→
P
1
Q
with
fourdegeneratebresoverthepointsgivenby
P
i
=(
−
b
i
:
a
i
)
∈
P
1
(
Q
)
for
i
∈{
1
,
2
,
3
,
4
}
.
Infact,theglueingof
P
2
Q
×
A
1
Q
to
P
2
Q
×
A
1
Q
throughthemap
(2.1)
((
X
:
Y
:
T
)
,U
)
7→
((
X
:
Y
:
U
2
T
)
,
1
/U
)
givestheprojectivebundle
(
1
)
P
=
P
(
O
2
⊕
O
(
−
2
))
over
P
1
Q
and
S
maybeseenasahyper-
surfaceinthatbundle.
Over
Q
(
i
)
,if
ξ
∈{−
i
,
i
}
,themap
A
Q
(
i
)
→
S
1
Q
(
i
)
givenby
u
7→
((
ξ
:1:0
)
,U
)
extends
toasection
σ
ξ
of
π
.Thesurface
S
Q
(
i
)
contains10exceptionalcurves,thatisirreducible
curveswithnegativeself-intersection.Eightofthemaregivenin
S
Q
(
i
)
bythefollowing
equations
D
jξ
:
L
j
(
π
(
P
))=
0and
X
−
ξY
=
0
for
ξ
∈{−
i
,
i
}
and
j
∈{
1
,
2
,
3
,
4
}
;thelastonescorrespondtothesection
σ
ξ
andaregiven
bytheequations
E
ξ
:
T
=
0and
X
−
ξY
=
0
.
Here
X
,
Y
and
T