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ON MANIN'S CONJECTURE FOR A FAMILY OF CHÂTELET SURFACES

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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

  • ????

  • ??


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Language English

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

:
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
areseenassectionsof
O
P
(
1
)
.Letusdenoteby
G
theGaloisgroupof
Q
(
i
)
over
Q
andby
z
7→
z
thenontrivialelementin
G
.Thenwehave
E
ξ
=
E
ξ
and
D

=
D

for
ξ
∈{−
i
,
i
}
and
j
∈{
1
,
2
,
3
,
4
}
.Weshallalsowrite
D
j
+
(resp.
D
j

,
E
+
,
E

)for
D
j
i
(resp.
D
j

i
,
E
i
,
E

i
).Theintersectionmultiplicitiesofthesedivisorsaregivenby
(
E
ξ
,E
ξ
)=

2
,
(
D
ξ
,D
ξ
)=

1
,
(
D
ξ
,D

ξ
)=
1
,
(
E
ξ
,D
ξ
)=
1
,
jjjjj(
1
)
Wedenehere
P
(
O
2

O
(

2
))
astheprojectivebundleassociatedtothesheaveofgradedcommutative
algebrasSym
(
O
2

O
(
2
))
.Inotherwordsthebreoverapointisgivenbythelinesinthebreofthevectorbundle
andnotbythehyperplanes.

4
RÉGISDELABRETÈCHE,TIMBROWNINGandEMMANUELPEYRE

where
ξ
∈{−
i
,
i
}
,and
j
∈{
1
,
2
,
3
,
4
}
,allotherintersectionmultiplicitiesbeingequalto0
(see[
CTSSD2
,p.73]).ThegeometricPicardgroupof
S
,thatisPic
(
S
)
,isisomorphicto
Pic
(
S
Q
(
i
)
)
andisgeneratedbytheseexceptionaldivisorswiththerelations
(2.2)
[
D
j
+
]+[
D
j

]=[
D
k
+
]+[
D
k

]
for
j,k
∈{
1
,
2
,
3
,
4
}
and
(2.3)
[
E
+
]+[
D
j
+
]+[
D
k
+
]=[
E

]+[
D
l

]+[
D
m

]
whenever
{
j,k,l,m
}
=
{
1
,
2
,
3
,
4
}
.UsingthefactthatPic
(
S
)=(
Pic
(
S
Q
(
i
)
))
G
itiseasyto
deducethatPic
(
S
)
hasrank2.
Itfollowsfromadjunctionformulathattheclassoftheanticanonicallinebundleisgiven
yb44ω
S

1
=
2
E
+
+
D
j
+
=
2
E

+
D
j

.
XXj
=
1
j
=
1
Lemma
2.1
.
Usingthetrivialisationdescribedby
(2.1)
,the
5
-tupleoffunctions
(
T,UT,U
2
T,X,Y
)
givesabasisof
G
(
S,ω
S

1
)
.
1−Proof
.Let
C
beagenericdivisorin
|
ω
S
|
.Then
C
isasmoothirreduciblecurve;let
g
C
beitsgenus.Accordingtotheadjunctionformula,wehavethat2
g
C

2
=
ω
S
.
(
ω
S

ω
S
)=
0.
Thus
g
C
=
1.Theexactsequenceofsheaves
0
−→
O
S
−→
ω
S

1
−→
ω
S

1

O
C
−→
0
givesanexactsequence
0
−→
H
0
(
S,
O
S
)
−→
H
0
(
S,ω
S

1
)
−→
H
0
(
C,ω
S

1
|
C
)
−→
H
1
(
S,
O
S
)
.
But
S
isgeometricallyrationaland
H
1
(
S,
O
S
)=
{
0
}
.Wegetthat
h
0
(
S,ω
S

1
)=
1
+
h
0
(
C,ω
S

1
|
C
)
.
1−Let
D
=
ω
S
|
C
.Wehavethatdeg
(
D
)=
4anddeg
(
ω
C

D
)=

4since
ω
C
=
0.Applying
RiemannRochtheoremto
C
,wegetthat
h
0
(
D
)=
deg
(
D
)+
2
g
C

2
=
4
and
h
0
(
S,ω
S

1
)=
5.Sincethesections
T,UT,U
2
T,X
and
Y
arelinearlyindependent,and
extendtoasectionof
O
P
(
1
)
,wegetabasisof
G
(
S,ω
S

1
)
.
Lemma
2.2
.
Thelinearsystem
|
ω
S

1
|
hasnobasepointandthebasisgiveninlemma2.1
givesamorphismfrom
S
to
P
4
Q
,theimageofwhichisthesurface
S

givenbythesystemof
equations
(
X
0
X
2

X
12
=
0
X
32
+
X
42
=(
aX
0
+
bX
1
+
cX
2
)(
a

X
0
+
b

X
1
+
c

X
2
)

MANIN'SCONJECTUREFORCHÂTELETSURFACES

5

erehwa
=
a
1
a
2
,b
=
a
1
b
2
+
a
2
b
1
,c
=
b
1
b
2
,
a

=
a
3
a
4
,b

=
a
3
b
4
+
a
4
b
3
,c

=
b
3
b
4
.
Theinducedmap
ψ
:
S

S

istheblowingupoftheconjugatesingularpointsof
S

given
by
P
ξ
=(
0:0:0:1:

ξ
)
with
ξ
2
=

1
and
ψ

1
(
P
ξ
)=
E
ξ
.
Proof
.Thisfollowsfromthefactthatthemapfrom
S
to
P
4
Q
inducesthemaps
((
x
:
y
:
t
)
,u
)
7−→
(
t
:
ut
:
u
2
t
:
x
:
y
)
from
S
1
to
P
4
Q
and
((
x

:
y

:
t

)
,v
)
7−→
(
v
2
t

:
vt

:
t

:
x

:
y

)
from
S
2
to
P
4
Q
.
Remark
2.3
.Thesurface
S

isanIskovskikhsurface[
CoTs
];itisasingularDelPezzo
surfaceofdegree4withasingularityoftype2
A
1
and
ψ
:
S

S

isaminimalresolutionof
singularitiesfor
S

.
3.Pointsofboundedheight
Over
Q
oreven
Q
(
i
)
,theonlygeometricalinvariantof
S
isthecross-ratio
a
3
a
1
a
3
a
2
a
4
a
1
a
4
a
2
α
=
b
3
b
1

b
3
b
2

Q
.

b
4
b
1

b
4
b
2