as Galois groups over the rational numbers
GaborWiese
Abstra t
Ideas from Khare's and Wintenberger's arti le on the proof of Serre's
onje ture for odd ondu tors are used to establish that for a xed
prime l in nitely many of the groupsPSL
2 (F
l r
)(for r running) o ur
asGaloisgroupsovertherationalssu hthatthe orrespondingnumber
eldsareunrami edoutsideaset onsistingof l,thein nitepla eand
onlyoneotherprime.
1 Introdu tion
Theaim ofthisarti leisto provethefollowingtheorem.
Theorem 1.1. Let l be aprime and s apositive number. Then there
exists aset T of rational primesof positive density su h that for ea h
q2T there existsamodularGalois representation
:G
Q
!GL
2 (F
l )
whi h isunrami edoutsidef1;l;qgandwhose proje tive imageisiso-
morphi toPSL
2 (F
l r
)for somer>s.
Corollary 1.2. Let lbeaprime. Then for in nitely many positive in-
tegersrthegroupsPSL
2 (F
l r
)o urasaGaloisgroupovertherationals.
UsingSL
2 (F
2 r
)
=PSL
2 (F
2 r
)oneobtainsthefollowingreformulation
forl=2.
Corollary 1.3. The group SL
2 (F
2 r
) o urs asaGalois group over the
rationalsfor in nitely many positive integersr.
This ontrastswithworkbyDieulefait, ReverterandVila([D1 ℄,[D2℄,
[RV℄ and [DV℄) whoprovedthat the groupsPSL
2 (F
l r
)and PGL
2 (F
l r
)
o uras Galois groups over Q for xed (small) r and in nitely many
primesl.
Dieulefait([D3℄)hasre entlyobtainedadi erent,rather elementary
proofofCorollary1.2undertheassumptionl5withadi erentrami -
ationbehaviour,namelyf1;2;3;lg.Hisproofhasthevirtueofworking
withafamilyofmodularforms thatdoesnotdependonl.
Intheauthor's PhDthesis [W ℄some omputationaleviden e onthe
statementof Corollary1.3wasexhibited. Morepre isely,it wasshown
that all groups SL
2 (F
2 r
) o ur as Galois groups over Q for 1 r
77, extending results by Mestre (see [S℄, p. 53), by omputing He ke
eigenformsofweight2forprimelevelover nite eldsof hara teristi 2.
However,at that time all attempts to provethe orollary failed, sin e
theauthor ouldnotruleouttheoreti allythatallGaloisrepresentations
atta hedtomodularformswithimage ontainedinSL
2 (F
2 r
)forrbigger
thansome xedbound andnotin anySL
2 (F
2 a
)forajr,a6=r,havea
dihedralimage.
Thepresent paper hasundergone somedevelopments sin ethe rst
writing. However,the oreoftheproofhasremainedthesame. Ituses
apro edureborrowedfrom thegroundbreakingpaper[KW℄ byKhare
and Wintenberger. The representations that we will onstru t in the
proofarealmost-intheterminologyof[KW℄-good-dihedral.Theway
wemaketheserepresentationbylevelraisingisadoptedfrom apartof
theproofof [KW℄,Theorem3.4. Althoughthepaper[KW℄isnot ited
intheproof,thispaperowesitsmereexisten etoit.
Meanwhile, the results of the present arti le have been generalised
to the groups PSp
2n (F
l
r) for arbitrary n by Khare, Larsen and Savin
(see [KLS℄). Theirpaper[KLS℄ also inspireda slight strengthening of
themain resultof thepresentarti le omparedtoanearlyversion.
A knowledgements. Theauthor would liketothankBas Edixhoven
forveryuseful dis ussionsandAlexanderS hmidtforan elegantargu-
mentused intheproofofLemma3.1.
Notation. We shall use the following notation. By S
k (
1
(N)) we
mean the omplex ve tor spa e of holomorphi usp forms on
1 (N)
of weight k. The notation S
k
(N; ) is used for the ve tor spa e of
holomorphi usp forms of level N, weight k for the Diri hlet har-
a ter . If is trivial, wewrite S
k
(N) for short. We x an algebrai
losure Q of Q and for every prime p an algebrai losure Q of Q
p
and we hoose on e and for all an embedding Q ,! Q
p
and a ring
surje tion Z
p
! F
p
, subje t to whi h the following onstru tions are
made. ByG
Q
wedenotetheabsoluteGaloisgroupoftherationalnum-
bers. For a rational prime q, we let D
q and I
q
be the orresponding
de ompositionand inertiagroupoftheprimeq,respe tively. Givenan
eigenformf 2S
k (
1
(N))one anatta hto itbywork ofShimuraand
DeligneaGaloisrepresentation
f;p :G
Q
!GL
2 (Q
p
)withwell-known
properties. Choosingalatti e, redu ingand semi-simplifying,onealso
obtainsarepresentation
f;p :G
Q
!GL
2 (F
p
). Wedenote the ompo-
sitionG
Q
!GL
2 (F
p
)PGL
2 (F
p )by
proj
f;p
. AllGaloisrepresentations
inthis paperare ontinuous. By
p r
wealwaysmean aprimitivep r
-th
rootofunity.
2 On Galois representations
The basi idea of the proof is to obtain the groups PSL
2 (F
l r
) as the
imageofsome proj
g;l
. Inordertodeterminethepossibleimagesof proj
g;l ,
wequotethefollowingwell-knowngrouptheoreti resultduetoDi kson
(see[Hu℄,II.8.27).
Proposition2.1(Di kson). Letlbeaprime andH a nitesubgroup
ofPGL
2 (F
l
). Then a onjugateof H isisomorphi tooneofthefollow-
inggroups:
nitesubgroups ofthe uppertriangular matri es,
PSL
2 (F
l r
)orPGL
2 (F
l r
)for r2N,
dihedral groups D
r
for r2N notdivisible by l,
A
4 ,A
5 orS
4 .
WenextquotearesultbyRibet showingthattheimagesof proj
g;l for
anon-CMeigenformg are\almostalways"PSL
2 (F
l
r)orPGL
2 (F
l r)for
somer2N.
Proposition 2.2 (Ribet). Let f = P
n1 a
n q
n
2 S
2
(N; ) be anor-
malisedeigenform oflevel N andsome hara ter whi h isnot aCM-
form.
Then for almost all primesp, i.e. allbut nitely many, the image of
therepresentation
proj
f;p :G
Q
!PGL
2 (F
p )
atta hed to f is equal to either PGL
2 (F
p
r) or to PSL
2 (F
p
r) for some
Proof. Redu ingmodulo p,Theorem 3.1of [R1℄ gives,foralmostallp,
that the image of
f;p
ontainsSL
2 (F
p
). This already provesthat the
proje tive image is of the form PGL
2 (F
p r
) or PSL
2 (F
p r
) by Proposi-
tion2.1.
InviewofRibet'sresult,therearetwotaskstobesolvedforproving
Theorem 1.1. The rst task is to avoidthe \ex eptional primes", i.e.
thoseforwhi htheimage isnotasin theproposition. These ond one
istoobtainatthesametimethatthe eldF
p r
is \big".
WewillnowusethisresultbyRibet inorder toestablishthesimple
fa t that for a given prime l there exists a modular form of level a
poweroflhavingonly nitelymany\ex eptionalprimes". Thefollowing
lemma anbeeasilyveri edusinge.g.WilliamStein'smodularsymbols
pa kagewhi hispartofMagma([Magma℄).
Lemma 2.3. In any of the following spa es there exists a newform
withoutCM: S
2 (2
7
),S
2 (3
4
), S
2 (5
3
), S
2 (7
3
),S
2 (13
2
).
Proposition 2.4. Letlbeaprime. Put
N :=
8
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
: 2
7
; if l=2;
3 4
; if l=3;
5 3
; if l=5;
7 3
; if l=7;
13 2
; if l=13;
l; otherwise.
Then there exists an eigenform f 2 S
2
(N) su h that for almost all
primesp,i.e. forallbut nitely many,the imageofthe atta hedGalois
representation
f;p
is of the form PGL
2 (F
p r
) or PSL
2 (F
p r
) for some
r2N.
Proof. If l 2 f2;3;5;7;13g, we appeal to Lemma 2.3 to get an eigen-
form f without CM.If lis notin that list,then there is aneigenform
in S
2
(l)and it iswell-knownthat it doesnothaveCM,sin e thelevel
issquare-free. Hen e,Proposition2.2givesthe laim.
Wewill be ableto nd aneigenformwith image asin the pre eding
propositionwhi his\bigenough"byapplyingthefollowinglevelraising
resultbyDiamondandTaylor. Itisaspe ial aseofTheoremAof[DT ℄.
Theorem 2.5 (Diamond, Taylor). Let N 2 N and let p> 3 be a
prime notdividing N. Let f 2S
2
(N; ) be anewformsu h that is
irredu ible. Let,furthermore, q-N beaprimesu hthatq 1 modp
andtr(
f;p (Frob
q ))=0.
Then thereexistsanewformg2S
2 (Nq
2
; )~ su hthat
g;p
=
f;p .
Corollary2.6. AssumethesettingofTheorem2.5. Then thefollowing
statementshold.
(a) The modpredu tionsof and ~areequal.
(b) The restri tion of
g;p to I
q
, the inertia group at q, is of the form
0 q
witha hara terI
q
!Z[
p r℄
,!Z
p [
p r℄
of order p r
for
somer>0.
( ) For all primes l6= p;q, the restri tion of
g;l to D
q
, the de ompo-
sition group at q, is irredu ible and the restri tion of
g;l to I
q is
of the form
0 q
withthe hara terI
q
!Z[
p r℄
F
l (
p r
)
of
the sameorder p r
asin (b).
Proof. (a)Thisfollowsfrom
g;p
=
f;p .
(b) As q 2
pre isely divides the ondu tor of
g;p
, the rami ation
at q is tame and the restri tion to I
q
is of the form
1
0 2
for non-
trivial hara ters
i : I
q
! Z
p
. Their order must be a power of p,
sin e their redu tions mod p vanish by assumption. Due to q 1
modp,lo al lass eldtheorytellsusthat
i
annotextendtoG
Qq ,as
their orders would divide q 1. Hen e, the image of
g;p j
D
q : D
q
!
GL
2 (Q
p
)!PGL
2 (Q
p
)isa nitedihedralgroup. Consequently,
g;p j
D
q
isan unrami edtwistofthe indu edrepresentationInd GK
D
q
( ) withK
the unrami ed extension of degree 2 of Q
q
and : G
K
! Z[
p r℄
a
totallyrami ed hara ter oforder p r
for somer >0. Conjugation by
g;p (Frob
q
)ex hanges
1 and
2
. Itiswell-knownthatthis onjugation
alsoraises to theq-th power. Thus, we nd
2
= q
1
andwithout loss
ofgenerality =
1 .
( )With thenotationsand thenormalisation usedin [CDT ℄,p.536,
thelo al Langlands orresponden ereads(forl6=q)
(
g;l jW
q )
ss
= Q
l
Q WD (
g;q );
if g orresponds to the automorphi representation
g
=
g;p . In
parti ular,knowingby(b)that
g;p j
I
q is
0 q
, itfollowsthat
g;l j
I
q
is alsoof that form. As is ofp-powerorder, it annot vanish under
redu tion mod l if l 6= p. Hen e,
g;l j
I
q
is of the laimed form. The
3 Proof and remarks
Inthisse tionweproveTheorem1.1and ommentonpossiblegeneral-
isations.
Lemma 3.1. Let l be a prime and s 2 N. Then there is a set of
primespof positive densitysu hthat pis1mod 4andsu hthat if the
group GL
2 (F
l t
) possesses an element of order p for some t 2 N, then
2jtandt>s.
Proof. Wetakethesetofprimespsu hthatpissplitinQ(i)andinertin
Q(
p
l ). ByChebotarev'sdensitytheoremthissethasapositivedensity.
The rst onditionimposed onpmeansthat 4divides theorderof F
p
andthese ondonethattheorderofthe2-SylowsubgroupofF
p
divides
theorderoflinF
p
. Iftheorderofg2GL
2 (F
l t
)isp,thenF
l 2t
ontains
anelementoforderp(aneigenvalueofg). Thus,l 2t
1isdivisiblebyp,
when etheorderoflinF
p
divides2tand onsequently2dividest. The
onditiont>s anbemetbyex luding nitelymanyp.
Proof ofTheorem 1.1. Letus xtheprimelandthenumbersfromthe
statementof Theorem1.1. Wewill nowexhibit amodularformg su h
that proj
g;l
hasimage equaltoPSL
2 (F
l t
)witht>s. Westart withthe
eigenform f 2 S
2 (l
) provided by Proposition 2.4. Next, we let p be
anyofthein nitelymanyprimes fromLemma 3.1su h thattheimage
of proj
f;p
isPSL
2 (F
p r
)orPGL
2 (F
p r
) withsomer. We wanttoobtaing
bylevelraising. Thenext lemma will yield aset of primes of positive
densityatwhi hthelevel anberaisedin awaysuitabletous.
Lemma 3.2. Under the above notations and assumptions, the set of
primesqsu hthat
(i) q 1 modp,
(ii) q splits inQ( i;
p
l) and
(iii) proj
f;p (Frob
q
) lies in the same onjuga y lass as proj
f;p
( ), where
isany omplex onjugation,
hasapositive density.
Proof. Theproof is adapted from [KW℄, Lemma 8.2. LetK =Q bethe
number eldsu hthatG
K
=ker(
proj
f;p
). De neLasQ(
p
;i;
p
l ). Condi-
tions(i)and(ii)mustbeimposedonthe eldLandCondition(iii)onK.
WeknowthatGal(K =Q)iseitherPSL
2 (F
p
r)orPGL
2 (F
p
r). Inthefor-
mer asethelemmafollowsdire tlyfromChebotarev'sdensitytheorem,
astheinterse tionL\KisQ, sin ePSL (F r)isasimplegroup. Inthe
latter asetheinterse tionL\K=M isanextensionofQ ofdegree2.
Asp1 mod4byassumption, proj
f;p
( )isin Gal(L=M)
= PSL
2 (F
p r
),
sin edet(
f;p
( )) = 1is asquare mod p. Consequently, any q satis-
fying Condition(iii) issplit in M=Q. Againasp1 mod4, omplex
onjugation xes thequadrati sub eld of Q(
p
), when e any prime q
satisfyingConditions(i) and (ii)is also split in M=Q. Hen e, we may
againappealto Chebotarev'sdensitytheorem,provingthelemma.
To ontinue the proof ofTheorem 1.1 weletT bethe set of primes
provided by Lemma 3.2. Let q 2 T. Condition (iii) assures that
f;p (Frob
q
)hastra e0,sin eitisas alarmultipleofthematrixrepre-
senting omplex onjugation,i.e.
1 0
0 1
. Hen e,weareinthesituation
to apply Theorem 2.5 and Corollary 2.6. We, thus, get an eigenform
g 2 S
2 (l
q 2
; ) with a Diri hlet hara ter of order apowerof p(its
redu tionmodpis trivial)su h that
g;l j
Dq
isirredu ibleand
g;l j
Iq is
ofthe form
0 q
with anon-trivial hara terof order p r
. Hen e,
in parti ular, the image proj
g;l (G
Q
) ontains an element of order p, as
0 q
annotbes alarduetop-q 1.
Wenext showthat proj
g;l (G
Q
) is not adihedral group. It annot be
y li either be ause of irredu ibility. If it were dihedral, then proj
g;l
wouldbearepresentationindu edfroma hara terofaquadrati exten-
sionR =Q. IfRwererami edatq,then proj
g;p (I
q
)wouldhaveevenorder,
butithasorderapowerofp. So,Rwould beeitherQ(
p
l )orQ(
p
l ).
AsbyCondition(ii)qwouldsplitinR ,thisimpliesthat proj
g;p j
D
q would
be redu ible, but it is irredu ible. Consequently, proj
g;l (G
Q
) is either
PSL
2 (F
l t
) or PGL
2 (F
l t
) for some t. By what we have seen above,
PGL
2 (F
l
t)then ontainsanelementoforderp. Hen e,sodoesGL
2 (F
l t)
andtheassumptionsonpimplyt>sand2jt.
Weknow,moreover,thatthedeterminantof
g;l (Frob
w
)foranyprime
w-lqisw k 1
(w)forsome xedk (theSerre weightof
g;l
). As (w)
isofp-powerorder,F
l t
ontainsasquarerootofit. Thesquarerootsof
elementsofF
l
areall ontainedinF
l
2 andthusalsoin F
l
t,astiseven.
Hen e, proj
g;l (Frob
w
)2 PSL
2 (F
l t
). As every onjuga y lass ontainsa
Frobenius element,theproofofTheorem1.1is nished.
Remark 3.3. One andevelopthebasi ideaused herefurther, inpar-
ti ular, in order to try to establish an analogue of Theorem 1.1 su h
thattherepresentationsramifyatagiven nitesetofprimesS andare
Remark 3.4. Itisdesirabletoremovetherami ationatl. Forthat,one
would needthat proj
g;l
isunrami edat l. This,however,seemsdiÆ ult
toestablish.
Bibliography
[Magma℄ W.Bosma,J.J.Cannon,C.Playoust.TheMagmaAlgebraSystemI:
The UserLanguage. J.Symboli Comput.24(1997),235{265.
[CDT℄ B. Conrad, F. Diamond, R. Taylor. Modularity of ertain potentially
Barsotti-Tate Galois representations. J. Am. Math. So . 12(2) (1999),
521{567-
[DT℄ F.Diamond,R.Taylor.Non-optimallevelsofmodlmodularrepresenta-
tions. Invent.math.115(1994),435{462.
[D1℄ L.V.Dieulefait. Newforms,innertwists,andtheinverse Galoisproblem
forproje tivelineargroups.JournaldeTheoriedesNombresdeBordeaux
13(2001),395{411.
[D2℄ L.V.Dieulefait.GaloisrealizationsoffamiliesofProje tiveLinearGroups
via uspforms.
[D3℄ L. V. Dieulefait. A ontrol theorem for the images of Galois a tions on
ertainin nitefamiliesofmodularforms.Thisvolume,pp.??.
[DV℄ L. V.Dieulefait, N.Vila.Proje tive lineargroupsas Galoisgroupsover
Q viamodularrepresentations.J.Symboli Computation30(2000),799{
810.
[Hu℄ B.Huppert.Endli heGruppen I.Grundlehrendermathematis henWis-
sens haften134.Springer-Verlag,1983.
[KLS℄ C. Khare, M. Larsen, G. Savin. Fun toriality and the inverse Galois
problem. Preprint,2006.arXiv:math.NT/0610860
[KW℄ C.Khare,J.-P.Wintenberger.Serre's modularity onje ture: the aseof
odd ondu tor(I).Preprint,2006.
[RV℄ A. Reverter, N. Vila. Some proje tive linear groups over nite elds as
Galois groupsover Q. ContemporaryMath.186(1995),51{63.
[R1℄ K.A.Ribet.Onl -adi representationsatta hedtomodularformsII.Glas-
gowMath.J.27(1985),185{194.
[R2℄ K.A.Ribet.ImagesofsemistableGaloisrepresentations. Pa i Journal
ofMathemati s181No.3(1997),277{297.
[S℄ J.-P. Serre. Topi s in Galois theory. Resear h Notes in Mathemati s 1.
JonesandBartlettPublishers,Boston,MA,1992.
[W℄ G.Wiese.ModularFormsofWeightOneOverFinite Fields.PhDthesis,
UniversiteitLeiden,2005.
