Lectures on Modular Galois Representations Modulo Prime Powers [Lecture notes]

33 Pages·2011·0.299 MB·English

by  Gabor Wiese

Lectures on
Modular Galois Representations Modulo Prime Powers
GaborWiese
9thDecember2011
Abstract
This is a sketch of the content of my three lectures during the PhD School Modular Galois
Representations Modulo Prime Powers, held in Copenhagen from 6/12/2011 until 9/12/2011,
organisedbyIanKiming. ThanksIan!
1 Modular Forms Modulo Prime Powers
Modularforms,intheirclassicalappearance(19thcentury! Eisenstein,Weierstraß,Jacobi,Poincaré,
etc.) and in the way one usually gets to know them during one’s studies, are objects of Complex
Analysis: holomorphicfunctionssatisfyingacertaintransformationrule. Manyhaveanevidentnum-
bertheoreticsignificance(theywerestudiedbecauseofthis!),likethen-thFouriercoefficientofthe
EisensteinseriesE beingσ (n) = dk−1. But,itisahighlynon-trivialstepto‘transport’
k k−1 P0<d|n
modular forms from Analysis to Algebra, i.e. to identify an algebraic structure, or, even stronger, an
integralstructure,onthecomplexvectorspaceofmodularforms. ThiswasachievedbyHecke,Eich-
ler and Shimura. Without that we would not be able to do anything of what we are doing this week,
anditisprobablyfairtosaythatwithoutthatFermat’slasttheoremwouldnothavebeenproved. So,
thisfirstlectureismainlyconcernedwithintegralstructuresonmodularforms. Finally,itwillbeused
tointroducemodularformsmoduloprimepowers,asanapplication.
Areferencewheremostofthecontentofthislectureisworkedoutaremylecturenotes[4].
1.1 Heckealgebrasandgeneralq-expansions
Definition 1.1. Let M (N) be the C-vector space of modular forms of weight k and level N (either
k
Γ (N)orΓ (N)–doesn’tmatterforus). ByS (N)wedenotethecuspidalsubspace.
1 0 k
LetH (N)betheC-subalgebraofEnd (M (N))generated(asC-algebra)bytheHeckeoper-
k C k
atorsT forn ∈ N.
n
Let T (N) be the subring of End (M (N)) generated (as a ring, i.e. as a Z-algebra) by the
k C k
HeckeoperatorsT forn ∈ N.
n
BothH (N)andT (N)arecalledHeckealgebraofweightk andlevelN.
k k
1
It is well-known that the Hecke algebra H (N) (and thus also T (N)) is commutative. As it
k k
is commutative, in M (N) there are modular forms that are eigenvectors for all Hecke operators
k
T : These are called (Hecke) eigenforms. Let f = ∞ a (f)qn be such an eigenform (we write
n Pn=0 n
q = q(z) = e2πiz). Wesaythatitisnormalisedifa (f) = 1.
1
OnecancomputedirectlythattheleveloneEisensteinseriesE areHeckeeigenforms(forallk).
k
One also trivially gets an eigenform if the space of modular (cuspidal) forms is 1-dimensional. This
provesthattheRamanujan∆ ∈ S (1)isaHeckeeigenform.
12
Thefollowingsimplelemma,whichisadirectconsequenceofthedescriptionofHeckeoperators
onFourierexpansionsofmodularforms,turnsouttobethekeytoeverythingthatfollows.
Lemma 1.2. Suppose f = ∞ a (f)qn ∈ M (N) be a modular form of weight k and level N.
Pn=0 n k
Thenforalln ≥ 1wehavea (T f) = a (f).
1 n n
Wenowdefineabilinearpairing,whichIcallthe(complex)q-pairing,as
M (N)×H (N) → C, (f,T) 7→ a (Tf).
k k 1
Proposition 1.3. Suppose k ≥ 1. The complex q-pairing is non-degenerate. In particular, we have
theisomorphism
Φ : Mk(N) ∼= HomC(Hk(N),C), f 7→ Φ(f), whereΦ(f)(T) = a1(Tf).
ItisusefultopointoutthatΦ(f)mapsT toa (T f) = a (f).
n 1 n n
TheinverseΨofΦisgivenbyφ 7→ a + ∞ φ(T )qn,wherea isauniquelydefinedcomplex
0 Pn=1 n 0
number.
Proof. This follows from Lemma 1.2 like this. If for all n we have 0 = a (T f) = a (f), then
1 n n
f = 0 (this is immediately clear for cusp forms; for general modular forms at the first place we
can only conclude that f is a constant, but since k ≥ 1, non-zero constants are not modular forms).
Conversely,ifa (Tf) = 0forallf, thena (T(T f)) = a (T Tf) = a (Tf) = 0forallf andall
1 1 n 1 n n
n, whence Tf = 0 for all f. As the Hecke algebra is defined as a subring in the endomorphism of
M (N),wefindT = 0,provingthenon-degeneracy.
k
Letφ ∈ Hom (H (N),C). ItisobviousthatΨ(φ)isamodularformf suchthata (f) = φ(T )
C k n n
foralln ≥ 1. Notethatthecoefficientsa (f)forn ≥ 1uniquelydeterminea (f),asthedifference
n 0
oftwoformshavingthesamea (f)forn ≥ 1wouldbeaconstantmodularformofthesameweight
n
andsoisthe0-functionbytheassumptionk > 0. However,Idonotknowageneralformulahowto
writedowna (f)(but,itcanbecomputedinallcases).
0
Theperfectnessoftheq-pairingisalsocalledtheexistenceofaq-expansionprinciple.
TheHeckealgebraisthelineardualofthespaceofmodularforms.
2
So, from the knowledge of the Hecke algebra we can recover the modular forms via their q-
expansionsastheC-linearmapsH (N) → C. Itisthispointofviewthatwillgeneralisewell!
k
But, more is true: We can identify normalised eigenforms as the C-algebra homomorphisms
amongtheH (N) → C:
k
Corollary1.4. Letf inM (N)beanormalisedeigenform. Then
k
T f = a (f)f foralln ∈ N.
n n
Moreover,ΦfromProposition1.3givesabijection
{NormalisedeigenformsinM (N)} ↔ Hom (H (N),C).
k C−alg k
Proof. Letλ betheeigenvalueofT onanormalisedeigenformf. Then:
n n
a (f) = a (T f) = a (λ f) = λ a (f) = λ ,
n 1 n 1 n n 1 n
provingthefirststatement. Furthermore:
Φ(f)(T T ) = a (T T f) = a (T a (f)f) = a (f)a (f) = Φ(f)(T )Φ(f)(T ),
n m 1 n m 1 n m m n n m
aswellas(usingthatT istheidentityofH (N)):
1 k
Φ(f)(T ) = a (f) = 1.
1 1
ThisprovesthatΦ(f)isaringhomomorphism(notethatitsufficestocheckthemultiplicativityona
setofgenerators–giventheadditivity).
Conversely,ifΦ(f)isaringhomomorphism,then
a (Tf) = Φ(Tf)(T ) = a (TT f) = Φ(f)(TT ) = Φ(f)(T)Φ(f)(T ) = Φ(f)(T)a (f)
n n 1 n n n n
foralln ≥ 1showingthatTf = λf withλ = Φ(f)(T)(notethatweagainhavetoworryaboutthe
0-thcoefficient,but,asbefore,itsufficesthattheothercoefficientsagreetoconcludethatthe0-thone
doesaswell).
1.2 ExistenceofintegralstructuresonHeckealgebras
Note that by definition T (N) is a subring of H (N). The main point is to see that T (N) is an
k k k
integral structure of H (N). We first prove this in the level 1 case, which requires least machinery.
k
Then, we prove it in general by citing the Eichler-Shimura theorem, as well as facts on modular
symbols.
3
1.2.1 Proofinlevel1usingEisensteinseries
Theinputtothisproofarethefollowingstandardfactsfrommodularformscourses:
Lemma1.5. (a) TheEisensteinseriesE ∈ M (1)andE ∈ M (1)haveaFourierexpansionwith
4 4 6 6
integralcoefficientsand0-thcoefficientequalto1. Ramanujan’s∆ ∈ M (1)isacuspformwith
12
integralFourierexpansionand1-stcoeffientequalto1.
(b) Letf ∈ M (N)beamodularformwithanintegralFourierexpansion. ThenT (f)alsohasan
k n
integralFourierexpansion.
(c) For any k, we have M = ∆·M ⊕CEαEβ, where α,β ∈ N are any elements such that
k+12 k 4 6 0
k+12 = 4α+6β(whichalwaysexistsincekiseven–otherwisewe’redealingwiththe0-space).
ThiscanbeusedtoconstructaVictor-MillerbasisofM (1)(say,itsdimensionisn),thatisany
k
basisoftheC-vectorspaceM (1)consistingofmodularformsf ,f ,...,f withintegralFourier
k 0 2 n−1
coefficientssuchthat
a (f ) = δ
i j i,j
forall0 ≤ i,j ≤ n−1.
How to construct such a basis? We do it inductively. For k = 4,6,8,10,14 the existence is
obvious, sincethespaceM (1)is1-dimensionalandtheEisensteinseriesdoesthejob. Fork = 12,
k
we start with E2 = 1−1008q +... and ∆ = q +..., so that we can take f = E2 +1008∆ and
6 0 6
f = ∆.
1
SupposenowthatwehaveaVictor-Millerbasisf ,...,f ofM (N). Fori = 0,...,n−1,let
0 n−1 k
g := ∆f andg := EαEβ. ThisisnotaVictor-Millerbasis,ingeneral,butcanbemadeintoone.
i+1 i 0 4 6
Notefirstthata (g ) = 1forall0 ≤ i ≤ nandthata (g ) = 0forall0 ≤ i ≤ nandall0 ≤ j < i.
i i j i
Graphically,itlookslikethis:
g = 1+ •q+ •q2+... ...+ •qn−1+ •qn
0
g = q+ •q2+... ...+ •qn−1+ •qn
1
g = q2+... ...+ •qn−1+ •qn
2
.
.
.
g = qn−1+ •qn
n−1
g = qn
n
IthinkthatitisnowobvioushowtomakethisbasisintoaVictor-Millerone.
Proposition 1.6. Let {f ,...,f } be a Victor-Miller basis of M (1). Then the Hecke operators
0 n−1 k
T ,writtenasmatriceswithrespecttotheVictor-Millerbasis,haveintegralentries.
m
4
Proof. Inordertowritedownthematrix,wemustdetermineT f forall0 ≤ i ≤ n−1intermsof
m i
thebasis. But,thisistrivial: If
T f = a +a q+a q2+···+a qn−1+...,
m i i,0 i,1 i,2 i,n−1
thenT f = n−1a f ,sothea arejusttheentriesofthematrix. Theyareintegral,asT f has
m i Pj=0 i,j j i,j m i
integralFouriercoefficients(usingherethatallthef do).
i
Nowwedrawourconclusions:
Corollary 1.7. The natural map C⊗ T (1) → H (1) is an isomorphism. In particular, T (1) is
Z k k k
freeasZ-module(i.e.abeliangroup)ofrankequaltotheC-dimensionofH (1).
k
WesaythatT (1)isanintegralstructureinH (1).
k k
Proof. Let us identify End (M (1)) with Mat (C) (with n the dimension of M (1)) by writing
C k n k
downtheHeckeoperatorswithrespecttoaVictor-Millerbasis.
By Proposition 1.6, we have that T (1) lies in Mat (Z). Let us write this more formally as a
k n
(ring)injection
ι : T (1) ֒→ Mat (Z).
k n
RecallthatCisaflatZ-module,hence,tensoringwithCoverZpreservesinjections,yielding
∼
id⊗ι : C⊗ZTk ֒→ C⊗ZMatn(Z) = Matn(C),
wherethelastisomorphismcanbeseenasC⊗Z(Z⊕Z⊕Z⊕Z) ∼= (C⊗ZZ)⊕(C⊗ZZ)⊕(C⊗Z
Z)⊕(C⊗Z Z) ∼= C⊕C⊕C⊕C. Theimageofid⊗ιliesinHk(1)andcontainsallTm, whence
theimageisHk(1),provingtheisomorphismC⊗ZTk ∼= Hk(1).
ItfollowsimmediatelythatT (1)isafreeZ-moduleofrankequaltothedimensionofH (1).
k k
What happened? The only non-trivial thing we used is that we could write down our Hecke
operators as matrices with integral entries. For higher levels this also works, but, I do not know of a
proofaseasyasthisone. We’llderiveitfromtheEichler-Shimuraisomorphism.
1.2.2 GeneralproofusingEichler-Shimura
Inthelevel1situationweobtainedHeckeoperatorswithintegralmatrixentriesbyprovingtheexis-
tence of a ‘good basis’ consisting of modular forms with integral Fourier coefficients and exploiting
thefactthatHeckeoperatorspreservethesubsetofmodularformswithintegralFouriercoefficients.
Inthe general level case, itis easier to obtain an integralstructurenot in the space of modular forms
directly, but, in an other C-vector space, a certain group cohomology space (or, a modular symbols
space–seebelow).
Wedonotdefinegroupcohomologyhere. Anaccountisgiveninmylecturenotes[4].
ThegroupSL (Z)actsonapolynomialf(X,Y)(intwovariables)fromtheleftasfollows:
2
a b f (X,Y) := f (X,Y) a b = f(aX +cY,bX +dY).
(cid:0)(cid:0)c d(cid:1) (cid:1) (cid:0) (cid:0)c d(cid:1)(cid:1)
5
ByR[X,Y] wedenotetheR-moduleofpolynomialsintwovariableswhicharehomogeneousof
k−2
degreek−2(fork ≥ 2)withcoefficientsinanycommutativeringR. ItispossibletodefineHecke
operators T for n ∈ N on the group cohomology space H1(Γ (N),R[X,Y] ) (also for Γ (N))
n 1 k−2 0
andontheparabolicsubspaceH1 (Γ (N),R[X,Y] ).
par 1 k−2
Theorem1.8(Eichler-Shimura). Letk ≥ 2. Thentherearenaturalisomorphisms
M (N)⊕S (N) ∼= H1(Γ,C[X,Y] )
k k k−2
and
S (N)⊕S (N) ∼= H1 (Γ,C[X,Y] ),
k k par k−2
whicharecompatiblewiththeHeckeoperators,whereΓisΓ (N)orΓ (N)(justasbefore).
1 0
The‘naturalisomorphism’isactuallygivenbyintegration(notsodifficult!). Thelecturenotes[4]
containadescriptionandaproof(whichisprobablynotthemostelegantone).
Corollary 1.9. The Hecke algebra H (N) (resp. T (N)) is isomorphic to the C-subalgebra (resp.
k k
thesubring)ofEnd (H1(Γ,C[X,Y] ))generatedbyT forn ∈ N.
C k−2 n
Proof. A Hecke operator on M (N)⊕S (N) can be written as a block matrix (T,T′) where T′ is
k k
therestrictionofT toS (N). Sending(T,T′)toT definesahomomorphismfromtheHeckealgebra
k
onH1(Γ,C[X,Y] )totheoneonM (N),whichisclearlysurjectiveasallgenerators(theT )are
k−2 k n
hit. Itisinjective,becauseifT iszero,thensoisT′.
From the standard resolution for definining group cohomology it is very easy to deduce that
H1(Γ,Z[X,Y] ) isanintegralstructureofH1(Γ,C[X,Y] )inthesensethatitisasubgroup
k−2 free k−2
and
C⊗ZH1(Γ,Z[X,Y]k−2)free ∼= H1(Γ,C[X,Y]k−2).
IfM isanyfinitely generatedZ-module, thenitis thedirectsumof afreeZ-moduleandthe torsion
submodule: M ∼= M ⊕M , where M = M/M . Note that the Hecke operators on
free torsion free torsion
H1(Γ,Z[X,Y] )sendtorsionelementstotorsionelements, and, thusgiverisetoHeckeoperators
k−2
onH1(Γ,Z[X,Y] ) bymoddingoutthetorsionsubmodule.
k−2 free
Now, we can draw the same conclusion as in the level 1 case: The Hecke operators T can be
n
written as matrices with integral entries, hence, T (N) ≤ Mat (Z), where n is the C-dimension of
k n
H1(Γ,C[X,Y] ),whichisequaltotheZ-rankofH1(Γ,Z[X,Y] ).
k−2 k−2
So,againbytheflatnessofCasZ-module,weobtain,preciselyasearlier:
Theorem1.10(Shimura??). C⊗ZTk(N) ∼= Hk(N).
Note that our proof requires k ≥ 2. For k = 1, I am not aware of a proof along the above
lines. However, itisknownthattheresultofthetheoremistruenevertheless. Thisoneprovesusing
the algebraic geometric description of modular forms due to Katz, which is beyond the scope of this
lecture.
6
1.2.3 Generalproofusingmodularsymbols
Personally,Ilikegroupcohomologymuchbetterthanmodularsymbols(atleast,ifonedefinesmod-
ular symbols in the way I am going to do in this section, namely, as an abstract formalism) because
workingwithgroupcohomologyonehasallthetoolsfromthattheoryatone’sdisposal.
However,modularsymbols(theformalism)ispreciselywhatisimplementedinMagmaandSage
(by William Stein, principally). Moreover, the definitions are so short that they easily fit into this
lecture(atleastthetypedversion),whereasagooddefinitionofgroupcohomologydoesn’t.
Recall that the projective line over Q can be seen as Q∪{∞} and that it carries the natural left
SL (Z)-actionbyfractionallinearcombinations: a b x = ax+by,where∞istreatedintheobvious
2 (cid:0)c d(cid:1) y cx+dy
way,namely,as 1.
0
Definition1.11. LetRbeacommutativeringandwriteΓforΓorΓ (N),aswellasV = R[X,Y]
0 k−2
forsomek ≥ 2. WedefinetheR-modules
M := R[{α,β}|α,β ∈ P1(Q)]/h{α,α},{α,β}+{β,γ}+{γ,α}|α,β,γ ∈ P1(Q)i
R
and
B := R[P1(Q)].
R
WeequipbothwiththenaturalleftΓ-action. Furthermore,welet
M (V) := M ⊗ V and B (V) := B ⊗ V
R R R R R R
fortheleftdiagonalΓ-action.
(a) WecalltheΓ-coinvariants
M (N;R) := M (V) = M (V)/h(x−gx)|g ∈ Γ,x ∈ M (V)i
k R Γ R R
thespaceofmodularsymbolsoflevelN andweightk.
(b) WecalltheΓ-coinvariants
B (N;R) := B (V) = B (V)/h(x−gx)|g ∈ Γ,x ∈ B (V)i
k R Γ R R
thespaceofboundarysymbolsoflevelN andweightk.
(c) Wedefinetheboundarymapasthemap
M (N;R) → B (N;R)
k k
whichisinducedfromthemapM → B sending{α,β}to{β}−{α}.
R R
(d) The kernel of the boundary map is denoted by CM (N;R) and is called the space of cuspidal
k
modularsymbolsoflevelN andweightk.
7
We now give the definition of the Hecke operator T for a prime ℓ on Γ (N) (the definition on
ℓ 0
Γ (N)isslightlymoreinvolved). TheT forcompositencanbecomputedfromthosebytheusual
1 n
formulae. Amatrix a b ∈ Mat (Z)withnon-zerodeterminantactsonM (N;R)bythediagonal
(cid:0)c d(cid:1) 2 k
actiononthetensorproduct. Letx ∈ M (N;R). Weput
k
T x = δ.x,
ℓ X
δ∈Rℓ
where
R := { 1 r |0 ≤ r ≤ ℓ−1}∪{ ℓ 0 }, ifℓ ∤ N
ℓ (cid:0)0 ℓ(cid:1) (cid:0)0 1(cid:1)
R := { 1 r |0 ≤ r ≤ ℓ−1}, ifℓ | N.
ℓ (cid:0)0 ℓ(cid:1)
It is very easy to see that M (N;Z) = M (N;Z)/M (N;Z) is an integral structure
k free k k torsion
inM (N;C),soHeckeoperatorsonM (N;C)canbewrittenasmatriceswithintegralentries.
k k
Modular symbols (over C) describe the first homology of Γ for the module C[X,Y] (or the
k−2
first homology of the modular curve Y – this comes with a caveat because we must pay attention
Γ
whetherweshouldnotusecompactlysupportedcohomologyatsomeplaces;ifweworkwithX and
Γ
cuspidal modular symbols, everything is simpler). As homology and cohomology are dual to each
other(atleastingoodsituations),wehave:
Proposition1.12. Thereisanon-degeneratepairing
H1(Γ,C[X,Y] )×M (N;C) → C.
k−2 k
ItfollowsthattheHeckealgebraonH1(Γ,C[X,Y] )isisomorphictotheoneonM (N;C),
k−2 k
where the isomorphism is simply given by transposing the matrices (wrt. to a fixed basis, say, of
M (N;R))becausethetwospacesaredualtoeachotherbythevirtueofthepairing.
k
Consequently, we can again prove the isomorphism C⊗Z Tk(N) ∼= Hk(N) by using the Hecke
operators on M (N;Z) (which, of course, can be represented by matrices with integral entries),
k free
againfork ≥ 2.
1.3 ExploitingintegralstructuresonHeckealgebras
WearenowexploitingconsequencesofC⊗ZTk(N) ∼= Hk(N).
Corollary1.13. (a) T (N)isafreeZ-moduleofrankequaltotheC-dimensionofH (N),whichis
k k
equaltotheC-dimensionofM (N)byProposition1.3.
k
(b) HomZ(Tk(N),C) ∼= HomC(Hk(N),C) ∼= Mk(N).
(c) TheZ-algebrahomomorphismsinHom (T (N),C)correspondbijectively(underthemapping
Z k
ofthepreviousitem)tothenormalisedHeckeeigenformsofM (N).
k
8
Proof. ThatthenaturalmapsareisomorphismsisimmediatelyclearifwewriteT (N) = Z⊕···⊕
k
Z.
Now,wedonotneedH (N)anymore. WewillonlyworkwithT (N).
k k
ThemodularformsinM (N)correspondtothegrouphomomorphismsT (N) → C.
k k
ThenormalisedeigenformsinM (N)correspondtotheringhomomorphismsT (N) → C.
k k
We include a short interlude on commutative algebra(s). Recall that a ring is called Artinian if
every descending ideal chain becomes stationary. This is the case for F ⊗ T (N) because it is a
p Z k
finite dimensional F -vector space, so that ideals are subspaces, and, of course, chains of subspaces
p
thus have to become stationary for dimension reasons. For the same reason, also Q ⊗ T (N) is
Z k
Artinian,butT (N),ofcourse,isnot!
k
Proposition1.14. LetRbeanArtinianring.
(a) EveryprimeidealofRismaximal.
(b) ThereareonlyfinitelymanymaximalidealsinR.
(c) LetmbeamaximalidealofR. Itistheonlymaximalidealcontainingm∞.
(d) Letm 6= nbetwomaximalideals. Foranyk ∈ Nandk = ∞theidealsmk andnk arecoprime.
(e) The Jacobson radical m is equal to the nilradical and consists of the nilpotent ele-
Tm∈Spec(R)
ments.
(f) Wehave m∞ = (0).
Tm∈Spec(R)
(g) (ChineseRemainderTheorem)Thenaturalmap
R −a−7→−−(..−.,−a+−m−−∞−,.−..→) R/m∞
Y
m∈Spec(R)
isanisomorphism.
(h) Foreverymaxmimalidealm,theringR/m∞ islocalwithmaximalidealmandishenceisomor-
phictoR ,thelocalisationofRatm.
m
TheHeckealgebraT (N)satisfiestheassumptions(andhencetheconclusions)ofthefollowing
k
proposition.
Proposition 1.15. Let T be a Z-algebra which is free of finite rank r as a Z-module. Let T :=
Q
Q⊗ T.
Z
(a) Z ⊆ Tisanintegralringextension.
9
(b) TisequidimensionalofKrulldimension1,meaningthateverymaximalidealmofTcontainsat
leastoneminimalprimeidealpandthereisnoprimeidealstrictlyincludedbetweenpandm.
(c) T/misafinitefieldofdegreeatmostr overtheprimefield,T/pisanorderinanumberfieldof
degreeatmostr overQ.
(d) TQ isanArtinQ-algebraofdimensionr. Assuchitsatisfies: TQ ∼= Qp(cid:1)TQprime(TQ)p(localisa-
tionatp).
(e) Theembeddingι : T ֒→ T inducesabijection(viapreimages)betweenthe(finitelymany)prime
Q
idealsofT (whichareallmaximal)andtheminimalprimeidealsofT. Theinverseisgivenby
Q
extension.
The proofs of the two propositions are not difficult. We will now exploit them for our purposes.
LetuswriteT (N) := Q⊗ T (N)(similarlytotheuseinthepreviousproposition).
k Q Z k
Wenowconsiderringhomomorphismsf : T (N) → Cinmoredetail.
k
Proposition1.16. Letf : T (N) → Cbearinghomomorphismandletp beitskernel.
k f
(a) p isaminimalprimeidealofT (N).
f k
(b) Theimageoff isanorderZ (thecoefficientringoff)inanumberfieldQ (thecoefficientfield
f f
off),whichcanbeexplicitlydescribedasZ = Z[f(T ) | n ≥ 1]andQ = Q(f(T ) | n ≥ 1).
f n f n
Moreover,[Q : Q] ≤ dim M (N).
f C k
(c) f : T (N) → C extends to a Q-linear map T (N) → C, whose kernel is the maximal
k k Q
ideal which is the extension of p (in accordance to the correspondence in Proposition 1.15).
f
Conversely, every f : T (N) → C arises by restriction from a Q-algebra homomorphism
k
T (N) → C.
k Q
(d) Let f : T (N) → C be a normalised Hecke eigenform and φ ∈ Aut (C) be a field automor-
k Q Q
phism. Theng := φ◦f isanothernormalisedHeckeeigenform,havingthesamekernel. Inthis
case,wesaythatf andg areAut (C)-conjugated.
Q
Conversely, suppose that f,g : T (N) → C have the same kernel. Then they are Aut (C)-
k Q Q
conjugated. Hence, the Aut (C)-conjugacy classes are in bijection with the maximal ideals of
Q
T (N) .
k Q
(e) The local factors in Tk(N)Q ∼= Qp(cid:1)Tk(N)Qprime(Tk(N)Q)p correspond to the AutQ(C)-conju-
gacyclasses.
Proof. Aroughsketchonly. Nothingisdifficultandeverythingcanbedoneasanexercise! Ofcourse,
thekernelpisanideal. ItisprimebecauseT (N)/p isasubringofC(hence,anintegraldomain),
k f
which is equal to the image of f. As T (N) is generated by the T , the image is generated by the
k n
valuesf(T ),i.e.isequaltoZ . ByProposition1.15,Z isanorderintheintegersofanumberfield,
n f f
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