Hecke operators Quasi-modular forms attached to elliptic curves: Journal of Number Theory

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Journal of Number Theory

www.elsevier.com/locate/jnt

Quasi-modular forms attached to elliptic curves:

Hecke operators

Hossein Movasati

InstitutodeMatemáticaPuraeAplicada,IMPA,EstradaDonaCastorina,110, 22460-320,RiodeJaneiro,RJ,Brazil

a r t i c l e i n f o a bs t r a c t

Article history:

Received30August2012 Receivedinrevisedform18May 2015

Accepted19May2015 Availableonline3July2015 CommunicatedbyDavidGoss

Keywords:

Quasi-modularforms Heckeoperators

InthisarticlewedescribeHeckeoperatorsonthedifferential algebraofgeometricquasi-modularforms.Asanapplication for each natural number d we construct a vector field in six dimensions which determines uniquely the polynomial relationsbetween theEisensteinseries ofweight 2,4 and 6 andtheirtransformationundermultiplicationoftheargument by d,andinparticular,it determinesuniquely themodular curveofdegreedisogeniesbetweenellipticcurves.

© 2015ElsevierInc.All rights reserved.

1. Introduction

Thetheory ofquasi-modular formswas firstintroducedbyKanekoand Zagierin[5]

duetoitsapplicationsinmathematicalphysics.Onecandescribequasi-modularformsin theframeworkofthealgebraicgeometryofellipticcurves,andinparticular,theRamanu- jandifferentialequationbetweenEisensteinseriescanbederivedfromtheGauss–Manin connection of families of elliptic curves, see for instance [7] and [9]. We call this the Gauss–Manin connection in disguise. The terminology arose from a private letter of

E-mailaddress:hossein@impa.br.

URL: http://www.impa.br/~hossein.

http://dx.doi.org/10.1016/j.jnt.2015.05.021 0022-314X/© 2015ElsevierInc.All rights reserved.

Pierre Deligne to the author, see [4]. In the present article we describe Hecke oper- ators for such quasi-modular forms, and certain differential ideals related to modular curves.In[3]theauthorsdescribeadifferentialequationinthej-invariantoftwoellip- ticcurveswhichistangenttoallmodularcurvesofdegreedisogeniesofelliptic curves.

ThisdifferentialequationcanbederivedfromtheSchwarziandifferentialequationofthe j-functionandthelattercanbecalculatedfromtheRamanujandifferentialequationbe- tweenEisensteinseries.ThissuggeststhattheremustbearelationbetweenRamanujan differential equationand modularcurves. In thisarticle we alsoestablish this relation.

Anothermotivation behindthis workisto prepare thegroundfor similartopics inthe caseofCalabi–Yauvarieties,see[8].

ConsidertheRamanujan ordinarydifferentialequation

R:

⎧⎪

⎨

⎪⎩

˙

s1= ₁₂¹(s²₁−s2)

˙

s2= ¹₃(s1s2−s3)

˙

s₃= ¹₂(s₁s₃−s²₂)

˙

s_k =∂s_k

∂τ (1)

whichissatisfiedbytheEisensteinseries:

s_i(τ) =a_iE_2i(q) :=a_i

⎛

⎝1 +b_i ∞ n=1

⎛

⎝

d|n

d²ⁱ⁻¹

⎞

⎠qⁿ

⎞

⎠,

i= 1,2,3, q=e^2πiτ, Im(τ)>0 (2) and

(b1, b2, b3) = (−24,240,−504), (a1, a2, a3) = (2πi,(2πi)²,(2πi)³).

ThealgebraofmodularformsforSL(2,Z) isgenerated bytheEisensteinseries E₄ and E6 andallmodularforms forcongruencegroupsarealgebraic overthefieldC(E4,E6), seeforinstance[14].Inasimilar waythealgebraofquasi-modularforms forSL(2,Z) is generatedbyE2,E4 andE6,seeforinstance [6,9],andwehave:

Theorem1. Fori= 1,2,3andd∈N, thereis ahomogeneous polynomial Id,i of degree i·ψ(d),whereψ(d):=d

p(1+¹_p)istheDedekindψfunctionandprunsthroughprimes pdividingd,in theweightedring

Q[ti, s1, s2, s3], weight(ti) =i, weight(sj) =j, j= 1,2,3 (3) andmonicinthevariablet_i suchthatt_i(τ):=d²ⁱ·s_i(d·τ),s₁(τ),s₂(τ),s₃(τ)satisfythe algebraicrelation:

Id,i(ti, s1, s2, s3) = 0.

Moreover,fori= 2,3thepolynomial Id,i doesnotdepend on s1.

ThenoveltyofTheorem 1ismainlyduetothecasei= 1.Weconsidersi,ti,i= 1,2,3 as indeterminatevariables andforsimplicity wedonotintroducenewnotationinorder to distinguish them from the Eisenstein series. We regard (t,s) = (t1,t2,t3,s1,s2,s3) as coordinates of the affine variety A⁶k, where k is any field of characteristic zero and not necessarily algebraically closed. Ramanujan’s ordinary differential equation (1) is consideredasavectorfieldinA³k withthecoordinates(s1,s2,s3).It canbeshownthat the curve given by Id,2 =Id,3 = 0 inthe weighted projective space P^(2,3,2,3)_C with the coordinates(t2,t3,s2,s3) isasingularmodelforthemodularcurve

X0(d) := Γ0(d)\(H∪Q), where Γ0(d) :={

a1 a2

a₃ a₄

∈SL(2,Z)|a3≡0 (mod d)}. Computing explicit equations for X₀(d) in terms of the variables j₁ = 1728_t3^t32

2−t²3 and j2= 1728_s3^s32

2−s²₃ hasmanyapplications innumbertheoryand ithasbeen donebymany authors,seeforinstance [13]andthereferencestherein.

Let Rt, respectively Rs, be the Ramanujan vector field in A³k with coordinates (t1,t2,t3), respectively(s1,s2,s3). InA⁶k = A³k ×A³k with the coordinatessystem (t,s) we considerthevectorfield:

R_d :=R_t+d·R_s.

LetT:=A³k\{t³₂−t²₃= 0}andletVd betheaffinesubvarietyofT×Tgivenbytheideal I_d,1,I_d,2,I_d,3⊂k[s,t,_t3¹

2−t²3,_s3¹ 2−s²3].

Theorem 2.Thevector fieldR_d istangent totheaffinevarietyV_d.

I do not know the complete classification of all R_d-invariant algebraic subvarieties of A⁶k.Weconsider R_d as adifferentialoperator:

k[t, s]→k[t, s], f →Rd(f) :=df(Rd).

From Theorem 2andthefactthatV_d isirreducible(seeSection9),itfollowsthat:

R^j_d(Id,i)∈RadicalId,1, Id,2, Id,3, i= 1,2,3, j∈N∪ {0}. Note that the ideal I_d,1,I_d,2,I_d,3 ⊂ k[s,t,_t3¹

2−t²3,_s3¹

2−s²3] may not be radical. We can compute I_d,i’s using the q-expansion of Eisenstein series, see Section11. This method works only for small degrees d. However, for an arbitrary d we can introduce some elements intheradicaloftheidealgenerated byI_d,i,i= 1,2,3.Let

J_d,i= det

⎛

⎜⎜

⎜⎝

α₁ α₂ · · · α_md,i Rd(α1) Rd(α2) · · · Rd(αmd,i)

... ... · · · ... R^m_d^d,i−1(α1) R^m_d^d,i−1(α2) · · · R^m_d^d,i−1(αmd,i)

⎞

⎟⎟

⎟⎠,

wherefori= 1,αj’sarethemonomials:

t^a_i⁰s^a₁¹s^a₂²s^a₃³, i·ψ(d) =ia₀+a₁+ 2a₂+ 3a₃, a₀, a₁, a₂, a₃∈N0 (4) andfori= 2,3,αj’saretheabovemonomialswitha1= 0.Here,md,i isthenumberof monomialsαj’s.ThepolynomialJd,i isweightedhomogeneousofdegree

i·ψ(d) +i·ψ(d) + 1 +i·ψ(d) + 2 +· · ·+i·ψ(d) +md,i−1

=md,i·i·ψ(d) +m_d,i·(m_d,i−1)

2 .

Theorem3.Wehave

Jd,i∈RadicalId,1, Id,2, Id,3, i= 1,2,3, andso

J_d,i(d²ⁱ·s_i(d·τ), s₁(τ), s₂(τ), s₃(τ)) = 0, i= 1,2,3.

Our proofs of Theorems 1, 2, 3 use the notion of geometric quasi-modular forms, Heckeoperatorsand thefactthattheaffinevarietyA³k\{t³₂−t²₃= 0}isafinemoduli of ellipticcurves enhancedwith elements intheirdeRham cohomologies, see Theorem 4.

It mightbe possible to modify ourproofs in order to avoidany reference to Algebraic Geometry,however,onemaylose themotivation formanyarguments whichmainlylie onthegeometryofvector fields.Moreover,geometric approachtoquasi-modular forms givesusaconvenientcontextinordertoworkoverrationalnumbersandthismightlead ustoconnectionsofarithmeticofellipticcurvesandquasi-modular forms.

Throughoutthetextwewillstateourresultsover afieldk ofcharacteristiczeroand notnecessarilyalgebraicallyclosed.Suchresultsarevalidifandonlyifthesameresults are valid over thealgebraic closure ¯k of k. ByLefschetz principle, see forinstance [11]

p. 164, it is enough to prove such results over the complex numbers. For a variety T definedoverk,T(k) denotesthesetofk-rationalpointsofT.

Thearticleisorganizedinthefollowingway.InSection2andSection3werecallthe definitionoffullquasi-modular formsintheframeworkofboth algebraicgeometry and complex analysis.In Section 4we describe somefacts relating isogenies and algebraic de Rham cohomology of elliptic curves. Using isogeny of elliptic curves we introduce geometricHeckeoperatorsinSection5andinSection6wedescribetheirtranslationinto holomorphic Hecke operators. For our discussion of modular curves we need a refined version of Hecke operators that we discuss in Section 7. Theorem 1, Theorem 2 and Theorem 3 are respectively proved in Section 8, Section9 and Section10. Finally, in Section11wegivesomeexamples.

The main idea behind the proof of Theorem 3 is due to J.V. Pereira in[10]. Here, I wouldliketothankhimforteachingmesuchanelegantandsimpleargument.Thanks goto J. Sijsling forhis useful commentsfor thefirst draft of thepresent text. Finally, I wouldlike tothanktherefereewhosecriticalcommentsimprovedthetext.

2. Geometricquasi-modularforms

In this section we recall some definitions and theorems in [6,7]. The reader is also referred to [9] for a complete account of quasi-modular forms ina geometric context.

Note thatin[9] thet parameter is infact (₁₂¹t1,12₁₂¹2t2,8₁₂¹3t3). Letk be any fieldof characteristic zero and let E be an elliptic curve over k. The first algebraic de Rham cohomologyofE,namelyH_dR¹ (E),isak-vectorspaceofdimensiontwoandithasaone dimensional spaceF¹ consisting ofelements representedby regulardifferential 1-forms onE.Letus define

T:= Spec(k[t₁, t₂, t₃, 1 t³₂−t²₃])

Theorem 4. (See[9], §5.5.) The affine varietyT isthe finemoduli of thepairs (E,ω), whereE isanellipticcurveandω∈H_dR¹ (E)\F¹.For(t1,t2,t3)∈T(k),thecorrespond- ing pair(E,ω)isgivenby

E: 3y²= (x−t₁)³−3t₂(x−t₁)−2t₃, ω= 1 12

xdx

y . (5)

From now on an element of T(k) is denoted either by (t1,t2,t3) or (E,ω). We can regard ti as a rule which for any pair (E,ω) as above it associates an element ti = t_i(E,ω)∈k. Wewill alsouset_i as anindeterminate variableor anelement ink,being clear from the text whichwe mean. Form an evennumber, afull quasi-modular form f ofweightm anddifferentialordernis ahomogeneouspolynomialofdegree ^m₂ inthe k-algebra

M :=k[t1, t2, t3], weight(ti) =i, i= 1,2,3,

with deg_t1f ≤n.Thesetofsuchquasi-modular formsisdenotedbyM_mⁿ.

Forapair(E,ω)∈T(k) wehavealsoauniqueelementα∈F¹ satisfyingα,ω= 1, where ·,· is the intersection form in the de Rham cohomology, see for instance [9]

§2.10. For this reason we sometimes use (E,{α,ω}) instead of (E,ω). The algebraic groupsGa:= (k,+) andGm:= (k− {0},·) actfromtherightonT(k):

(E, ω)•k:= (E, kω), k ∈Gm, (E, ω)◦k:= (E, ω+kα), k∈Ga

and so theyactfrom theleft onM.It canbe shownthatM_mⁿ is invariantunderthese actionsand thefunctionst_i:T→k,i= 1,2,3 satisfy

k◦t₁=t₁+k, k◦t_i=t_i, i= 2,3 k∈Ga, (6) k•ti =k⁻²ⁱti, i= 1,2,3 k∈Gm. (7)

LetRbetheRamanujanvectorfieldinT.ItistheuniquevectorfieldinTwhichsatisfies

∇Rα=−ω,∇Rω= 0,where∇istheGauss–Maninconnectionoftheuniversalfamilyof ellipticcurvesover T,see forinstance [9]§2.Thek-algebraoffull quasi-modularforms hasadifferentialstructure whichis givenby:

M_mⁿ →M_m+2ⁿ⁺¹, t →R(t) :=

3 i=1

∂t

∂t_iRi,

whereR=3 i=1R_i∂t^∂

i istheRamanujanvectorfield.

3. Holomorphicquasi-modularforms

Now,letusassumethatk=C.Theperioddomain isdefinedto be P :=

x1 x2

x3 x4

|xi∈C, x1x4−x2x3= 1, Im(x1x3)>0

. (8)

WeletthegroupSL(2,Z) actfromtheleftonP byusualmultiplicationofmatrices.In P we considerthevectorfield

X =−x2

∂

∂x₁−x4

∂

∂x₃ (9)

which is invariant under the action of SL(2,Z) and so it induces a vector field in the complex manifold SL(2,Z)\P. For simplicity we denote it again by X. The Poincaré upperhalfplaneHisembeddedinP inthefollowing way:

τ→

τ −1

1 0

andsowe haveacanonicalmap H→SL(2,Z)\P. Theorem5.(See[9]§8.4 and§8.8.) Theperiod map

pm:T(C)→SL(2,Z)\P t →

1

√−2πi

δα

δω

γα

γω

is a biholomorphism, where {δ,γ} is a basis of H1(E,Z) with δ,γ =−1. Under this biholomorphism theRamanujan vector fieldis mapped toX.The pull-backof ti by the composition

H→SL(2,Z)\P ^pm→⁻¹T(C)→A³_C, (10) istheEisenstein seriesaiE2i(e^2πiτ)in(2).

The algebra of full holomorphic quasi-modular forms is the pull-back of k[t1,t2,t3] underthecomposition(10).Wecanalsointroduceitinaclassicalwayusingfunctional equations plus growth conditions: a holomorphic function f on H is called a (holo- morphic) quasi-modular form of weightm and differentialorder n ifthe following two conditionsaresatisfied:

1. Thereareholomorphicfunctionsfi(z),i= 0,1,. . . ,nonHsuchthat (cz+d)^−mf(Az) =

n i=0

n i

cⁱ(cz+d)⁻ⁱfi(z), ∀A= a b

c d

∈SL(2,Z). (11)

2. fi(z),i= 0,1,2,. . . ,nhavefinitegrowthswhen Im(z) tendsto+∞,i.e.

Im(z)→+∞lim fi(z) =a_i,∞<∞, a_i,∞∈C.

Fortheproofoftheequivalenceofbothnotionsofquasi-modularformssee[9]§8.11.We havef₀=fandtheassociatedfunctionsf_iareunique.Infact,f_iisaquasi-modularform of weightm−2i anddifferential order n−i andwith associated functionsf_ij :=f_i+j. It isusefultodefine

f||mA:= (detA)^m−1 n i=0

n i

( −c

det(A))ⁱ(cz+d)^i−mfi(Az), A=

a b c d

∈GL(2,R), f ∈M_mⁿ. (12)

Inthisway, theequality(11)iswrittenintheform

f =f||mA,∀A∈SL(2,Z) (13)

and wehave

f||mA=f||m(BA), ∀A∈GL(2,R), B∈SL(2,Z), f ∈M_mⁿ, (14) f^k||kmA= (detA)^k−1(f||mA)^k, ∀A∈GL(2,R), k∈N. (15) It can be proved that the algebra of full quasi-modular forms is generated by the EisensteinseriesE2i,i= 1,2,3.Forfurtherdetailsonholomorphicquasi-modularforms see [6,7,9].

4. Isogenyofellipticcurves

Let (E₁,0₁) and (E₂,0₂) be two elliptic curves over the field k. Here, 0_i ∈ E_i(k), i = 1,2 is theneutral element of the groupEi(k). We say that E1 is isogenous to E2

overkifthereisanon-constantmorphismofalgebraiccurvesoverkf :E1→E2 which sends 01 to 02. It canbe shown thatf induces amorphism of groupsE1(k)→ E2(k).

We also say that f is an isogeny between E1 and E2 over k. For all points p∈ E(¯k) exceptafinitenumber,#f⁻¹(p) isafixed numberwhichwedenote itbydeg(f).Here, wehave consideredf as amap from E1(¯k) toE2(¯k).Since f is amorphism ofgroups, for apoint q ∈ f⁻¹(p) themap x → x+q induces abijection f⁻¹(0₂) ∼=f⁻¹(p). We concludethat for allpoints p∈ E₂(¯k), theset f⁻¹(p) has deg(f) points(and hence f hasnoramification points).

Proposition1. Letf :E1→E2 bean isogeny ofdegree d.Thenfor allω,α∈H_dR¹ (E2) wehave

f^∗ω, f^∗α=d· ω, α.

Here,·,·:H_dR¹ (E)×H_dR¹ (E)→kistheintersectionforminthedeRhamcohomol- ogy,see[9]§2.10.

Proof. Itisenoughtoprovethepropositionoveranalgebraicallyclosedfield.Sincethe aboveformulais k-linear inboth ω andα, itis enoughto proveitinthecase ω= ^dx_y , α=^xdx_y ,wherex,yaretheWeierstrasscoordinatesofE2.Since^dx_y ,^xdx_y = 1,wehave toprovethatf^∗(^dx_y ),f^∗(^xdx_y )=d.Letf⁻¹(02)={p1,p2,. . . ,pd}.Thedifferentialform f^∗(^dx_y) isagainaregulardifferentialformandf^∗(^xdx_y ) haspolesofordertwoateachpi. ConsiderthecoveringU ={U0,U1}of E1,where U0=E1\f⁻¹(02) andU1 isanyother opensetwhichcontainsf⁻¹(02).Thedifferentialformsf^∗(^xi_y^dx),i= 0,1 aselementsin H_dR¹ (E1) arerepresentedbythepairs

(d˜x

˜ y ,d˜x

˜

y ), (xd˜ x˜

˜ y ,xd˜˜ x

˜ y −1

2d(y˜

˜ x)),

wherex˜=f^∗x,y˜=f^∗y.Wehave ^d˜_y˜^x∪^˜xd˜_y˜^x={ω₀₁}, whereω₀₁= ⁻¹₂ ^d˜_x˜^x andso f^∗(dx

y ), f^∗(xdx

y )=d˜x

˜ y ,˜xd˜x

˜ y =

d i=1

Residue(−1 2

d˜x

˜ x, pi) =

d i=1

1 =d. 2 Proposition2.Wehave:

1. Let f : E1 → E2 be an isogeny defined over k.The induced map f^∗ : H_dR¹ (E2) → H_dR¹ (E1)isan isomorphism.

2. Let [d]_E : E → E be the multiplication by d∈ N map. We have [d]^∗_E : H_dR¹ (E) → H_dR¹ (E),ω →d·ω.

Proof. Inthe complexcontext,E=C/τ,1 and[d]E isinduced byC→C, z →d·z.

Moreover, abasisof theC^∞ de Rham cohomology isgiven by dz, d¯z. This proves the

second part ofthe proposition.For thefirst partwetakethe dual isogenyanduse the firstpart. 2

For E an ellipticcurve overan algebraicallyclosed fieldk ofcharacteristic zero, the numberof isogeniesf :E1 →E ofdegreedand uptocanonicalisomorphismsis equal to σ1(d):=

c|dc. Toprovethiswemayworkinthecomplexcontextandassumethat E = C/τ,1. The numberof such isogenies is the number of subgroups of order dof (Z/dZ)²,whichisknowntobe σ1(d).

5. GeometricHeckeoperators

In this section all the algebraic objects are defined over k unless it is mentioned explicitly. Let dbe apositiveinteger. The Hecke operatorTd acts on thespace of full quasi-modular formsas follows:

Td:M_mⁿ →M_mⁿ, Td(t)(E, ω) =1

d

f:E1→E,deg(f)=d

t(E1, f^∗ω), t∈M_mⁿ

where the sumrunsthroughallisogenies f :E1 →E of degreeddefined over¯k.Since (E,ω) andt aredefined over k, T_d(t)(E,ω) is invariantunder Gal(¯k/k) andso it is in the fieldk. This impliesthatT_d(t) is definedover k. ThestatementT_d ∈k[t₁,t₂,t₃] is not at all clear. In order to provethis, we assumethat k =C and we provethe same statementforholomorphicquasi-modular forms,see Section6.Thefunctionalequation of T_dt with respect to theaction of thealgebraic groupsGm and Ga canbe proved in thealgebraic contextasfollows:

Proposition 3.The actionof Gmcommutes with Heckeoperators, that is,

k•T_d(t) =T_d(k•t), t∈M , k∈Gm (16) and theactionof Ga satisfies:

k◦Td(t) =Td((d·k)◦t), t∈M , k∈Ga. Proof. Thefirstequalityistrivial:

(k•Td(t))(E, ω) =Td(t)(E, kω) = 1 d

t(E1, f^∗(kω))

= 1 d

(k•t)(E₁, f^∗(ω)) =T_d(k•t)(E, ω)

ForthesecondequalityweuseProposition 1:

(k◦Td(t))(E, ω) =Td(t)(E, ω+kα) = 1 d

t(E1, f^∗(ω+kα))

=1 d

t(E₁, f^∗(ω) +d·kf^∗(1

dα)) =1 d

((d·k)◦t)(E₁, f^∗(ω))

=Td(d·k◦t)(E, ω). 2

Wecanalsodefine theHecke operatorsinthefollowingway:

Td(t)(E, ω) =d^m−1

g:E→E1,deg(g)=d

t(E1, g_∗ω), t∈M_mⁿ

where thesum runsthroughall isogeniesg :E →E₁ ofdegree ddefinedover ¯k.Both definitionsofTd(t) areequivalent:foranisogenyf :E1→E ofdegreeddefinedover¯k wehaveauniquedualisogenyg:E→E1 suchthat

f ◦g= [d]E, g◦f = [d]E1.

ThereforebyProposition 2wehaved·g_∗ω= [d]^∗_E1(g_∗ω)=f^∗ω andso t(E1, f^∗ω) =t(E1, d·g_∗ω) =d^mt(E1, g_∗ω).

It can be shown that the geometric Eisenstein modular form G_k (see [9] §6.5) is an eigenformwitheigenvalue

σk−1(d) :=

c|d

c^k−1

fortheHeckeoperatorTd,thatis

T_dG_k =σ_k−1(d)G_k, d∈N, k∈2N seeforinstance[6].

ThedifferentialoperatorR:M →M andtheHeckeoperatorTd commute,thatis R◦Td=Td◦R, ∀d∈N.

For the proof we may assume that k = C. In this way using Theorem 5 it is enough to provethesame statementfor holomorphicquasi-modular forms, seefor instance [7]

Proposition4.

6. HolomorphicHeckeoperators

In this section we want to use the biholomorphism in Theorem 5 and describe the Hecke operators on holomorphic quasi-modular forms. Let us take k = C and let Mat_d(2,Z) bethesetof2×2 matriceswithcoefficientsinZandwith determinantd.

Proposition 4. The d-th Hecke operator on the vector space of quasi-modular forms of weight mand differentialorder nisgivenby

T_d:M_mⁿ →M_mⁿ, T_df =

A

f||mA,

whereArunsthroughthesetSL(2,Z)\Matd(2,Z)and||isthedoubleslashoperator(12) forquasi-modular forms.

Proof. Letus consider two points(Ei,{αi,ωi}), i= 1,2 in the moduli spaceT. Letus also considerad-isogenyf :E1→E2with

f^∗ω2=ω1, f^∗α2=d·α1.

Wecantakeasymplecticbasisδ1,γ1 ofH1(E1,Z) andδ2,γ2 ofH1(E2,Z) suchthat f_∗[δ₁, γ₁]^tr=A[δ₂, γ₂]^tr,

where A∈Mat_d(2,Z) andtrmeanstranspose ofamatrix.Fromanother sidewehave [α₁, ω₁]B =f^∗[α₂, ω₂], whereB=

d 0 0 1

Therefore, if the period matrix associated to (Ei,{αi,ωi},{δiγi}), i = 1,2 is denoted respectivelybyx andxthen

xB=Ax.

Using Theorem 5,thed-thHeckeoperatoractsonthespaceofSL(2,Z)-invariantholo- morphic functionsonP by:

TdF(x) = 1 d

A

F(AxB⁻¹), x∈ P

whereA=

a₁ b₁ c₁ d₁

runsthroughSL(2,Z)\Mat_d(2,Z).Letf ∈M_mⁿ beaholomorphic quasi-modular form definedon theupperhalfplane. Bydefinitionthere isageometric modularformf˜∈C[t1,t2,t3] suchthatf isthepull-backoff˜bythecomposition(10).

LetFbetheholomorphicfunctiononPobtainedbythepush-forwardoff˜bytheperiod mapandthen itspull-backbyP →SL(2,Z)\P.Wehave

T_df(τ) =T_dF

τ −1

1 0

= 1 d

A

F(A

τ −1

1 0

B⁻¹)

= 1 d

A

F(

Aτ −1

1 0

d⁻¹(c₁τ+d₁) −c₁ 0 (c1τ+d1)⁻¹d

)

= 1 d

A

d^m(c1τ+d1)^−m n i=0

n i

(−c1)ⁱ(c1τ+d1)ⁱd⁻ⁱfi(A(τ))

=

A

f||mA. 2

Inthepassage fromthesecondto thirdequalitywe haveused A

τ −1

1 0

=

Aτ −1

1 0

c1τ+d1 −c1

0 (detA)(c1τ+d1)⁻¹

.

Forthepassage from thethirdto fourthequalitywe haveused thefunctional equation of f˜(and hence the SL(2,Z)-invariant function F) with respect to the actions in (6) and(7),see [7]Proposition6.Onecantaketherepresentatives

{A_i}:=

_d

c b 0 c

c|d, 0≤b < c

forthequotientSL(2,Z)\Matd(2,Z) andso Tdf(τ) = 1

d

cc=d, 0≤b<c

c^mfcτ+b c

. (17)

Inasimilarway tothecaseofmodularforms (see[1]§6)onecancheck that Tp◦Tq=

d|(p,q)

d^m−1T^pq

d2. Ifwewritef =_∞

n=0f_nqⁿ thenwehave:

(Tdf)n=

c|(d,n)

c^m−1f^nd

c2.

In particular if we set n = 0 then the constant term of Td(f) is f0σ_m−1(d). If f is an eigenvector of Td and the constant term of f is non-zero then the corresponding eigenvalueisσm−1(d).

7. RefinedHeckeoperators

Let Wd be the set of subgroups of _d^Z_Z × _d^Z_Z of order d and Sd be the set (up to isomorphism)ofabelianfinitegroupsoforderdandgeneratedbyatmosttwoelements.

Wehaveacanonicalsurjective mapW_d→S_d. Proposition 5.Wehave bijections

SL(2,Z)\Matd(2,Z)∼=Wd, (18) SL(2,Z)\Matd(2,Z)/SL(2,Z)∼=Sd, (19) both givenby

a b c d

→ Z² Z(a, b) +Z(c, d).

If dis square-freethenboth sides ofthesecond bijectionare singlepoints.

Proof. First note that the induced maps are both well-defined. The first bijection is alreadyprovedandusedinSection6.TheactionofSL(2,Z) fromtheleftonMatd(2,Z) corresponds to thebase change inthelatticeZ(a,b)+Z(c,d) in theright handsideof the bijection. The action of SL(2,Z) fromthe right on Matd(2,Z) corresponds to the isomorphismoffinite groupsintheright handsideofthebijection. 2

Any element inS_d is isomorphic to thegroup _d^Z

2Z ×_d1^Z_d2Z forsome d₁,d₂ ∈ Nwith d=d²₂d₁.Intherighthandsideof(19)thecorresponding elementisrepresentedbythe matrix

d₁d₂ 0 0 d₂

. Inthe geometric context, this means thatany isogenyof elliptic curves E1→E2 over analgebraicallyclosed fieldcanbe decomposedinto E1 α

→E1

→β

E₂, whereαisthemultiplicationbyd₂ andβ isadegreed₁ isogenywithcyclic center.

Note that

σ1(d) =

d=d²₂d1

ψ(d1).

We concludethatwehaveanaturaldecompositionofboth geometricand holomorphic Hecke operators:

Tdt=

d=d²₂d1

d^−m−2₂ ·T_d⁰₁t, t∈M_mⁿ (20)

where inthegeometric context

T_d⁰:M_mⁿ →M_mⁿ, T_d⁰(t)(E, ω) = 1 d

f:E1→E,deg(f)=d,ker(f) is cyclic

t(E1, f^∗ω),

andintheholomorphiccontext

T_d⁰:M_mⁿ →M_mⁿ, T_d⁰f =

A∈(SL(2,Z)\Matd(2,Z))⁰

f||mA.

Here,(SL(2,Z)\Mat_d(2,Z))⁰ isthefiberofthemap

SL(2,Z)\Matd(2,Z)→SL(2,Z)\Matd(2,Z)/SL(2,Z) overthematrix

d 0 0 1

.Thefactord^−m₂ in(20)comesfromthefunctionalequationof t with respect to theaction of Gm and thesecond part ofProposition 2. Note thatin the geometric context using the Galois action Gal(¯k/k), we cansee that T_d⁰ is defined overthefieldkandnotitsalgebraicclosure.WecallT_d⁰therefinedHeckeoperator.The refinedHeckeoperatorT_d⁰ willbeusedinthenextsections.Notethatifdissquarefree thenT_d⁰=T_d.

8. ProofofTheorem 1

Foraholomorphicquasi-modular formofweightmweassociatethepolynomial

P_f⁰(x) :=

A∈(SL(2,Z)\Matd(2,Z))⁰

(x−d·f||mA) =

ψ(d)

j=0

P_f,j⁰ x^j.

Proposition6.P_f,j⁰ is afullquasi-modular formof weight (ψ(d)−j)·m.

Proof. ThecoefficientP_f,j⁰ ofx^j isahomogeneouspolynomialwithrationalcoefficients andofdegreeψ(d)−j in

T_d⁰(f^k), k= 1,2, . . . , ψ(d)−j, weight(T_d⁰(f^k)) =k,

whereT_d⁰:M →M istherefinedd-thHeckeoperatordefinedinSection7.Here,wehave used(15).Forinstance,thecoefficientofx^ψ(d)−1is−d·T_d⁰fandthecoefficientofx^ψ(d)−2 is ^d₂²(T_d⁰f)²−^d₂T_d⁰f². Now theassertionfollows from the factthattheHecke operator T_d⁰sends aquasi-modular formofweightmtoaquasi-modularform ofweightm. 2

Using the fact that the algebra of quasi-modular forms over Q is isomorphic to Q[E₂,E₄,E₆] we concludethatP_f⁰(x) isahomogeneous polynomialof degreeψ(d)·m inthering

Q[x, E2, E4, E6], weight(E2k) =k, k= 1,2,3, weight(x) =m.

Thegeometric definitionofthepolynomialP_f⁰(x) is:

P_f⁰(x)(E, ω) =

(x−f(E₁, g^∗ω)), (21) where theproductistakenoveralldegreedisogeniesg:E1→Ewith cyclickernel.

ProofofTheorem1. Letusregardsi’sasholomorphicfunctionsontheupperhalfplane and ti’sas variables.Wedefine

I_d,i:=P_s⁰_i(t_i), i= 1,2,3.

For i = 1,2,3, T_d⁰s^k_i is a homogeneous polynomial of degree ki in Q[s₁,s₂,s₃], weight(s_i)=i,andhence,fromProposition 6itfollowsthatI_d,i(t_i,s₁,s₂,s₃) isahomo- geneouspolynomialofdegreei·ψ(d) intheweightedring(3). Wehave

si||2i

d 0 0 1

=d²ⁱ⁻¹si(dτ)

and so I_d,i(d²ⁱ·s_i(d·τ),s₁(τ),s₂(τ),s₃(τ)) = 0. By definition I_d,i is monic in t_i for i= 1,2,3.Finally,notethatfori= 2,3,T_d⁰s^k_i isahomogeneouspolynomialofdegreeki inQ[s2,s3],weight(si)=i,andso,Id,i(ti,s1,s2,s3) doesnotdependons1. 2

9. ProofofTheorem 2

Sincetheperiodmappmisabiholomorphism,itisenoughtoprovethesamestatement forthepush-forwardofRandVd undertheproduct oftwoperiodmaps:

pm×pm:T×T→SL(2,Z)\P ×SL(2,Z)\P.

Firstwedescribethepush-forwardofVd.UsingTheorem 4andthecomparisonofHecke operators inboth algebraicandcomplexcontext,wehave:

Vd ={((E1, ω1),(E2, ω2))∈T×T | ∃f :E1→E2, f^∗ω2=ω1, ker(f) is cyclic of orderd}.

Let us now consider the elliptic curves Ei,i = 1,2 as complexcurves. For ad-isogeny f :E1→E2suchthatker(f) iscyclic,wecantakeasymplecticbasisδ1,γ1ofH1(E1,Z) and δ₂,γ₂ ofH₁(E₂,Z) suchthat

f_∗δ1=d·δ2, f_∗γ1=γ2.

Therefore, if the period matrix associated to (E_i,{α_i,ω_i},{δ_iγ_i}), i = 1,2 is denoted respectivelybyxandy then

x=πd(y) :=

y1 dy2

d⁻¹y₃ y₄

.

Therefore, the push-forward of V_d under pm×pm and then its pull-back to P × P is givenby:

V_d^∗={(πd(y), y)|y∈ P}.

Thepush-forwardofthevector fieldR_d bypm×pmandthen itspull-backinP × P is givenbythevectorfield

R^∗=d(y₂ ∂

∂y1

+y₄ ∂

∂y3

)−(x₂ ∂

∂x1

+x₄ ∂

∂x3

)

where wehaveusedthe coordinates(x,y) forP × P.Now, itcanbe easilyshownthat theabovevectorfieldR^∗ istangenttoV_d^∗.

Remark1.ThelocusV_d^∗ containstheonedimensionallocus:

H˜ :={

τ −d d⁻¹ 0

,

τ −1

1 0

|τ ∈H}. (22) Notealsothatthepush-forwardoftheRamanujanvectorfieldRistangenttotheimage ofH→ P and restrictedto this locusit is _∂τ^∂ . Therefore, R^∗ istangentto the locusH˜ andrestrictedtothere isagain _∂τ^∂ .

10. ProofofTheorem 3

FromTheorem 1itfollowsthatI_d,1isalinearcombinationofthemonomials(4)and I_d,i, i = 2,3 is alinear combination of thesame monomials with a₁ = 0.The proof is aslightmodification ofan argumentinholomorphic foliations,see [10]. Weprove that Jd,i’srestricted toVd are identicallyzero.Weknow thatId,i is alinearcombination of αj’swithC(infactQ)coefficients:

I_d,i= c_jα_j.

SinceRdistangenttothevarietyVd,weconcludethatR^r_d(

cjαj)=

cjR^r_dαjrestricted toVd iszero.ThisinturnimpliesthatthematrixusedinthedefinitionofJd,irestricted toVd hasnon-zerokernelandsoitsdeterminantrestrictedtoVd iszero.

11. Examplesandfinal remarks

In order to calculate I_d,i’s using Theorem 3 we can proceed as follows: We use the Gröbner basis algorithm and find the irreducible components of the affine vari- ety given by the ideal Jd,1,Jd,2,Jd,3 and among them identify the variety Vd. In

practice this algorithm fails even for the simplest case d = 2. In this case we have deg(J2,1)= 42,deg(J2,2)= 40,deg(J2,3)= 69 andcalculatingtheGröbnerbasisof the ideal Jd,1,Jd,2,Jd,3 isahugeamountof computations.Weusetheq-expansionofti’s and wecalculateId,i,i= 1,2,3,d= 2,3.Wehavewrittenpowersof ti inthefirst row andthecorrespondingcoefficientsinthesecondrow.Formoreexamplesseetheauthor’s web-page.¹

I2,1:

t³₁ t²₁ t1 1

1 −6s₁ 12s²₁−3s₂ −8s³₁+ 6s₁s₂−2s₃

I2,2:

t³₂ t²₂ t2 1 1 −18s₂ 33s²₂ 484s³₂−500s²₃

I2,3:

t³₃ t²₃ t3 1

1 −66s3 −1323s³₂+ 1452s²₃ 10 584s³₂s3−10 648s³₃

I_3,1:

t⁴1 t³1 t²1 t1 1

1 −12s1 54s²₁−24s2 −108s³₁+ 144s1s2−64s3 81s⁴₁−216s²₁s2−48s²₂+ 192s1s3

I3,2:

t⁴₂ t³₂ t²₂ t2 1

1 −84s2 246s²₂ 63 756s³₂−64 000s²₃ 576 081s⁴₂−576 000s2s²₃

I3,3:

t⁴₃ t³₃ t²₃ t3 1

1 −732s3 −169 344s³2+ 171 534s²3 11 007 360s³2s3−11 009 548s³3 −502 020 288s⁶2+ 939 266 496s³2s²3−437 245 479s⁴3

ThepolynomialsI_d,i,i= 2,3 haveintegralcoefficients.Thisfollowsfromtheformula(21) andthefactthatgeometricmodularformscanbedefinedoverafieldofcharacteristicp.

Our computations show thatId,1 has also integralcoefficients. However, the notionof a geometric quasi-modular form is only elaborated over a field of characteristic zero (see [9]). The proof of such a statement for Id,1 would need a reformulation of the definitionofgeometric quasi-modularforms.

One of the fascinating applications of modular curves X₀(d) is formulated in the Shimura–Taniyama conjecture, nowthe modularity theorem.Itstates thatany elliptic curveoverQmustappearinthedecompositionoftheJacobianofX0(d),wheredisthe conductoroftheellipticcurve(see[12]forthecaseofsemi-stableellipticcurvesand[2]

forthecaseofallellipticcurves).Itwouldbeinterestingtoknowwhetherthedifferential equationapproachofthepresentarticletomodularcurveshassomeimplicationsinthis direction.

References

[1]TomM.Apostol,ModularFunctionsandDirichletSeriesinNumberTheory,secondedition,Grad.

TextsinMath.,vol. 41,Springer-Verlag,NewYork,1990.

1 http://w3.impa.br/~hossein/WikiHossein/files/Singular%20Codes/2014-08- QuasiModularFormsHeckeOperators.txt.