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Linear differential equations with finite differential Galois group

van der Put, M.; Sanabria Malagon, C.

Published in: Journal of algebra DOI:

10.1016/j.jalgebra.2020.01.023

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van der Put, M., & Sanabria Malagon, C. (2020). Linear differential equations with finite differential Galois group. Journal of algebra, 553, 1-25. https://doi.org/10.1016/j.jalgebra.2020.01.023

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Contents lists available atScienceDirect

Journal

of

Algebra

www.elsevier.com/locate/jalgebra

Linear

differential

equations

with

finite

differential

Galois

group

M. van der Puta, C. Sanabria Malagónb, J. Topa,∗

aBernoulli Institute, University of Groningen, the Netherlands bUniversidad de Los Andes, Bogotá DC 111711, Colombia

a r t i c l e i n f o a b s t r a c t

Article history:

Received15October2019 Availableonline20February2020 CommunicatedbyGunterMalle

MSC:

34M15 34M50

Keywords:

DifferentialGaloistheory Inverseproblem

Invariantcurves Schwarzmaps

Evaluationofinvariants

For a finite irreducible subgroup H ⊂ PSL(Cn) and an irreducible,H-invariantcurveZ⊂ P(Cn) suchthatC(Z)H = C(t), a standard differential operator Lst ∈ C(t)[dtd] is constructed.Forn = 2 this isessentiallyKlein’s work. For

n> 2 an actualcalculationofLstisdonebycomputingan evaluationofinvariantsC[X1,. . . ,Xn]H→ C(t) andapplying ascalarformofatheoremofE. Compointina“Procedure”. Also in some cases where Z is unknown evaluations are produced.

This new method is tested for n = 2 and for three irreducible subgroups of SL3. This supplements [18]. The

theorydevelopedhererelatestoandcontinuesclassicalwork ofH.A. Schwarz,G. Fano,F. KleinandA. Hurwitz.

©2020TheAuthor(s).PublishedbyElsevierInc.Thisisan openaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introductionandsummary

LetC denoteanalgebraicallyclosedfieldofcharacteristiczero.Letk beC(z) andlet

k denotethealgebraicclosureofk.BothfieldsareprovidedwiththeC-linearderivation

* Correspondingauthor.

E-mail addresses:m.van.der.put@rug.nl(M. van der Put),c.sanabria135@uniandes.edu.co

(C. Sanabria Malagón),j.top@rug.nl(J. Top).

https://doi.org/10.1016/j.jalgebra.2020.01.023

0021-8693/©2020TheAuthor(s). PublishedbyElsevierInc. ThisisanopenaccessarticleundertheCC BY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

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f → f with z = 1. Thepositive butnotexplicit orconstructive answerto theinverse

problem ofGaloistheoryis:

Forany finite groupG there isaGaloisextension⊃ k with groupG.

Indeed, a proof for the complex case uses analytic tools, in particular the “Riemann

Existence Theorem”.The proof for any fieldC as above is deduced from the complex

case.Thereisanextensiveliteratureonsolvingtheinverseproblemexplicitly forcertain

finite groups.

A finite Galoisextension ⊃ k canbe given as thesplitting fieldof apolynomialP

ink[T ].Sometimes,amoreefficient wayistodescribe ⊃ k asthePicard–Vessiotfield

of alineardifferential operatorL in k[∂] with ∂ = d

dz. From apolynomialP for ⊃ k

onecaneasily computeadifferentialoperatorL for ⊃ k,see[18,§1] and [8,§2]. The

other directionisfarmorecomplicated(see(ii)below).

Letπ denotetheprofiniteGaloisgroupofk/k.Thereisawellknownbijectionbetween

themonicdifferentialoperatorsL∈ k[∂] ofordern,suchthatallsolutionsarealgebraic

over k,andtheC-vectorspacesV ⊂ k ofdimensionn whicharestable underπ.

Indeed,oneassociatestoL theπ-stablespace{f ∈ k |L(f )= 0} (i.e.,the contravari-antsolutionspace).Ontheotherhand,lettheπ-stableV ⊂ k havebasisb1,. . . ,bn over

C.There isauniqueoperatorL= ∂n+ a

n−1∂n−1+· · · + a1∂ + a0 withallai∈ k such

thatallL(bj)= 0.Theuniquenessandtheπ-stabilityofV implythatallai∈ k.

AmoreabstractwaytocomparedifferentialequationsandGaloisextensions⊃ k is

thefollowing.ThecategoryDiffk/k thatwestudyhere,hasasobjectsthefinite

dimen-sionaldifferentialmodulesM overk whichbecometrivialoverthefieldk.Thiscondition

onM isequivalenttoM havingafinitedifferentialGaloisgroup.Themorphismsinthis

category arethek-linearmapsthatcommutewithdifferentiation.

LetReprπdenotethecategoryofthe(continuous)representationsofπ onfinite

dimen-sionalC-vector spaces.The functor Diffk/k → Reprπ,whichassociates to adifferential

module M its(covariant) solutionspaceker(∂,k⊗kM ), isknowntobe an equivalence

of (Tannakian) categories.

The aim of this paper is to make this equivalence of categories explicit for special

cases.There aretwodirections toconsider:

(i)Computeadifferentialoperatorconnectedto agivenrepresentation ofagivenfinite

groupandsomeadditionaldata.

(ii)DescribeorconstructthePicard–VessiotfieldforagivenmoduleM∈ Diffk/k,when

M is representedbyadifferentialoperatorL.

Werecallsomeearlierresultson(i)and(ii).

Regarding (i): The Schwarz’ list (see [18] for a modern version) and Klein’s theorem (e.g., see [1] and [2])are classicalresults forthe specialcase of ordern = 2. Werecall

thestatementofKlein’stheorem:

for eachof theirreducible subgroups G⊂ PSL(C2) (so G∈ {D

n,A4,S4,A5}),there is

a standard order two differential operator Lst having G as projective differential

Ga-lois group. It has the universal property that any order two differential operatorwith

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Inthe casen = 3 Hurwitz’ paper [12] produces examples. This methodwas refined in[18]. Klein’stheorem is generalizedin,e.g.,[2,20–22]. Notmuchseemsto havebeen doneforn> 3.Here (Section3)we treatthegeneralcase.

Regarding (ii): This was initiated by J. Kovacic inhis paper [14] dealing with n = 2. Therearemanysubsequentpapers[23,10,11] consideringsmalln.Forgeneraln thereis

workofE. Compointand M.F.Singer[6,7]. Thepaper[4] discussestheparticular case

ofhypergeometricdifferential equations.

We now describe the present paper, which is mainly concerned with (i) but also

contributesto(ii) byexploitinginvarianttheory forfinitegroupsandCompoint’swork

[6].

Section2associatestoadifferentialoperatorL∈ k[∂] withallsolutionsink,

geomet-ricobjects: aPicard–Vessiot curve, aFanocurve, Schwarzmaps, projective differential

Galoisgroupsand anevaluationof invariants.

In Section3 Klein’s theorem for order two is generalized, resulting ina subtle

con-struction of astandard differential operator Lst (Theorem 3.1). The data forthis

con-struction are a finite irreducible subgroup H ⊂ PSL(Cn), an H-invariant irreducible

curveZ ⊂ P(Cn) such thatthenormalization ofZ/H has genus zeroandavariable z

withC(Z/H)= C(z).IntheconstructionofLst thegroupH isreplacedbyasubgroup

˜

H ⊂ SL(Cn) whichisminimal suchthatH˜ → H issurjective.

The“universalproperty” ofLst isthefollowing:

anydifferential operatorL with projectivedifferential GaloisgroupisomorphictoH

andFanocurveisomorphictoZ isaweakpullbackofLst(see3.1and3.7).Thisclarifies

andextendstheworkof[2,20–22].

Section4.Forordern= 2 theFanocurveisbydefinitionP (C2) andthecomputation

of the standard operators Lst is easy and produces the classical operators. For n >

2 however, the construction of Lst as described in Section 3 does not in an obvious

way result in a computation of this operator. A new method for the computation of

Lst is introduced. Wederive a“scalar version” of Compoint’stheorem (see4.2) which

is roughly the following. Let the homogeneous polynomials f1,. . . ,fN be generators

forthe ringof invariants C[X1,. . . ,Xn]H˜. Anevaluation ofthe invariantsis asuitable

homomorphism ev : C[X1,. . . ,Xn]H˜ → C(t) and the Picard–Vessiot field of Lst is

K := C(t)[X1,. . . ,Xn]/(f1− ev(f1),. . . ,fN− ev(fN)).

Our“Procedure”4.3computingLstworksasfollows.Asetofhomogeneousgenerators

f1,. . . ,fN and theirrelations are taken(ifpossible) from theliterature. Thegiven

H-invariant irreducible curve Z ⊂ P(Cn) with C(Z)H = C(t) effectively produces an

essentiallyuniqueevaluation,see4.6.From theexplicitpresentationofK onecomputes

thederivationD onK extending dtd.ThenoneobtainsthemonicoperatorL∈ C(t)[dtd]

of degree n with kernel CX1+· · · + CXn, where Xi denotes the image of Xi in K.

FinallyLst isobtainedbynormalizingL suchthatitscoefficientof(dtd)n−1 iszero.

OurProcedurecanbeseenasthe“opposite”ofanalgorithm,byM.vanHoeijand

J.-A.Weil[11],whichcomputesforagivendifferentialoperator,theassociated evaluation

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Section5.For ordern= 2,weshow how toobtainevaluations oftheinvariantsand

apply theProcedure to produce the knownstandard operators.For the groupG168

PSL(C3) andthe Klein curve Z ⊂ P2

C, adirect computationof the standardoperator

from its construction in §3 fails. However, evaluation and the Procedure produce the

standardoperator.

The LIST, copied from [18], contains all possibilities, determined by the Riemann

Existence Theorem,oforder3differentialoperatorsoverC(z) (up toequivalence)with

group G168 and singular locus {0,1,∞}. In most of these cases one does not know a

stable Z ⊂ P2

C such that the normalization of Z/G168 has genus zero. The methods

of [18] producedexplicitthirdorderequations forabouthalfofthecases.Forthesame

casesournewmethodofevaluationandtheProcedureproducesmoreeasilythestandard

equations.

In[18] nostandardequationforthegroupHSL3

72 wasfound.Ournewmethodsproduce

anequation.

In Section 6 standard equations for A5 ⊂ SL3 are studied. Moreover, properties in

relationwiththepreimageASL2

5 ⊂ SL2(C) ofA5⊂ PSL(C2) andthelistsofdifferential

operators in[18] arediscussed.

2. Objectsassociatedtoa differentialoperatorL over k = C(z) withfinite

differential Galoisgroup

L hastheformdn

z+ an−1dn−1z +· · · + a0 withallai∈ C(z),dz= dzd andallsolutions

are supposedto bealgebraicover C(z).AssociatedtoL is:

(1) ThePicard–Vessiot field K⊃ C(z) withitsGaloisgroupG.

(2) The (contravariant) solution space V ⊂ K of L with the action of G on it. The

image ofG⊂ GL(V ) intoPGL(V ) willbe denotedbyGproj andiscalled theprojective

differential Galoisgroup.

(3) The Picard–Vessiot curve Xpv is the smooth, irreducible, projective curve over C

with function fieldK. G acts onXpv and there is anisomorphism Xpv/G = Pz1. Here

Pz1denotes theprojectivelinewithfunctionfieldC(z).

(4)Evaluationof theinvariants.OneconsidersaC-linearhomomorphismφ: C[X1,. . . ,

Xn] → K which sends the variables X1,. . . ,Xn to a basis of V . The C-linear action

of G on C[X1,. . . ,Xn] is defined by the G-invariance of CX1 +· · · + CXn and the G-equivariance of φ. This makes G into asubgroup of GL(n,C).The homomorphism

φ induces ahomomorphism ev : C[X1,. . . ,Xn]G → KG = C(z) which wewill call the evaluationoftheinvariants.WriteC[X1,. . . ,Xn]G= C[f1,. . . ,fN] wheref1,. . . ,fNare

homogeneousgeneratorsandev mapseachfi to anelement inC(z).

NowsupposethattheactionofG onV isknownandisirreducible,i.e.,noproperlinear

subspace= (0) ofV isinvariantunderG.IfwedefinetheactionofG onCX1+· · ·+CXn

such thatanequivariant φ withφ(CX1+· · · + CXn)= V exists,then thisφ is unique

uptomultiplicationbyascalarc∈ C∗.Asaconsequence,theevaluationmapisunique

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(5) The Fano curve. H ⊂ ker(φ), the“homogeneous kernel”, is the ideal generated by

thehomogeneouselements inker(φ).For n= 2 onehas H = 0.For notationalreasons

wewillcall P (V )= P1 itselftheFano curve inthiscase.

Suppose that n > 2, then H defines an irreducible curve in Pn−1, invariant under

the action of G. Indeed, H is the homogeneous ideal induced by the kernel J of the

corresponding homomorphism C[X2

X1,. . . ,

Xn

X1] → K. It is a curve since K/C has

tran-scendence degree 1. The curve in Pn−1 defined by H will be denoted by Xf ano and

will be called the Fano curve. This curve was indeed consideredby Fano inhis

1900-paper[9].Wenote thatXf ano canhavesingularities.Fromthedefinitiononeseesthat

C(Xf ano)= C(xx21,. . . ,xxn1),where x1,. . . ,xn isabasisof V ⊂ K.

(6)The Schwarzmap.ThehomomorphismC[X1,. . . ,Xn]/H→ K inducesamorphism

ofcurvesXpv → Xf ano whichisG-equivariant.After dividingbyG weobtaina

multi-valued map Schw : P1

z = Xpv/G· · · → Xf ano called the Schwarz map. Forn = 2 it is

thewellknownclassicalSchwarzmap.

After dividing byG we obtainqSchw : P1

z = Xpv/G → Xf ano/Gproj which canbe

calledthequotientSchwarzmap.WenotethatXf ano/Gproj canhavesingularities.The

relationbetweenXpv andXf ano isingeneralnotobvious.

Lemma2.1. SupposethatqSchw : P1

z = Xpv/G→ Xf ano/Gproj isbirational.Letc(G)⊂ G bethe groupof themultiples of the identity belonging toG. Since c(G) acts trivially on the curve Xf ano, the map Xpv → Xf ano factors over Xpv/c(G). The morphism Xpv/c(G)→ Xf ano isbirational.

Proof. One has K = C(Xpv) ⊃ Kc(G) ⊃ C(Xf ano). The group Gproj = G/c(G)

acts faithfully on Kc(G) = C(X pv/c(G)) and (Kc(G))G proj = C(z). Since C(Xf ano) = C(x2 x1,· · · , xn x1),thegroupG

projactsfaithfullyonC(X

f ano).ByassumptionC(Xf ano)G proj

= C(z).ThereforeKc(G)= C(X

f ano). 

3. Constructionofastandardoperatorandpullbacks

Theorem3.1. Letthefollowingdatabegiven:

(i)aC-vector spaceV withn:= dim V ≥ 2, (ii)an irreduciblefinitesubgroup H ⊂ PSL(V ),

(iii)anirreducibleH-invariantcurveZ⊂ P(V ) suchthat thenormalisationofZ/H has

genus 0and

(iv)avariablez suchthat C(Z)H= C(z).

Thedata V,H,Z,z determineadifferential operator Lst= ( d dz) n+ a n−2(dzd)n−2+· · · + a0∈ C(z)[ d dz] such that

(a)The C-vectorspaceW :={f ∈ C(z) |Lstf = 0} hasdimensionn (i.e.,allsolutions arealgebraic).

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(b) LetG⊂ SL(W ) denotethedifferential Galoisgroup ofLst.ThereisaC-linear iso-morphismφ: W → V such thattheprojectivedifferential GaloisgroupGproj ismapped isomorphicallytoH andtheFanocurveXf ano⊂ P(W ) ofLstismappedisomorphically to Z⊂ P(V ).

Remarks 3.2.(0). It is a standardfact thatin (b) onehas G ⊂ SL(W ); see, e.g., [17,

Exc. 1.35 5(b)-(c)].

(1). The operatorLst will be called the standard operator for the data V,H,Z,z.For

n = 2,one hasZ = P (V ). FurtherZ/H isidentified with Pz1 andso C(Z/H)= C(z). OneknowsthatthepossibilitiesforH areDn,A4,S4,A5.Thevariablez ischosensuch thatz = 0,1,∞ arethebranchofZ→ Z/H.ThusLst dependsessentiallyonlyonH.

(2).IntheproofofTheorem3.1wewilluseagroupH˜ ⊂ SL(V ) whichmapssurjectively

to H and isminimal withrespect to thisproperty. Thekernelof H˜ → H has theform

· 1|λm= 1} foracertaindivisorm ofn.

(3). Inthe construction of Lst only the data V,N,Z,z are used. It canbe shownthat

theoperatorLst isactuallydetermined bytheproperties(a)and(b)inTheorem3.1.

(4).TheactionofH onZ isfaithful.Indeed,sinceH isirreducibleandZ isH-invariant,

Z is not contained in a proper projective subspace of P (V ). By induction on i, one

finds for i= 1,. . . ,n, elements z0,. . . ,zi ∈ Z such thatz0,. . . ,zi is notcontained ina

projectivesubspaceofdimension< i.Further,foreachj,onecanreplacezjbyinfinitely

manyelements z˜j∈ Z suchthatz0,. . . ,zj−1,z˜j,zj+1,. . . ,zn hasthesameproperty.

Supposethath∈ H acts asidentity onZ. Thenh∈ PGL(V ) has adiagonal matrix

withrespect toabasisofV correspondingtoanysequencez0,. . . ,zj−1,z˜j,zj+1,. . . ,zn.

This impliesthath= 1. 

Proof. We start the construction of Lst. The above data yield inclusions C(z) =

C(Z)H ⊂ C(Z) ⊂ C(z). The variable z is given in the data and the embedding C(Z) ⊂ C(z) is unique up to an automorphism of C(Z) over C(z), i.e., an element

of H. We would like to identify V with the solution space in C(z) of the standard

operatortobe constructed.However,V doesnotlieinC(Z).

One chooses any  ∈ V,  = 0. For any v ∈ V one considers the restriction of the

rational function v on P (V ) toZ (this makes sense becauseZ isnot containedin the

hypersurface= 0).Write V forthefunctionsonZ obtainedinthisway,so V ⊂ C(Z).

TheC-vectorspace V isnotinvariantunderH,or whatisthesame,itisnotinvariant

underπ.Thefollowinglemma isthekeyingredient oftheconstruction.

Lemma 3.3.There existsanelement f ∈ C(z)∗ suchthat fV isinvariantunder π.The

canonicalmapP (V )→ P(fV), givenbyv→ f ·v

 isequivariantfortheaction of π.

Proof. ThegroupH is˜ supposedtohavethepropertiesofRemarks3.2.Foreachσ∈ H,

onedenotesbyσ an˜ elementinH with˜ imageσ.Nowσ(V

)= V ˜ σ =  ˜ σ· V .Theterm  ˜ σ

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a1-cocycle. ByHilbert90, there is anelement f ∈ C(Z) such that σff · (σ˜)m = 1 for

allσ∈ H.

For the case m = 1 we conclude that fV ⊂ C(Z) is invariant under H (and thus

also under π). For the case m > 1 we claim that the equation Tm− f is irreducible

overC(Z).Assumingthis claim,thefieldC(Z)(fm) with fmm= f isaGaloisextension

of C(z) since for every σ ∈ H one has σff is an mth power inC(Z). We mayembed

C(Z)(fm) intoC(z) and concludethatfmV isinvariantunderπ.

Now weprove the claim. If theequation Tm− f is reducible over C(Z),then there

exists a proper divisord of m and an element g ∈ C(Z) withgd = f . Theexpression

E(˜σ):= σgg · (˜σ )m/dhasthepropertyE(˜σ)d= 1.Onecanconsider foreachσ∈ H the

elements σ˜ ∈ ˜H such that E(˜σ) = 1. This defines a proper subgroup of H which˜ has

imageH.Thiscontradictstheassumptions onH.˜

Thelaststatementofthelemmafollowsfrom σ(fv)= σff ·σ · fσv . 

ThemonicoperatorL ofordern overC(z),definedbyker(L,C(z))= W := f·V has

itscoefficientsinC(z),sinceW isinvariantunderπ. ThisoperatorL isnotyetunique

sincewehavemadechoicesfor andf .

ThestandardoperatorLstisdefinedtobe theoperatoroftheformLst= (dzd)n+ 0·

(dzd)n−1+· · · , obtainedfrom theabove L by ashift dzd → dzd + a for suitable a = hh withh∈ C(z)∗.

Wefinish theproof of Theorem3.1by statingthefollowingproperties:

(1).Lst doesnotdependonthechoicesof and f inLemma3.3.

(2).ThesolutionspaceofLst hastheformg· W forcertaing∈ C(z)

.

(3).LetG⊂ SL(g · W ) denotethedifferentialGaloisgroupofLst.Fromg· W = g · f ·V

oneobtainsanaturalidentificationoftheprojectivespacesP (V ) andP (g· W ) andafter

thisidentification onehasGproj= H andtheFanocurve ofL

st isZ.

Statement(1)followseasilyfromLemma3.4andObservation3.5,part (1).Statements

(2)and(3)follow fromtheconstructionofLst. 

Lemma3.4. LetL1,L2 bemonicdifferentialoperatorsoverC(z) suchthatalltheir

solu-tionsare algebraic.LetV1,V2⊂ C(z) denote thetwosolution spaces.The followingare

equivalent:

(a).L1 isobtained fromL2 byashift dzd → dzd + a forsome elementa∈ C(z). (b).Thereexistsf ∈ C(z)∗ suchthatV2= f V1.

Proof. (a)⇒(b).LetL1beobtainedfromL2bytheshift dzd → dzd + a.Onewritesa= f

 f

withf insomedifferentialfieldcontainingC(z).OnefindsV2= f V1.SinceV1,V2⊂ C(z) oneactuallyhasf ∈ C(z)∗.

(b)⇒(a).IfV2= f V1,then clearlyL1= f−1◦ L2◦ f.Sincef−1◦dzd ◦ f = dzd +f

 f,one

hasthatL1 isobtainedfrom L2 bytheshift dzd → dzd +f



f. Note that f

f ∈ C(z) since

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Observations 3.5. (1). Forgeneral monic differentialoperators L1,L2 oforder n,

prop-erty (a)ofLemma3.4is calledprojective equivalence.Ifboth L1 andL2 havetheform

dn

z + 0· dn−1z +· · · ,thenprojectiveequivalence impliesequality.

(2).Theimplication(b)⇒(a)inLemma3.4holdsforgeneraldifferentialoperators.

How-ever(a)⇒(b)isingeneralfalsesincetheequationf = af witha∈ C(z),neednothave

asolutiononC(z)∗.

(3). For differential modulesM1,M2 there is asomewhatdifferent notion ofprojective

equivalence definedby:there isa1-dimensionalmoduleE suchthatM1⊗ E ∼= M2.

(4). Projective equivalence ofsubgroups G1,G2⊂ GL(V ) means thatGproj1 = G

proj

2

PGL(V ). Projectiveequivalenceof operatorsimpliesprojective equivalenceoftheir

dif-ferential Galoisgroupsbuttheconverseis false. 

Definition 3.6.Consider a homomorphism φ : C(z)[dzd] → C(x)[dxd] of the form: z →

φ(z) ∈ C(x)\ C and dzd → φ(z)1 (dxd + b) with b ∈ C(x). Let L ∈ C(z)[dzd]. A weak pullback of L isanoperatoroftheforma· φ(L) witha∈ C(x)∗.Therestrictionofφ to C(z)→ C(x) iscalled thepullbackfunction.

Proposition 3.7. Let L∈ C(s)[dsd] be an operatorof order n suchthat all solutions are

algebraic andletM⊂ C(s) denoteitssolution space.Thedifferential GaloisgroupG of L is asubgroup ofGL(M ).

Suppose that Gproj ⊂ PSL(M) is irreducible. According to §2, part (5) and (6), L determinessomeXf ano⊂ P(M) andC(Xf ano)G

proj

isasubfieldofC(s).Choosez such that C(z)= C(Xf ano)G

proj .

Then L is a weak pullback of the standard operator Lst determined by thedata M , H = Gproj⊂ PSL(M) and Z = X

f ano⊂ P(M) andthevariable z.

Proof. By construction the standard operator Lst has solution space h· W for some

h ∈ C(z)∗, where W = fM

m for suitable f ∈ C(z)

and m ∈ M, m = 0. Further, the

inclusion C(z) = C(Xf ano/Gproj)⊂ C(s) determines the pullback function.Using 3.4

and 3.6oneverifiesthatthispullbackfunctionappliedtoLst producesL. 

Remark 3.8.A standard differential equation forgiven H ⊂ PSL(V ),Fano curve Z

P (V ) andvariablez canbeaproperpullbackofanotherstandardequation.Thisoccurs

essentially only when H is a proper subgroup of a finite automorphism group of (the

desingularizationof) Z.

Example3.9. AcalculationofthestandardoperatorLst,usingtheaboveconstruction,is

possible.Onehastocomputethef inLemma3.3andonehastocomputethederivation

on C(Z)[f ] in order to compute themonic differential operatorL with solutionspace

fV ⊂ C(Z)[f].Furtheracomputationofageneratorof C(Z)Gproj isneeded. However

forthecasen= 2 thecalculationiswellknown([1,3])andrathereasy.Weillustratethis forthecaseH = A4⊂ PSL2and itspreimage H = A˜ SL4 2 inSL2.

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ForZ = P1 weusehomogeneouscoordinatesx,y and thefunctionfieldis C(t) with t= yx.Accordingto[5],theinvariantsundertheactionofASL2

4 aregenerated by:

Q3 = xy(x4− y4), Q4 = (x4+

−12x2y2+ y4)· (x4−12x2y2+ y4), Q

6 = (x4+

−12x2y2+ y4)3+ (x4−12x2y2+ y4)3. There isonerelationQ2

6− Q43− 4Q34 = 0.

Thefield ofthe homogeneousinvariants of degreezero isgenerated overC by Q6

Q2 3 and

Q3 4

Q4

3 andthere isonerelation(

Q6 Q2 3) 2= 1+ 4Q3 4 Q4

3. Hencewe cantakez =

Q6

Q2

3 where x,y in

this expressionis replaced byx,tx. This expressesz as rationalfunction int ofdegree

12.Thus dzdt isalsoknown.

Now V = Cx+ Cy, take  = x, then V = C1+ Ct. Then f ∈ C(t) should satisfy (σx˜x)2 = f

σf. An explicit choice for f turns out to be

1

t where t := dt

dz. Then the

Picard–Vessiot field is C(t)[√t]. The operator that we want to compute has solution

spaceC1

t+ C t

t.ThisleadstothestandardoperatorforcaseA4.Theotherstandard

operators forn= 2 canbecomputed inasimilar way. This“classical”calculation fails

forn> 2 andoneneedsthenewmethod“evaluationof invariantsandProcedure”(see

§4,3). This newmethodwill be appliedin§5 foranother computationof thestandard

operatorsforn= 2 and forcaseswithn> 2.

Observation3.10.The singularpointsof thestandardequations. LetL∈ C(z)[dzd] bean

operatoroforder n such thatitsPicard–Vessiot fieldisafinite extension ofC(z). The

singularpointsofL,whicharenotapparent,arethebranchpointsofXpv→ Pz1.Indeed,

suppose that z = 0 is not abranch point, then the solutionsof L live at any point p

abovez = 0. Thefraction fieldof OXpv,p canbe identified with C((z)) and contains n

independentsolutionsofL.Itfollows thatthesingularityisat mostapparent.

Inthe specialcase Lst andG = Gproj = H,one canidentifyXpv with the

normal-ization Z of˜ Z ⊂ P(V ) and the non apparent singularpoints are the branchpoints of

˜

Z→ P1

z. Inthegeneralcase,thecyclicextensionXpv→ ˜Z canbe responsibleformore

singularitiesofLst.

4. Compoint’stheoremandevaluationofinvariants

Notationandassumptions:

Suppose that the differential equation y = Ay over k = C(z) has a reductive

dif-ferential Galois group G ⊂ GLn(C). The differential algebra R := k[{Xi,j},D1] (with

D = det(Xi,j) ) isdefinedby(Xi,j )= A· (Xi,j).

LetI beamaximaldifferentialidealinR andK thePicard–Vessiotfieldobtainedas

fieldoffractions ofR/I.

GLn(C) actsontheC(z)-algebraR bysending thematrixofvariables (Xi,j) to the

matrix(Xi,j)· g for any g ∈ GLn(C). Then G isidentified with the g ∈ GLn(C) such

thatgI = I.

Thealgebra of invariantsC[{Xi,j}]G is generated over C by homogeneouselements

f1,. . . ,fN (since G is reductive). The natural map R → K induces a homomorphism eve: C[{Xi,j}]G→ C(z) whichiscalled theevaluationoftheinvariants.

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Theorem 4.1 (E. Compoint 1998). The ideal I ⊂ R generated by the elements {f1

eve(f1),. . . ,fN− eve(fN)} isamaximal differential ideal.

The proofof Compoint’s theorem,[6], hasbeen simplifiedin[3] andTheorem 4.1is

almostidenticaltotheformulationin[3].Notethatalthough[3] formulatestheresultfor

C = C,the argumentiscompletely algebraic hencetheresultholdsforany C.Wewill

apply Compoint’stheorem forthecaseoffinitedifferential Galoisgroups.Moreover we

willneedaformulationintermsofdifferentialoperators(orscalardifferentialequations).

Notation andassumptions:

Let L = (dzd)n + a

n−1(dzd )n−1+· · · + a1dzd + a0 over C(z) have afinite differential

GaloisgroupG andPicard–VessiotfieldK⊂ C(z).

Considerthehomomorphismφ: R0= C(z)[X1,. . . ,Xn]→ K whichsendsX1,. . . ,Xn

toabasisofthesolutionspaceofL inK.G actsC(z)-linearonR0 byaC-linearaction

onCX1+· · · + CXn whichcoincideswiththeactionofG (orofπ)onthesolutionspace

of L.

The restriction of φ to C[X1,. . . Xn]G → C(z) is also called the evaluation of the

invariants and denoted by ev (seealso §2). Write C[X1,. . . ,Xn]G = C[φ1,. . . ,φr] for

certain homogeneouselements φk.

Corollary 4.2. The kernel of φ : C(z)[X1,. . . ,Xn] → K is generated by the elements

1− ev(φ1),. . . ,φr− ev(φr)}.

Proof. Write again R0 = C(z)[X1,. . . ,Xn] and R := C(z)[{Xij}

j=0,...,n−1

i=1,...,n ] where X j i

denotes formally thejth derivativeof Xi (alli,j).The map φ: R0 → K has aunique

extension φe : R → K defined by φe(Xij) = φ(Xi)(j) (all i,j). The restriction of φ to RG

0 → C(z) is calledev andtherestrictionofφeto RG→ C(z) is calledeve.

By Compoint’s theorem, the ideal ker(φe) ⊂ R is generated by the set {F −

eve(F ) | F ∈ RG}. We want to prove that the ideal ker(φ) ⊂ R0 is generated by

{F − ev(F ) |F ∈ RG0}.Wewill construct aC(z)-algebrahomomorphism Ψ: R → R0 whichhasthefollowingproperties:Ψ(r)= r forr∈ R0;Ψ(Xi0)= Xi;φ◦ Ψ= φeandΨ

is G-equivariant.

Consider an element ξ ∈ ker φ. Then also ξ ∈ ker φe and ξ is a finite sum



c(F )· (F − eve(F )) with F ∈ RG and c(F ) ∈ R. Applying Ψ to this expression

yields ξ = Ψ(c(F ))· (Ψ(F ) − Ψ(eve(F )). Since Ψ is G-equivariant Ψ(F ) ∈ RG0.

Moreover Ψ(eve(F )) = ev(Ψ(F )). This implies that ξ lies in the ideal generated by

the{F − ev(F )|F ∈ RG

0} intheringR0.

Construction of Ψ. Define a C-linear derivation E : R0 → R0 by E(z) = 1 and, for

i= 1,. . . ,n,E(Xi)∈ R0hasthepropertythatφE(Xi)= φ(Xi).WenotethatE exists

since the map φ : R0 → K is surjective. Then D := #G1 g∈GgEg−1 : R0 → R0 is a

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DefinetheC(z)-algebrahomomorphismΨ: R→ R0byΨ(Xij)= Dj(Xi) foralli,j.

The first two propertiesof Ψ areobvious. Further φ(Ψ(Xij))= φ(Dj(Xi)) = φ(Xi)(j)

(for all i,j; the case j = 1 given earlier implies the general case) and so φ◦ Ψ = φe.

FinallyΨ isG-equivariantbecauseD isG-equivariantandtheactionsofG onthevector

spacesCX1j+· · · + CXj

n,forj = 0,. . . ,n− 1,areidentical. 

Theexplicitdescriptionofthekernelofφ giveninCorollary4.2providesanimportant

stepinthecomputationof astandardoperator,as willnowbe explained.

Procedure 4.3.Constructing the differential operator from an evaluation. Let an

irre-ducible finite group G ⊂ GL(Cn) be given. The group G acts on C[X

1,. . . ,Xn] by

identifyingCn withCXj.SupposethatC[X1,. . . ,Xn]G= C[f1,. . . ,fN] withknown

homogeneouselementsf1,. . . ,fN.

ConsideraC-algebrahomomorphismh: C[X1,. . . ,Xn]G→ C(z) suchthattheimage

of h generates the field C(z) over C. We will call such h again an evaluation of the invariants. The aim is to compute a differential operator L = dn

z + an−1dn−1z +· · · + a1dz+ a0 over C(z) thatinduces the groupG and suchthattheevaluation ev defined

aboveCorollary 4.2isequalto h.

The C(z)-algebra R := C(z)[x1,. . . ,xn] = C(z)[X1,. . . ,Xn]/I, where I = (f1

h(f1),. . . ,fN− h(fN)),hasfinitedimensionoverC(z) (byobservingthatC(z)[X1,. . . ,

Xn] is finiteoverC(z)[f1,. . . ,fN]).

(a).WeassumethatR isafield.Wenotethatintheoppositecase,R cannotbeaPicard–

Vessiotfield forasuitable operatorover C(z). Theaction of G on Cx1+ . . . + Cxn is

inducedbytheirreducibleactionofG onCX1+· · ·+CXn.HenceeitherCx1+. . .+Cxn=

(0) oritisisomorphictoCX1+· · ·+CXn.Theassumptionthattheimageofh generates

C(z) impliesthatCx1+· · · + Cxn= 0,henceithasdimensionn.

The derivation dzd has aunique extension to R which we call D.˜ There is aunique

operator L := ˜˜ Dn+ a

n−1D˜n−1+· · · + a1D + a˜ 0 (with all ai ∈ R) having kernel the n-dimensionalvectorspaceCx1+ Cx2+· · · + Cxn.ByuniquenessandtheG-invariance

of Cx1+ Cx2+· · · + Cxn,the operatorL is˜ G-invariantand therefore an−1,. . . ,a0

RG = C(z). Then L is˜ the differential operatorassociated to the evaluation h of the

invariants.

InordertofindL one˜ needstocomputetheD˜jx

i. Thisisdoneasfollows.

(b). By assumption R is a finite field extension of C(z) and I is a maximal ideal

of C(z)[X1,. . . ,Xn], which is the coordinate ring of the nonsingular variety An over C(z). The well knownJacobian criterion forsmoothness implies thatthe unitideal of

C(z)[X1,. . . ,Xn] isgeneratedbyI andthedeterminantsdet



∂(fj−h(fj)) ∂Xi

j∈J

i=1,...,nwhere J rangesover thesubsetsof{1,. . . ,N} with#J = n.

Sincetheelementsh(fj) belongtoC(z) wehave∂(fj∂X−h(fi j)) = ∂X∂fji.Afterrenumbering

we may suppose that DET = det∂fj

∂Xi

j=1,...,n

i=1,...,n is non zero. Then df1∧ · · · ∧ dfn = DET· dX1∧ · · · ∧ dXn.Thusforσ∈ G⊂ GL(Cn) onehasσ(DET )= det(σ)−1· DET .

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Since G is finite, there exists an integer m ≥ 1 with DETm ∈ C[X1,. . . ,Xn]G and h(DETm)∈ C(z).Since R isafield,itfollows thath(DETm)= 0.

TheextensionD of˜ dzd onR liftstoaderivationD onC(z)[X1,. . . ,Xn] withD(z)= 1

and suchthatD(I)⊂ I.TheliftD isnotuniquesinceonecanadd toeachD(Xi) any

element intheidealI.

The condition D(I)⊂ I with I = (f1− h(f1),. . . ,fN − h(fN)) can be rewritten as

thefollowing explicitformula

n  j=1 ∂fi ∂Xj · D(Xj )≡ h(fi) mod I, for i = 1, . . . , N.

Since we have assumed that R is a field, h(DETm) = 0 and this suffices for the

computationofthevector(DX1,. . . ,DXn)t satisfyingtheequation



∂fi ∂Xj



(DX1, . . . , DXn)t= (h(f1), . . . , h(fn))t.

Then D(fi− h(fi))∈ I foralli= 1,. . . ,N andD(I)⊂ I.Onethen computesformulas

forDi, i= 0,. . . ,n.Fromthis onededucesalinearcombinationL:= dn

z + an−1dnz−1+ · · ·+a1dz+a0suchthatL(xi)= 0 foralli.Thisrelationisuniquesincewehaveassumed

thatR isafieldandweknowthatx1,. . . ,xn areC-linearlyindependent.Itfollowsthat L is G-invariantandallaj∈ RG= C(z).Weconclude:

L is thedifferential operatorassociatedtotheevaluationh. 

Remarks 4.4.(1). Webriefly explain why4.3 is called “Procedure”rather than

“Algo-rithm”.AsuccessfulapplicationoftheProceduredependsonpropertiesoftheevaluation

h oftheinvariants.IftheR in4.3isknowntobe afield,thenL exists.Ifh isknownto

be the evaluation ofan operatorL, then theProcedurecomputes L up to(projective)

equivalence. Forsomechoicesofh theoperatorL doesnotexist.It canhappen, inthe

case that R is not afield, thatL exists buthas a differential Galois groupwhich is a

proper subgroupofG (see4.5).

(2). Suppose thatthe evaluation of the invariants h produces the operatorL. For the

change of h into , given by hλ(fi) = λdeg fih(fi) for all i and fixed λ such that a

powerλm∈ C(z) forsomeintegerm≥ 1,Procedurecomputesanewoperator,namely

λLλ−1.ThusifL= dnz+ an−1dzn−1+· · · + a1dz+ a0,thenthenewoperatorisobtained

from L by theshiftdz → dz−λ



λ (note that

λ

λ ∈ C(z)).The evaluationsh andhλ are

called essentiallythesame.

Examples 4.5. In some cases, the ideal I of Procedure 4.3 is not a maximal ideal of

C(z)[X1,. . . ,Xn] andtherefore R isnotbe field.Weconsider,as inExample3.9,G= ASL2

4 and C[x,y]G = C[Q3,Q4,Q6] with the relation Q26 − Q43 − 4Q34 = 0. For the evaluations h1 : (Q3,Q4,Q6)→ (z,0,z2) andh2: (Q3,Q4,Q6)→ (0,z2,2z3) theabove

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idealI isnotmaximal. Inboth cases,R isaproduct ofanumberofcopies ofthe field

C(z).

Procedure4.3appliedtotheevaluationh1leadstothefirst orderdifferentialoperator

dz−6z1 insteadofasecondorderoperator.Thisisinaccordancewiththeobservationthat

for suitablex0,y0 ∈ C∗,x0 = y0 onehas Q3(x0z1/6,y0z1/6)= z, Q4(x0z1/6,y0z1/6)= 0, Q6(x0z1/6,y0z1/6)= z2.MoreoverthedifferentialGaloisgroupisC6,thecyclicgroup

oforder6,whichcanbe seenas asubgroupof ASL2

4 .

TheProcedure doesnot produce anoperator forh2. Indeed,Q3 isa product of six

linearformsinthetwoC-linearlyindependentsolutionsx,y andh2(Q3)= 0 contradicts

thislinearindependence. 

Fortheexistenceandconstructionofanevaluationfrom aG-invariantcurveZ with

C(Z)G= C(z) wewillusethefollowinglemma anditsproof.

Lemma4.6. LetA be afinitely generatedgradedC-algebra.Assumethat A isadomain.

LetA((0)) denotethesubfield of thefield of fractionsA(0) of A consisting of the

homo-geneouselements of degree 0.

Assume that A((0)) = C(z). Then there exists a C-algebra homomorphism h : A

C[z] suchthat h inducestheidentification A((0))= C(z).

Proof. WriteA= C[f1,. . . ,fr] wherethef1,. . . ,frarehomogeneouselementsofdegrees d1,. . . ,dr∈ Z>0.Letv(i)= (v(i)1,. . . ,v(i)r) fori= 1,. . . ,r−1 denotefreegeneratorsof {(n1,. . . ,nr)∈ Zr|



nidi= 0}.Wemayandwillsupposethatthematrix{v(i)j}r−1i,j=1

isinvertible.Letm∈ Z=0 beitsdeterminant.

The elements {fv(i)1

1 · · · f

v(i)r

r | i = 1,. . . ,r− 1} generate the field A((0)) = C(z) over C and thus we can identify fv(i)1

1 · · · f

v(i)r

r with some αi ∈ C(z). First we

define ˜h : A → C(z) by ˜h(fr) = 1 and the ˜h(f1),. . . ,˜h(fr−1) are such that

˜

h(f1)v(i)1· · · ˜h(fr−1)v(i)r−1 = αi for i = 1,. . . ,r − 1. One observes that the ˜h(fi)

are Laurent polynomials in α1/m1 ,. . . ,αr−11/m. Thus the expressions ˜h(fi) havethe form R(z)· (z − a1)n1/m· · · (z − as)ns/m with R ∈ C(z),certain distinct a1,. . . ,as∈ C and

certainintegersni ∈ {0,. . . ,m− 1}.

Thealgebraicrelationsbetweenthef1,. . . ,fraregeneratedbyhomogeneousrelations.

Hence for any expression λ ∈ C(z)∗ we can consider the C-algebra homomorphism h

given as h(fi) = λdih(f˜ i) for i = 1,. . . ,r. Using the shape of the ˜h(fi) one observes

thatforsuitableλ allλdi˜h(f

i)∈ C[z].Thustherequiredh existsandcanbe seentobe

unique(uptoconstants) undertheconditionthatri=1deg h(fi) isminimal.

Werecallthath and˜h are“essentiallythesame”accordingto4.4. 

Corollary 4.7.Let be given an irreducible finite group G ⊂ SL(V ) and an irreducible

G-invariant curve Z ⊂ P(V ) suchthat the function field of Z/G is C(z). Lemma 4.6 producesan evaluationof theinvariantsh: C[V ]G → C[z] which inducesthe identifica-tionof thefunctionfield ofZ/G with C(z).

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This evaluation h isessentially thesame asthe evaluationof theinvariants induced by thestandardoperatorLst forthedataG andZ (see §3).

Proof. Let M⊂ C[V ] bethehomogeneousprimeidealofZ⊂ P(V ). ThenM∩ C[V ]G

definesthecurveZ/G andthehomogeneousalgebraofZ/G isA:= C[V ]G/(M∩C[V ]G).

Now oneapplies Lemma4.6 to A. Thelast statementfollows from theunicity of h up

to achangeh(fi)→ λdeg fih(fi) fori= 1,. . . ,r. 

5. ComputationswithProcedure 4.3

InSections5.1-5.3wepresentinconcretecasesthedifferentialoperatorobtainedfrom

Procedure4.3.Theevaluationsusedare computedasintheproof ofLemma4.6.

5.1. Finitesubgroups of SL2

For finite subgroups ofSL2 and their invariantswe usethe notationsand equations

from [5].ThestandardequationsforthesubgroupsDn,A4,S4,A5 ofPSL2 areclassical and wellknown,seeforinstance[1,2].

(1). Thegroup DSL2

n of order 4n is generated by

ζ 0 0 ζ−1

, 01 0−1with ζ = e2πi/2n. The

semi-invariants are generated byf3 = xy, f12 = x2n+ y2n,and f13 = x2n− y2n. The

invariantshavegenerators

F1= f3f13, F2= f12, F3= f32and relation F12− F22F3+ 4F3n+1= 0.

This leadsto thefollowing evaluations.

If n is odd, A((0)) is generated by F1 F3(n+1)/2 , F2 Fn 3 with relation ( F1 F3(n+1)/2 )2= F2 Fn 3 − 4. Define z by F1 F3(n+1)/2

= 2iz. This gives ˜h : (F1,F2,F3) → (2iz,2i(z2 − 1)1/2,1) and

h : (F1,F2,F3)→ (2iz(z2− 1)(n+1)/2,2i(z2− 1)(n+1)/2,(z2− 1)).

For 2|n, generators of A((0)) are F

2 1 F3n+1 , F2 F3n/2 satisfying F12 F3n+1 = ( F2 F3n/2 )2− 4. Corre-sponding evaluations are ˜h : (F1,F2,F3) → (2(z2− 1)1/2,2z,1) and h : (F1,F2,F3) → (2(z2− 1)1+n/2,2z(z2− 1)n/2,(z2− 1)).

Foralln the differentialoperatorisL= d2

z+z2z−1dz−4n2(z12−1). Thisbecomes,after

thetransformation z→ 2z − 1, thestandardequation

(d dz) 2+ 3 16z2 + 3 16(z− 1)2 n2+ 2 8n2z(z− 1) for Dn.

(2). For the group ASL2

4 , we continue the discussion from Example 3.9. Generators

for the invariants are thehomogeneous polynomials Q3,Q4,Q6 of degrees6,8,12 with

relation Q2 6 = Q43 + 4Q34. Here A((0)) = C(QQ62 3, Q3 4 Q4 3) = C(z) with z = Q6 Q2 3. This leads to the evaluations h : (Q˜ 3,Q4,Q6) → (1,(z

2−1

4 )1/3,z) and h : (Q3,Q4,Q6) → ((z24−1)2,(z24−1)3,(z24−1)4z).

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Thedifferential operatorisd2z+ 27z

2

+101

144(z2−1)2.Thisbecomesafterz→ 2z − 1 the standard

equation (d dz) 2+ 3 16z2 + 2 9(z− 1)2 3 16z(z− 1) for A4. (3). The group SSL2

4 has ring of invariants A := C[F1,F2,F3] with generators Fj of

degrees12,8,18,respectively.OnefindsA((0))= C(F

3 2 F2 1, F2 3 F3 1) withrelation F3 2 F2 1 = F2 3 F3 1 + 108. Put ˜h : (F1,F2,F3)→ (1,3· 22/3z,2· 33/2(z3− 1)1/2) and h : (F1,F2,F3)→ (2233(z3 1)3,2233z(z3− 1)2,2436(z3− 1)5).

The differential operator is d2z+

(7z3+101)z

64(z2+z+1)2(z−1)2. The equation has4 singular points

and is a pullback of the standard equation. Note that our choice of evaluation is not

‘minimal’,i.e., themap induced by˜h and h fromA((0)) to C(z) has imageC(z3). The

operatorisapullbackofthestandardoperator

(d dz) 2+ 3 16z2 + 2 9(z− 1)2 101 576z(z− 1) for S4. (4). The group ASL2

5 has ring of invariants A := C[f9,f10,f11], generators of degree

30,20,12,respectively,withrelationf2

9+ f103 − 1728f115 = 0.InthepresentcaseA((0))=

C(f92 f5 11, f3 10 f5 11) with f2 9 f5 11 = f3 10 f5

11 + 1728. This leads to the evaluations

˜

h : (f9,f10,f11) → (−123/2(z− 1)1/2,12· z1/3,1) and

h : (f9, f10, f11)→ (129z10(z− 1)8,−126z7(z− 1)5,−123z4(z− 1)3).

Thedifferentialoperatorisd2

z+864z

2−989z+800

3600z2(z−1)2 . Afterz→ 1−z,thisbecomesthestandard

operator ( d dz) 2 + 3 16z2+ 2 9(z− 1)2 611 3600z(z− 1) for A5. 5.2. G= G168⊂ SL3

5.2.1. Computationof thedifferential equation relatedtoKlein’s quartic

FortheuniquesimplegroupG⊂ SL(3,C) oforder168 weusenotationsandformulas

of[2,p. 50]. HereC[X1,X2,X3]G = C[F4,F6,F14,F21]/(rel),where F4,F6,F14,F21 are ofdegrees4,6,14,21.TheKleinquarticZ ⊂ P2isgivenbyF

4= 0 withF4:= 2(X1X23+

X2X33+ X3X13).

UnlikethecaseoffiniteirreduciblesubgroupsofSL2(compareExample3.9),adirect

computation of the standard operator for these data with the methods of Section 3

meets difficulties. How to compute f ∈ C(Z)∗ such that σ(f )f = σX1

X1 for all σ ∈ G?

How to compute the derivativesw.r.t. dt = dtd of abasis of the solution space W =<

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Wenow usethemethodsofSection4.ThegradedalgebraofZ/G is {C[X1, X2, X3]/(F4)}G= C[F6, F14, F21]/(F212 − 4F143 − 54F67); the fieldA((0)) = C(Z/G) equals C(F

6 21 F21 6 , F143 F7 6) with relation F216 F21 6 = (4 F143 F7 6 + 54) 3. Hence A((0))= C(t) witht= F3 14 F7 6 .Aresultingevaluationis h : (F4, F6, F14, F21)→ (0, t2(4t + 54)3, t5(4t + 54)7, t7(4t + 54)11).

Now Procedure 4.3 leads to an operator S0 with singularities t = 0,−272,∞. Its local

exponentsare1,2/3,1/3||1,1/2,3/2||− 3/7,−5/7,−6/7.

Thechanget=272z (hencedt=272dz)movesthesingularitiesto0,1,∞,withthe

samelocalexponents.Thecorrespondingoperatoris

S1:= d3z+ 1 zd 2 z+ 72z2+ 61z + 56 252z2(z− 1)2 dz− 6480z3+ 3945z2+ 13585z− 5488 24696z3(z− 1)3 .

The conjugateS2:= z−1(z− 1)−1S1z(z− 1) hasthe“classical”localexponentsand

coincides withtheformulasintheliterature[12,23,18]:

S2= d3z+ 7z− 4 z(z− 1)d 2 z+ 2592z2− 2963z + 560 252z2(z− 1)2 dz+ 72·11 73 z−4080524696 z2(z− 1)2 .

5.2.2. The Hessianof theKleinquartic

The Hessian is the G-invariant curve Z ⊂ P2 with equation F

6 = 0. The graded

algebraofZ/G isC[F4,F14,F21]/(F212 −4F143 + 8F14F47) andC(Z)G = C(t) witht=

F142

F7 4 .

A resultingevaluationis

h : (F4, F6, F14, F21)→ (t3(t− 2)2, 0, t11(t− 2)7, 2t16(t− 2)11).

Procedure4.3thenyields (afterachangeofvariables)theoperator

d3z+3(3z− 2) 2z(z− 1)d 2 z+ 3(116z− 35) 112z2(z− 1)dz+ 195 2744z2(z− 1).

5.2.3. Morethird orderoperatorswith groupG= G168

The third order operators over C(z), or more precisely, the differential modules of

dimension 3, with singularpoints 0,1,∞ anddifferential Galois groupG are classified

in[18],usingthe“transcendental”Riemann–Hilbertcorrespondence.Eachcaseisgiven

byabranchtype[e0,e1,e∞] andachoiceofoneofthetwoirreduciblecharacters χ23

of dimension3.TheLISTis:

[2,3,7], 1case, g = 3;[2,4,7], 1case, g = 10;[2,7,7], 1case,g = 19;

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[3,4,7]∗,2cases,g = 24; [3,7,7], 2cases,g = 33;[4,4,4]∗,2 cases,g = 22;

[4,4,7], 1case, g = 31; [4,7,7]∗,2cases,g = 40;[7,7,7],1 case,g = 49.

Formany casesinLIST these data leadto a computationof thethirdorder operator.

Thecaseswhere thisfailsareindicatedbya∗.

IngeneraltheFanocurvecorrespondingtoanelementinLISTisnotexplicitlyknown.

IfonecanidentifyforaniteminLISTtheG-invariant(Fano)curveZ⊂ P2,thisresults

in an evaluation and via Procedure 4.3 in a computation of the desired differential

operator. [2,3,7],[2,4,7] in LIST correspond to F4 = 0 and F6 = 0. [2] considered

smoothG-invariantZ⊂ P2 withquotientofgenus 0anddidnotfindnewexamples.

Weextendhis searchandconsider the(singular)curves aF3

4 + F62= 0.

If such acurve Z = Za leads to an evaluation h with˜ ˜h(F4) = 1 and ˜h(F6) = λ (so

λ2=−a) and˜h(F14)= t,then

h(F21)2= 4t3− 44λt2+ (126λ4+ 68λ2− 8)t + 54λ7− 938λ5+ 172λ3− 8λ. Thediscriminantofthispolynomialint equals−64(27λ2− 2)32+ 2)4,soλ= (−2)1/2

and λ = (2/27)1/2, or a = 2,a = −2/27 are special. Note that if the discriminant is

nonzerothenthequotientmapfromZawouldhaveatleast5 branchpoints.Bothspecial

casesleadtoquotientmapswithexactly3branchpoints.InfactZ−2/27isbirationalto

theKleinquartic(ofgenus3),andZ2 isbirationalto thecurve givenbyF6= 0.

Forλ= (−2)1/2onefinds˜h(F21)= 2(−t+ 9−2)(t+ 7−2)1/2andforλ= (2/27)1/2 wehave˜h(F21)=−2

3 243 (27t+

6)(−27t+ 356)1/2.Using4.3thecorresponding

opera-torsarefound.Theoperatorshavethreesingularpointsandthesolutionsaregeneralized

hypergeometricfunctions.Weremarkthattheabove“Frickepencil”Zawasalsostudied

by M. Kato (see [13, Prop. 2.3]),using Schwarz maps. In arather different way than

our’shefoundthetwospecialcasesas wellasthecorrespondingthirdorderdifferential

operators.

5.2.4. Computing theevaluationfordifferential operatorsin LIST

Anelement inLIST is given by atopological covering of P1\ {0,1,∞} withgroup

G= G168, producedbyatripleg0,g1,g∞∈ G satisfyingg0g1g∞= 1 andgenerating G.

Onemayhopethatfrom agiventripleonecanreadoffapartofanevaluationh ofthe

operator, namely the orders of the functions h(F4),h(F6),h(F14),h(F21) at the points 0,1,∞.

Inanumberofcasesknowledgeoftheseorderstogetherwiththerelationbetweenthe

fourinvariantssufficestocomputeasuitableh.

Weillustratethis fortheitem[2,4,7] in LIST :

Let x,y,z denote a basis of solutions for the differential equation we try to compute. As F4,F6,F14,F21 are explicit expressions in x,y,z, and one has (by [18, §5.2]) lower bounds12,−34,87 forthelocal exponentsatt= 0,1,∞,onededuces

(h(F4), h(F6), h(F14), h(F21)) =  f4 t2(t− 1)3, f6 t3(t− 1)4, f14+ g14t t7(t− 1)10, f21(t + 2400) t10(t− 1)15 

(19)

for constantsf4,f6,f14,g14,f21 (uniqueupto anappropriatescaling). Therelation be-tweentheFj’syields(f4,f6,f14,g14,f21)= (−74 ,−34 ,−1498 ,14,18).

Evaluations forseveral itemsin LIST. Thesameideausedfor[2,4,7] above, results

inevaluationsforvariousother itemsinLIST.Thenexttablepresentstheresults.The

first row gives the branchtype and the rational functions h(F4),h(F6),h(F14),h(F21).

The secondrow liststhelocal exponentsat0,1,∞ andtheaccessoryparameterμ (see

[18,§ 5.1]).Theoperatorisuniquelydeterminedbythesedata.

• [2,3,7] 0, −3 3 t3(t− 1)4, 2238 t7(t− 1)9, 23312 t10(t− 1)14 1 2,0, 1 2 || 1 3,− 1 3,0|| 8 7, 9 7, 11 7 || 12293 24696. • [2,4,7] −7 4t2(t− 1)3, 3 4t3(t− 1)4, (−149 + 2t) 8t7(t− 1)10, (t + 2400) 8t10(t− 1)14 1 2,0, 1 2 || 3 4,− 1 4,0|| 8 7, 9 7, 11 7 || 5273 10976. • [2,7,7] 14 t2(t− 1)3, 3 t2(t− 1)5, 4(−294 + 294t + t2) t6(t− 1)12 , 8(t− 2)(t2− 9604t + 9604) t9(t− 1)18 1 2,1, 1 2 || 6 7,− 5 7,− 3 7 || 8 7, 9 7, 11 7 || 1045 686. • [3,3,7] 0, 2 433 t4(t− 1)4, 21238 t9(t− 1)9, 217312(1− 2t) t14(t− 1)14 2 3,− 1 3,0|| 2 3,− 1 3,0|| 9 7, 11 7, 15 7 ||0. • [3,7,7] 0, 3 3 t4(t− 1)5, 38(9t− 8) t9(t− 1)12, 312(27t2− 36t + 8) t14(t− 1)18 2 3,− 1 3,0|| 6 7,− 5 7,− 3 7|| 10 7, 13 7, 19 7 || 830 1029. • [4,4,7] −14 t3(t− 1)3, −12 t4(t− 1)4, 25(8t2− 8t − 147) t10(t− 1)10 , 29(2t− 1)(4t2− 4t + 2401) t15(t− 1)15 3 4,− 1 4,0|| 3 4,− 3 4,0|| 9 7, 11 7, 15 7 ||0. • [7,7,7] 16 t3(t− 1)3, 512t2− 512t + 5 16t5(t− 1)5 , P6(t) 28t12(t− 1)12, (2t− 1)P8(t) 212t18(t− 1)18 6 7,− 5 7,− 3 7 || 6 7,− 5 7,− 3 7 || 9 7, 11 7, 29 7 ||0 whereP6(t)= 220t5(t− 3)+ 215t3(385t− 610)+ 27t(74441t− 457)+ 1 and P8(t)= 229t7(t−4)−223t5(8869t−27503)−215t3(7074623t−2338174)+28t(1963429t− 5413)− 1.

Remarks 5.1.(1). Fortheremainingitems inLIST theapproachabovedoesnot

deter-mineh (upto equivalence).

(2). For the case [3,4,4] a choice ofh leads to a proper subgroup of G (compare [18, §8.2.1 part(6)]).

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