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/).
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
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
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
(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).
(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
{λ· 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
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
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.
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
spaceC√1
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.
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
{φ1− 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
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 .
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 hλ, 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
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).
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).
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 =<
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 χ2,χ3
of dimension3.TheLISTis:
[2,3,7], 1case, g = 3;[2,4,7], 1case, g = 10;[2,7,7], 1case,g = 19;
[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)3(λ2+ 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+ 35√6)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 bounds−12,−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
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)]).