Computation of some Moduli Spaces of covers and ex

arXiv:math/0202125v1[math.NT]13Feb2002ComputationofsomeModuliSpaesofoversandexpliitSnandAnregularQ(T)-extensionswithtotallyrealbersEmmanuelHallouinandEmmanuelRiboulet-Deyris∗1stFebruary2008Summary.WestudyandomputeaninnitefamilyofHurwitzspaespa-rameterizingoversofP1CbranhedatfourpointsanddedueexpliitregularSnandAn-extensionsoverQ(T)withtotallyrealbers.IntrodutionInthispaper,westudyafamilyofoversoftheprojetivelinesuggestedtousbyGunterMalle,namelythoseoversofevendegreen≥6,ramiedoverfourpoints,withmonodromySnandhavingbranhyledesriptionC=(C1,C2,C3,C4)oftype:(n−2),3,2n−22,2n2MallesuspetedtheHurwitzurveshavegenuszeroforeveryn≥6andsomeoversinthefamilyhavetotallyrealbers.AsimilarfamilywassuggestedandextensivelystudiedbyDŁbesandFriedin[DF94℄.Unfortunately,theirHurwitzspaeshappentohaveaquadratigenusinnandonlyprovidetheexpetedregularextensionsfordegrees5and7(see[DF94℄,Theorem4.11).Theirworkusesbraidationformulae(see[FV91℄)andomplexonjugationationformulae(see[FD90℄,Proposition2.3).Inthispaper,werstfollowthelinesofDŁbesandFriedmethodandshowthatMalle’sexpetationswereright.Thentheseondhalfofthepaperisdevotedtotheexpliitalulationoftheonerneduniversalfamilyofovers.Tothisend,weuseanexpliitversionofHarbater’sdeformationtehniques([Har80℄)asproposedin[CG94,Cou99℄.Asfarasweknow,itisthersttimesuhanadvanedmethodisusedforomputinganinnitefamilyofovers.Wepresentnumerialresults,obtainedwithMagma,showingtheeienyoftheproposedmethodomparedtothelassialonesinvolvingBuhbergeralgorithm.Indeedouromputationreduestosolvinglinearsystems.Intherstsetion,wereallresultsfromthetheoryofHurwitzspaesand(non)-rigiditymethodsdevelopedin[FV91,Vl96,Ful69,MM99℄.Theseondsetionisdevotedtotheombinatoristudyofourfamilyandarithmetial∗GroupedeReherheenInformatiqueetMathØmatiqueduMirail(G.R.I.M.M.),Univer-sityofToulouseII,Frane.1onsequenesofitusingthemethodofDŁbesandFried.Inthethirdsetion,weshowtheexisteneoftotallyrealSnandAnresidualQ-extensionsinthatfamily.Relatedtothistotallyrealspeialization,wendaveryspeialpointoftheboundaryofourHurwitzspaethatshowsveryusefulwhenomputinganexpliitmodel.Thisisdoneinthelastsetion,byadeformationmethod.WenotiethatafalloutofouronstrutionistheexisteneoftotallyrealQ-extensionswithGaloisgroupSnandAn.Howeverasimplerandastuteonstrutionanbefoundin[Mes90℄.WethankJean-MarCouveignesfornumerousextremelyhelpfuldisussionsaboutthiswork.1.GeneralframeworkAssoiatedtoourfamilyofovers,thereisaoarsemodulispaealledHurwitzspae,denotedH′4(Sn,C).Itisaquasi-projetiveregular(notapriorion-neted)varietyoverQwiththefollowingproperties(see[FV91,Vl96,Ful69℄):•SineanyonjugaylassofSnisrational,H′4(Sn,C)isdenedoverQ.•LetF4betheongurationspaeof4points,e.g.(P1C)4\DwhereDdenotesthedisriminantvariety.Themap:φ:H′4(Sn,C)//F4(P1C)h//(z1,z2,z3,z4)wherez1,z2,z3,z4arethebranhedpoints(inthegivenorder)oftheoverorrespondingtothepointh,isaniteØtalemorphismdenedoverQ.•SineSnhasnoenter,theoversinourfamilyhavenoautomorphismsothemodulispaeH′4(Sn,C)isaneoneandforanyh∈H′4(Sn,C)theassoiatedoverphanbedenedoverQ(h)theeldofdenitionofthepointh.Asin4.2of[DF94℄,ratherthanlookingatthefullmodulispae,weon-entrateonaurveinit.Letusxthreepointsz1,z2,z3∈P1(Q)andonsidertheurveH′(z1,z2,z3)obtainedbythepullbak:H′(z1,z2,z3)//ϕH′4(Sn,C)φP1C\{z1,z2,z3}i//F4(P1C)(thelowerhorizontalmapiisz7→(z1,z2,z3,z)).Ifthethreexedpointsarerational,allthemapsaredenedoverQandtheurveH′(z1,z2,z3)isalsodenedoverQ.2.CombinatoristudyoftheHurwitzurveInthissetion,wegatherthemoreinformationweanabouttheoverϕ:H′(z1,z2,z3)−→P1Cr{z1,z2,z3}fromitsombinatoridesription,namelythemonodromy,theramiationtype,theonnetivityandthegenus.Thiswillbededued,fromanexpliitenumerationoftheNielsenlassassoiatedtoC(setion2.1)andfromtheationofbraidsonit(setion2.2).22.1.Nielsenlassesdesription.Letusxthebranhedpointsz=(z1,z2,z3,z4)∈F4(P1C)andanhomotopibaseofP1C\{z1,z2,z3,z4}.Usingthetopologiallassiationofovers,elementsoftheberφ−1(z)areinbijetionwithsniab(Sn,C)thestritabsoluteNielsenlassoftype(Sn,C),thatis:sniab(Sn,C)=(σ1,σ2,σ3,σ4)∈(Sn)4,σ1σ2σ3σ4=1,σi∈Ci∀ihσ1,σ2,σ3,σ4i=Sn,∼where(σ1,σ2,σ3,σ4)∼(σ′1,σ′2,σ′3,σ′4)meansthatthereexistsτ∈Snsuhthatσ′i=τσiτ−1forall1≤i≤4.WerstenumeratetheNielsenlass.Sineanytwo(n−2)-ylesareSn-onjugate,everyelementofsniab(Sn,C)hasarepresentativewithσ1=(1,...,n−2).Conjugatingbyapowerofσ1,wealsoassumethatσ2=(n−2,k,l).Wenowdistinguishthreeases.•Thease{k,l}={n−1,n}.Conjugatingbyapowerofσ1andby(n−1,n),everyelementinthatlasshasauniquerepresentativewithσ1=(1,...,n−2),andσ2=(n−2,n−1,n).Soσ1σ2=(1,...,n)andtheenumerationreduestondingallthepermutations(σ4,σ3)∈C4×C3suhthatσ4σ3=(1,...,n)andhσ1,σ2,σ3,σ4i=Sn.Weneedalemmausedseveraltimesfurther.ItdealswithrelationsinthediedralgroupDm.Lemma2.1Letm≤nbeevenandletcbeanm-yleofSn.Thereisabijetionbetweennontrivialylesofcm2(i.e.setsoftheform{x,cm2(x)}wherexbelongstothesupportofc),andthedeompositionsc=στwithσaprodutofm2transpositionsandτaprodutofm2−1transpositions.Thereforethereareexatlym2suhdeompositionsofc.Proof.Letxbeanelementofthesupportofcthenoneanverifythat:c=m2Y