THE OPEN BOOK SERIES 1 ANTS X Proceedings of the Tenth Algorithmic Number Theory Symposium Imaginary quadratic fields with isomorphic abelian Galois groups AthanasiosAngelakis and PeterStevenhagen msp THEOPENBOOKSERIES 1 (2013) TenthAlgorithmicNumberTheorySymposium msp dx.doi.org/10.2140/obs.2013.1.21 Imaginary quadratic fields with isomorphic abelian Galois groups AthanasiosAngelakis and PeterStevenhagen In1976,Onabediscoveredthat,incontrasttotheNeukirch-Uchidaresultsthat wereprovedaroundthesametime,anumberfieldK isnotcompletelycharac- terizedbyitsabsoluteabelianGaloisgroupA . Thefirstexamplesofnoniso- K morphicK havingisomorphicA wereobtainedonthebasisofaclassification K byKubotaofideleclasscharactergroupsintermsoftheirinfinitefamiliesof Ulminvariants,anddidnotyieldadescriptionofA . Inthispaper,weprovide K adirect“computation”oftheprofinitegroupA forimaginaryquadraticK,and K useittoobtainmanydifferentK thatallhavethesameminimalabsoluteabelian Galoisgroup. 1. Introduction TheabsoluteGaloisgroupG ofanumberfieldK isalargeprofinitegroupthat K we cannot currently describe in very precise terms. This makes it impossible to answerfundamentalquestionson G ,suchastheinverseGaloisproblemover K. K Still,Neukirch [7] provedthatnormalnumberfields arecompletelycharacterized by their absolute Galois groups: If G and G are isomorphic as topological K1 K2 groups, then K and K are isomorphic number fields. The result was refined 1 2 by Ikeda, Iwasawa, and Uchida ([8], [9, Chapter XII, §2]), who disposed of the restriction to normal number fields, and showed that every topological isomor- phism GK1(cid:0)(cid:24)! GK2 is actually induced by an inner automorphism of G(cid:81). The samestatementsholdifallabsoluteGaloisgroupsarereplacedbytheirmaximal prosolvablequotients. ItwasdiscoveredbyOnabe[10]thatthesituationchangesifonemovesafurther stepdownfromG ,toitsmaximalabelianquotientA DG =ŒG ;G (cid:141),which K K K K K istheGaloisgroupA DGal.Kab=K/ofthemaximalabelianextensionKab ofK. K MSC2010: primary11R37;secondary20K35. Keywords: absoluteGaloisgroup,classfieldtheory,groupextensions. 21 22 ATHANASIOSANGELAKISANDPETERSTEVENHAGEN EventhoughtheHilbertproblemofexplicitlygeneratingKab forgeneralnumber fieldsK isstillopenaftermorethanacentury,thegroup A canbedescribedby K classfieldtheory,asaquotientoftheideleclassgroupofK. Kubota[5]studiedthegroupX ofcontinuouscharactersonA ,andexpressed K K thestructureofthep-primarypartsofthiscountableabeliantorsiongroupinterms ofaninfinitenumberofso-calledUlminvariants. IthadbeenshownbyKaplansky [4,Theorem14]thatsuchinvariantsdeterminetheisomorphismtypeofacount- able reduced abelian torsion group. Onabe computed the Ulm invariants of X K explicitlyfor anumber ofsmall imaginaryquadratic fieldsK, andconcluded from 0 thisthatthereexistnonisomorphicimaginaryquadraticfieldsK andK forwhich theabsoluteabelianGaloisgroupsAK andAK0 areisomorphicasprofinitegroups. 0 ThismayevenhappenincasepswhereK andKp havedifferentclassnumbers,but theexplicitexampleK D(cid:81). (cid:0)2/,K0D(cid:81). (cid:0)5/ofthisthatoccursinOnabe’s maintheorem [10,Theorem 2]isincorrect. This isbecausethe valueofthe finite Ulminvariantsin[5,Theorem4]isincorrectfortheprime2incasethegrounpdfield isaspecialnumberfieldinpthesenseofourLemma3.2. Asithappens,(cid:81). (cid:0)5/ andtheexceptionalfield(cid:81). (cid:0)2/dohavedifferentUlminvariantsat2. Thenature ofKubota’s errorissimilarto anerrorin Grunwald’stheoremthat wascorrected byatheoremofWangoccurringinKubota’spaper[5,Theorem1]. Itisrelatedto thenoncyclicnatureofthe2-powercyclotomicextension(cid:81)(cid:26)(cid:81).(cid:16)21/. Inthispaper,weobtainOnabe’scorrectedresultsbyadirectclassfieldtheoretic approachthatcompletelyavoidsKubota’sdualizationandthepmachineryofUlm invariants. WeshowthattheimaginaryquadraticfieldsK ¤(cid:81). (cid:0)2/thataresaid tobeof‘typeA’in[10]sharea minimal absoluteabelianGaloisgroupthatcanbe describedcompletelyexplicitlyas A D(cid:90)y2(cid:2) Y (cid:90)=n(cid:90): K n(cid:21)1 Thenumericaldatathatwepresentsuggestthatthesefieldsareinfactverycommon amongimaginaryquadraticfields: Morethan97%ofthe2356fieldsofoddprime class number h Dp <100 are of this nature. We believe (Conjecture 7.1) that K there are actually infinitely many K for which A is the minimal group above. K Ourbeliefissupportedbycertainreasonableassumptionsontheaveragesplitting behavior of exact sequences of abelian groups, and these assumptions are tested numericallyinthefinalsectionofthepaper. 2. Galoisgroupsas(cid:90)y-modules TheprofiniteabelianGaloisgroupsthatwestudyinthispapernaturallycomewith atopologyforwhichtheidentityhasabasisofopenneighborhoodsthatareopen subgroupsoffiniteindex. Thisimpliesthattheyarenotsimply(cid:90)-modules,butthat IMAGINARYQUADRATICFIELDSWITHISOMORPHICABELIANGALOISGROUPS 23 theexponentiationinthesegroupswithordinaryintegersextendstoexponentiation with elements of the profinite completion (cid:90)y Dlim (cid:90)=n(cid:90) of (cid:90). By the Chinese (cid:0) remaindertheorem, wehaveadecomposition ofthenprofinite ring(cid:90)y DQ (cid:90) into p p a product of rings of p-adic integers, with the index p ranging over all primes. As(cid:90)y-modules,ourGaloisgroupsdecomposecorrespondinglyasaproductofpro- p-groups. It is instructive to look first at the (cid:90)y-module structure of the absolute abelian Galois group A(cid:81) of (cid:81), which we know very explicitly by the Kronecker-Weber theorem. Thistheoremstatesthat(cid:81)ab isthemaximalcyclotomicextensionof(cid:81), andthatanelement(cid:27) 2A(cid:81) actsontherootsofunitythatgenerate(cid:81)ab byexponen- tiation. Moreprecisely,wehave(cid:27).(cid:16)/D(cid:16)u forallrootsofunity,withuauniquely defined element in the unit group (cid:90)y(cid:3) of the ring (cid:90)y. This yields the well-known isomorphismA(cid:81)DGal.(cid:81)ab=(cid:81)/Š(cid:90)y(cid:3)DQp(cid:90)p(cid:3). For odd p, the group (cid:90)(cid:3) consists of a finite torsion subgroup T of .p(cid:0)1/-st p p rootsofunity,andwehaveanisomorphism (cid:90)(cid:3)DT (cid:2).1Cp(cid:90) /ŠT (cid:2)(cid:90) p p p p p because 1Cp(cid:90) is a free (cid:90) -module generated by 1Cp. For p D2 the same p p is true with T Df˙1g and 1C4(cid:90) the free (cid:90) -module generated by 1C4D5. 2 2 2 Takingtheproductoverallp,weobtain A(cid:81)ŠT(cid:81)(cid:2)(cid:90)y; (1) withT(cid:81)DQpTp theproductofthetorsionsubgroupsTp (cid:26)(cid:81)p(cid:3) ofthemultiplica- tivegroupsofthecompletions(cid:81)p of(cid:81). Morecanonically,T(cid:81) istheclosureofthe torsionsubgroupofA(cid:81)DGal.(cid:81)ab=(cid:81)/,andA(cid:81)=T(cid:81) isafree(cid:90)y-moduleofrank1. TheinvariantfieldofT(cid:81) inside(cid:81)ab istheunique(cid:90)y-extensionof(cid:81). EventhoughitlooksatfirstsightasiftheisomorphismtypeofT(cid:81) dependson thepropertiesofprimenumbers,oneshouldrealizethatinaninfiniteproductof finitecyclicgroups, theChineseremainder theoremallowsustorearrange factors inmanydifferentways. Onehasforinstanceanoncanonicalisomorphism Y Y T(cid:81)D Tp Š (cid:90)=n(cid:90); (2) p n(cid:21)1 as both of these products, when written as a countable product of cyclic groups of prime power order, have an infinite number of factors (cid:90)=`k(cid:90) for each prime power `k. Note that, for the product Q T of cyclic groups of order p(cid:0)1 (for p p p¤2),thisstatementisnotcompletelytrivial: Itfollowsfromtheexistence,bythe well-known theorem of Dirichlet, ofinfinitely many primesp that are congruent to1 mod`k,butnotto1 mod`kC1. 24 ATHANASIOSANGELAKISANDPETERSTEVENHAGEN Now suppose that K is an arbitrary number field, with ring of integers O. By classfieldtheory,A isthequotientoftheideleclassgroupC D(cid:0)Q0 K(cid:3)(cid:1)=K(cid:3) K K p(cid:20)1 p ofK bytheconnectedcomponentoftheidentity. Inthecaseofimaginaryquadratic fieldsK,thisconnectedcomponentisthesubgroupK(cid:3) D(cid:67)(cid:3)(cid:26)C comingfrom 1 K the unique infinite prime of K, and in this case the Artin isomorphism for the absoluteabelianGaloisgroupA ofK reads K A DKy(cid:3)=K(cid:3)D(cid:16)Y0K(cid:3)(cid:17)=K(cid:3): (3) K p p Here Ky(cid:3) DQ0 K(cid:3) is the group of finite ideles of K, that is, the restricted direct p p (cid:3) productofthegroupsK atthefiniteprimespofK,takenwithrespecttotheunit p groupsO(cid:3) ofthelocalringsofintegers. Forthepurposesofthispaper,whichtries p todescribeA asaprofiniteabeliangroup,itisconvenienttotreattheisomorphism K forA in(3)asanidentity—aswehavewrittenitdown. K Theexpression(3)issomewhatmoreinvolvedthanthecorrespondingidentity A(cid:81) D(cid:90)y(cid:3) for the rational number field, but we will show in Lemma 3.2 that the inertialpart ofA ,thatis, thesubgroupU (cid:26)A generatedbyallinertia groups K K K O(cid:3)(cid:26)C ,admitsadescriptionverysimilarto(1). p K DenotebyOyDQ O theprofinitecompletionoftheringofintegersOofK. In p p thecasethatK isimaginaryquadratic,theinertialpartofA takestheform K U D(cid:16)YO(cid:3)(cid:17)=O(cid:3)DOy(cid:3)=(cid:22) ; (4) K p K p sincetheunitgroupO(cid:3) ofOisthenequaltothegroup(cid:22) ofrootsofunityinK. K Apart from the quadratic fields of discriminant (cid:0)3 and (cid:0)4, which have 6 and 4 rootsofunity,respectively,wealwayshave(cid:22) Df˙1g,and(4)canbeviewedas K theanalogueforK ofthegroup(cid:90)y(cid:3)DA(cid:81). In the next section, we determine the structure of the group Oy(cid:3)=(cid:22) . As the K approach works for any number field, we will not assume that K is imaginary quadraticuntiltheveryendofthatsection. 3. Structureoftheinertialpart LetK beanynumberfield,andOyDQ O theprofinitecompletionofitsringof p p integers. DenotebyT (cid:26)O(cid:3) thesubgroupoflocalrootsofunityinK(cid:3),andput p p p T DYT (cid:26)YO(cid:3)DOy(cid:3): (5) K p p p p Theanalogueof(1)forK isthefollowing. IMAGINARYQUADRATICFIELDSWITHISOMORPHICABELIANGALOISGROUPS 25 Lemma3.1. Theclosureofthetorsionsubgroupof Oy(cid:3) isequaltoT ,andOy(cid:3)=T K K isafree(cid:90)y-moduleofrankŒK W(cid:81)(cid:141). Lesscanonically,wehaveanisomorphism Oy(cid:3)ŠT (cid:2)(cid:90)yŒKW(cid:81)(cid:141): K Proof. As the finite torsion subgroup T (cid:26)O(cid:3) is closed in O(cid:3), the first statement p p p followsfromthedefinitionoftheproducttopologyonOy(cid:3)DQ O(cid:3). p p ReductionmodulopinthelocalunitgroupO(cid:3) givesrisetoanexactsequence p 1(cid:0)!1Cp(cid:0)!O(cid:3)(cid:0)!k(cid:3)(cid:0)!1 p p (cid:3) that canbe splitby mapping theelements ofthe unit groupk of theresidue class p field to their Teichmüller representatives in O(cid:3). These form the cyclic group of p order #k(cid:3) D Np(cid:0)1 in T consisting of the elements of order coprime to p D p p char.k /. The kernel of reduction 1Cp is by [3, one-unit theorem, p. 231] a p finitelygenerated(cid:90) -moduleoffreerankŒK W(cid:81) (cid:141)havingafinitetorsiongroup p p p consistingofrootsofunityinT ofp-powerorder. Combiningthesefacts,wefind p that O(cid:3)=T is free over (cid:90) of rank ŒK W(cid:81) (cid:141) or, less canonically, that we have a p p p p p localisomorphism O(cid:3)ŠT (cid:2)(cid:90)ŒKpW(cid:81)p(cid:141) p p p for each prime p. Taking the product over all p, and using the fact that the sum of the local degrees at p equals the global degree ŒK W(cid:81)(cid:141), we obtain the desired globalconclusion. (cid:3) InordertoderiveacharacterizationofT DQ T forarbitrarynumberfieldsK K p p similarto(2),weobservethatwehaveanexactdivisibility`k k#T oftheorder p ofT byaprimepower`k ifandonlyifthelocalfieldK atpcontainsaprimitive p p `k-throotof unity,but not aprimitive`kC1-throotof unity. Wemayrewordthis as: TheprimepsplitscompletelyinthecyclotomicextensionK (cid:26)K.(cid:16) /,butnot `k inthecyclotomicextensionK (cid:26)K.(cid:16) /. Ifsuchpexistatallfor`k,thenthere `kC1 areinfinitelymanyofthem,bytheChebotarevdensitytheorem. Thus, T can be written as a product of groups .(cid:90)=`k(cid:90)/(cid:90) DMap.(cid:90);(cid:90)=`k(cid:90)/ K that are themselves countable products of cyclic groups of order `k. The prime powers`k >1thatoccurforK areallbutthoseforwhichwehaveanequality K.(cid:16) /DK.(cid:16) /: `k `kC1 ForK D(cid:81)allprimepowers`k occur,butforgeneralK,therearefinitely many primepowersthatmaydisappear. Thisisduetothefactthattheinfinitecyclotomic extension(cid:81)(cid:26)(cid:81).(cid:16)`1/withgroup(cid:90)(cid:3)` canpartially“collapse”overK. 26 ATHANASIOSANGELAKISANDPETERSTEVENHAGEN Todescribetheexceptionalprimepowers`k thatdisappearforK,weconsider, for`anodd prime,thenumber w.`/DwK.`/D#(cid:22)`1.K.(cid:16)`// of`-powerrootsofunityinthefieldK.(cid:16) /. Foralmostall`,thisnumberequals`, ` andwecall`exceptionalforK ifitisdivisibleby`2. Notethatnooddexceptional primenumbersexistforimaginaryquadraticfieldsK. Fortheprime`D2,weconsiderinsteadthenumber w.2/DwK.2/D#(cid:22)21.K.(cid:16)4// of 2-power roots in K.(cid:16) /DK.i/. If K contains i D(cid:16) , or if w.2/ is divisible 4 4 by8,wecall2 exceptional forK. NotethatptheonlyimaginaryquadraticfieldsK forwhich2isexceptionalare(cid:81).i/and(cid:81). (cid:0)2/. Thenumberw.K/ofexceptionalrootsofunityforK isnowdefinedas Y w.K/D w.`/: `exceptional Note that w.K/ refers to roots of unity that may or may not be contained in K itself,andthateveryprime`dividingw.K/occurswithexponentatleast2. The primepowers`k >1thatdonot occurwhenT iswrittenasadirectproductof K groups.(cid:90)=`k(cid:90)/(cid:90) arethestrict divisorsofw.`/atexceptionalprimes`,withthe exceptionalprime`D2givingrisetoaspecialcase. Lemma3.2. LetK beanumberfield, andw Dw.K/itsnumberofexceptional rootsofunity. Thenwehaveanoncanonicalisomorphismofprofinitegroups Y Y T D T Š (cid:90)=nw(cid:90); K p p n(cid:21)1 exceptwhen2isexceptionalforK andi D(cid:16) isnotcontainedinK. Inthisspecial 4 case,wehave Y Y T D T Š .(cid:90)=2(cid:90)(cid:2)(cid:90)=nw(cid:90)/: K p p n(cid:21)1 ThegroupTK isisomorphictothegroupT(cid:81) in(2)ifandonlyifwehavewD1. Proof. If`isodd,thetoweroffieldextensions K.(cid:16) /(cid:26)K.(cid:16) /(cid:26)(cid:1)(cid:1)(cid:1)(cid:26)K.(cid:16) /(cid:26)K.(cid:16) /(cid:26)(cid:1)(cid:1)(cid:1) ` `2 `k `kC1 isa(cid:90) -extension,andthestepsK.(cid:16) /(cid:26)K.(cid:16) /withk(cid:21)1inthistowerthat ` `k `kC1 areequalitiesareexactlythoseforwhich`kC1 dividesw.`/. Similarly,thetoweroffieldextensions K.(cid:16) /(cid:26)K.(cid:16) /(cid:26)(cid:1)(cid:1)(cid:1)(cid:26)K.(cid:16) /(cid:26)K.(cid:16) /(cid:26)(cid:1)(cid:1)(cid:1) 4 8 2k 2kC1 IMAGINARYQUADRATICFIELDSWITHISOMORPHICABELIANGALOISGROUPS 27 isa(cid:90) -extensioninwhichthestepsK.(cid:16) /(cid:26)K.(cid:16) /withk(cid:21)2thatareequal- 2 2k 2kC1 itiesareexactlythoseforwhich2kC1 dividesw.2/. TheextensionK DK.(cid:16) /(cid:26) 2 K.(cid:16) / that we have in the remaining case k D1 is an equality if and only if K 4 containsi D(cid:16) . 4 Thus,aprimepower`k >2thatdoesnotoccurwhenT iswrittenasaproduct K ofgroups.(cid:90)=`k(cid:90)/(cid:90) isthesameasastrict divisor`k >2ofw.`/atanexceptional prime `. The special prime power `k D2 does not occur if and only if i D(cid:16) is 4 inK. Notethatinthiscase,2isbydefinitionexceptionalforK. ItisclearthatreplacingthegroupQ (cid:90)=n(cid:90)from(2)byQ (cid:90)=nw(cid:90)has n(cid:21)1 n(cid:21)1 theeffectofremovingcyclicsummandsoforder`k with`kC1jw,andthisshows thatthegroupsgivenintheLemmaareindeedisomorphictoT . OnlyforwD1 K weobtainthegroupT(cid:81) inwhichallprimepowers`k arise. (cid:3) Lemmas3.1and3.2telluswhatOy(cid:3) lookslikeasa(cid:90)y-module. Inparticular,it showsthatthedependenceonK islimitedtothedegreeŒK W(cid:81)(cid:141),whichisreflected intherankof thefree(cid:90)y-partofOy(cid:3),andthe natureoftheexceptionalrootsofunity for K. For the group Oy(cid:3)=(cid:22) , the same is true, but the proof requires an extra K argument,andthefollowinglemma. Lemma3.3. ThereareinfinitelymanyprimespofK forwhichwehave gcd.#(cid:22) ;#T =#(cid:22) /D1: K p K Proof. For every prime power `k > 1 that exactly divides #(cid:22) , the extension K K D K.(cid:16) / (cid:26) K.(cid:16) / is a cyclic extension of prime degree `. For different `k `kC1 primepowers`k k#(cid:22) ,wegetdifferentextensions,soinfinitelymanyprimesp K ofK areinertinallofthem. Forsuchp,wehavegcd.#(cid:22) ;#T =#(cid:22) /D1. (cid:3) K p K Lemma3.4. WehaveanoncanonicalisomorphismT =(cid:22) ŠT . K K K Proof. Pickaprimep ofK thatsatisfiestheconditionsofLemma3.3. Then(cid:22) 0 K embeds as a direct summand in T , and we can write T Š(cid:22) (cid:2)T =(cid:22) as a p0 p0 K p0 K product of two cyclic groups of coprime order. It follows that the natural exact sequence Y 1(cid:0)! T (cid:0)!T =(cid:22) (cid:0)!T =(cid:22) (cid:0)!1 p K K p0 K p¤p0 canbesplitusingthecomposedmapT =(cid:22) !T !T !T =(cid:22) . Thismakes p0 K p0 K K K Q T =(cid:22) isomorphictotheproductof T andacyclicgroupforwhichthe K K p¤p0 p orderisaproductofprimepowersthatalready“occur”infinitelyofteninT . Thus K T =(cid:22) isisomorphictoaproductofexactlythesamegroups.(cid:90)=`k(cid:90)/(cid:90) thatoccur K K inT ,andthereforeisomorphictoT itself. (cid:3) K K ForimaginaryquadraticK,whereOy(cid:3)=(cid:22) constitutestheinertialpartU ofA K K K from(4),wesummarizetheresultsofthissectioninthefollowingway. 28 ATHANASIOSANGELAKISANDPETERSTEVENHAGEN Theorem3.5. LetK beanimaginaryquadraticfieldp. ThenthesubgroupTK=(cid:22)K ofU isadirectsummandofU . ForK ¤(cid:81).i/;(cid:81). (cid:0)2/,wehaveisomorphisms K K 1 U DOy(cid:3)=(cid:22) Š(cid:90)y2(cid:2).T =(cid:22) /Š(cid:90)y2(cid:2) Y (cid:90)=n(cid:90) K K K K nD1 ofprofinitegroups. p ForK equalto(cid:81).i/or(cid:81). (cid:0)2/,theprime2isexceptionalforK,andthegroups T =(cid:22) ŠT aredifferentastheydonothavecyclicsummandsoforder2and4, K K K respectively. 4. ExtensionsofGaloisgroups Intheprevioussection, allresultscouldeasily bestated andprovedforarbitrary numberfields. Fromnow on,K willdenote animaginary quadraticfield. Inorder todescribethefullgroupA from(3),weconsidertheexactsequence K 1(cid:0)!U DOy(cid:3)=(cid:22) (cid:0)!A DKy(cid:3)=K(cid:3)(cid:0) !Cl (cid:0)!1 (6) K K K K that describes the class group Cl of K in idelic terms. Here maps the class K of the finite idele .xp/p 2 Ky(cid:3) to the class of its associated ideal Qppep, with e Dord x . p p p Thesequence(6)showsthatU isanopensubgroupofA ofindexequaltothe K K classnumberh ofK. InviewofTheorem3.5,thisimmediatelyyieldsOnabe’s K discoverythatdifferentK canhavethesameabsoluteabelianGaloisgroup. p Theorem4.1. AnimaginaryquadraticnumberfieldK ¤(cid:81).i/;(cid:81). (cid:0)2/ofclass number1hasabsoluteabelianGaloisgroupisomorphicto G D(cid:90)y2(cid:2) Y (cid:90)=n(cid:90): n(cid:21)1 InOnabe’s paper[10, §5], thegroup G, whichis notexplicitlygivenbutcharac- terized by its infinitely many Ulm invariants, is referred to as ‘of type A’. We will refer to G as the minimal Galois group, as evepry absolute abelian Galois group of an imaginary quadratic field K ¤ (cid:81).i/;(cid:81). (cid:0)2/ contains a subgroup isomorphic to G. We will show that there are actuallymany more K having this absoluteabelianGaloisgroupthanthesevenfieldsK ofclassnumber1towhich theprecedingtheoremapplies. Now take for K any imaginary quadratic field of class number h >1. Then K Theorem3.5andthesequence(6)showthatA isanabeliangroupextensionof K Cl bytheminimalGaloisgroupG fromTheorem4.1. Iftheextension(6)were K split, we would find that A is isomorphic to G(cid:2)Cl ŠG; but it turns out that K K splittingatthislevelnever occursfornontrivialCl ,inthefollowingstrongsense. K