Galois Groups and Complex Multiplication Author(s): Michael Fried Source: Transactions of the American Mathematical Society, Vol. 235, (Jan., 1978), pp. 141-163 Published by: American Mathematical Society Stable URL: http://www.jstor.org/stable/1998211 Accessed: 24/05/2008 21:33 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=ams. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We enable the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact [email protected]. http://www.jstor.org TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 235, January 1978 GALOISGROUPS AND COMPLEXMULTIPLICATION(') BY MICHAELFRIED ABSTRACT.The Schurproblem for rationalfunctions is linked to the theory of complex multiplicationand thereby solved. These considerationsare viewed as a special case of a general problem, prosaically labeled the extensionof constantsproblem. The relationbetween this paper and a letter of J. Herbrandto E. Noether (publishedposthumously) is speculatively summarizedin a conjecturethat may be regardedas an arithmeticversion of Riemann'sexistence theorem. 0. Introduct.ion.Let F be a perfect field, and F a fixed algebraicclosure of F. We considerthe finite Galois extensionsof F that come from certaintypes of geometric situations. Let W (vw)Vbe a cover (finite, flat morphism)of quasi-projectivevarieties such that W, V, and p(V, W) are defined over F, and W and V are absolutelyirreducible. For X, a varietydefined over F, we let F(X) be the field of rationalfunctions on X defined over F. Therefore,we have a field extensionF(W) over F(V) (by abuse, F(W)/F(V)). If ( V, W) is a separablemorphism, then F(W)/F(V) is a finite separableextension, and we obtain - (0.1) I G (.''_W) /F' ( V)) -->G (F( IF( V)) 's>tG (F/lF) 1> where: 9W is the smallestGalois extensionof F(V) containingF(W) (the Galoisclosure of F(W)/F(V)); F = F( W, F) is the algebraicclosure of F in F(W , and; rest denotes the restriction of elements of the Galois group G( I)/F(V)) to F. We call F the extension of constantsobtained from W/V. The problem of the descriptionof G (F/F) arises in several well-known problems.For example.let G be a finite group which we desire to realize as the Galois group of some Galois extension of the rational field Q. Suppose tha: = Q; V is a Zariski open subset of P' (projectiven-space); and G(Q(W)/Q( V)), the (geometric)monodromy group is equal to G (a fact that may have come to us from analytic considerations,say in the mannerof [Fr, 1]). In this case the limitationtheorems of [Fr, 1, ?2] sometimesserve to show Receivedby the editorsFebruary 23, 1976. AMS (MOS) subjectclassifications (1970). Primary lOB15, lOD25, 12A55, 14D20, 14G05, 14H05,14H10; Secondary 10M05, 14D05, 14H25, 14H30, 14K22. (1) The researchfor this paper was partiallysupported by an Alfred P. Sloan Foundation Grantand by a grantfrom the Institutefor AdvancedStudy (Princeton) in Spring1973. r American Mathematical Society 1978 141 142 MICHAELFRIED that Q = Q. If so we may apply Bertini'stheorem and Hilbert'sirreducibility theoremto show that G is realizedas a Galois groupover Q. There are many diophantine (sic!) problems where our psychological motivationis not to have F turnout to be F, but ratherto have G (F/F) be as large as possible. The Schur problem and the theory of complex multi- plicationare two examplesof such problems.In ?2 we completethe solution of the Schur problemfor rationalfunctions (of prime degree) by noting the precise connectionsbetween these two problems(and the theory of division points of elliptic curves). We reviewquickly the contentsof this papersection by section. After preliminarynotation and definitionsin ? 1, ? L.Adiscusses Riemann's existence theorem and the descriptionof covers of PI through the use of branchcycles. We have added to the classicalinterpretation of branchcycles some important comments on the explicit (algebraic) computation of a descriptionof the branch cycles of a cover. In ?1.B we consider the exact conditionsunder which the points of a cover of P' lying over branch points provide rigidifyingdata (i.e., the identity is the only automorphismof the cover leavinginvariant each of these points). In ?2.A, after we explaincarefully the originalSchur problem, we consider the generalSchur problem which specializesto the Schurproblem for rational functions.The importantresults here are the translationbetween arithmetic (diophantine)conditions and geometricconditions concerning the arithmetic monodromygroup of a cover. In ?2.B, we show that, using the theory of elliptic curves,it is possible to solve the problemposed by the conditions on the arithmeticmonodromy group of a cover that arisesin the considerationof the Schurproblem for rationalfunctions of primedegree. Riemann'sexistence theorem says that the geometricmonodromry group of a cover of Pl is deterniinedby branchcycle data. In ?3 we considerall triples (Y, p,F) consistingof a cover Y . P' defined over a field F such that the cover has a given (a priori) descriptionof its branch cycles. Msingprevious notation, the arithmeticmonodromy group of the cover is G(f(Y")/F(P')). A conjectureis describedthat supportsa precise version of the statement:the arithmeticmonodromy group of the cover is deternminedby branchcycle data and the residueclass fields of points lying over the branchpoints of the cover. Here we use the rigidifying data of ?1.B to consider certain appropriate familiesof covers of P' relatedto (but not the same as) Hurwitzfaniulies. The section also relates the extent to which the resultsof ?2 are supportfor this conjecture. We would like to thank Armand Brumer for his suggestions on the expositionof portionsof this paperand the relatedresults of [Fr, 1, ?2]. Also, it is fairlyclear that Herbrandconsidered problems similar to those discussed GALOISGROUPS AND COMPLEXMULTIPLICATION 143 in thispaper. Since his letterto E. Noetherconsists of a sketchylist of private discoveriesit is difficultto makea precisecomparison with [He]. However, we makean attemptat the end of ?1 to give a technicalinterpretation of his remarks.It was MarvinTretkoff who suggestedthe relevanceof Herbrand's letterwhen we firsttried to exposevarious arithmetic versions of Riemann's existencetheorem during a stayat the Institutefor Advanced Study. 1. Preliminarieson coversand Riemann's existence theorem Let W ( V be a coverof absolutelyirreducible, normal, quasi-projective varieties such that W, V, q(V, W) aredefined over a perfectfield F. All functionfields are assumedto be embeddedin a fixed algebraicallyclosed field containingF. We retainthe notationsof the introductn andwe assumethat 9p(V, W) is a separablemorphism of degreen. Let F(-.'() be the Galoisclosure of the field extensionF(W)/F(V). Thereis a varietyover F, W,such that we havea coverW-* W-+ V and F(W) = ) [Mum,pp. 396-397].Also: W is calledthe normalizationof Win W); W is absolutelyirreducible if and only if F = F whereF is the algebraicclosure of F in F(W) and,the automorphismsof Was a coverof V whichare defined over F (denotedAut(W/ V, F)) are in one-onecorrespon- dencewith the elementsof G(FW)/F( V)). Let S, be the symmetricgroup on n letters.Our notation throughout this paperis: elementsof S,, act on the right of the integers 1, 2, ... , n; elements of Aut(W/V, F) (for any cover W-* V definedover F) act on the left of pointsof W; elementsof G(FiW)/F(V)) act on the rightof elementsof T(W), and, the identificationbetween the groups G(F(W)/F(V)) and Aut(Wi/V, F) is denotedby f0(p) = f(a- (p)) for p E W, f E F(W), and aoE Aut(W/ V, F). The group G(1$0)/F(V)) has a naturalpermutation representation: T(W/ V): G(F(W)/F(V)) -- S, (faithfuland transitive)given by the action on theright cosets of G(1rW)/F(W)). ForH, a subgroupof a groupH', we let NH,(H) be thenormalizer of H in H'. LAj 1.1. The group G(VW)/F(V)) is canonically identified with G(F(W)/F(V)) (the geometricmonodromy group of W/ V). Also, G(F/F) is identified w ith a subgroup of the quotient Ns(G (PW)/F(V)))/G(F(W)/F(V)) whereG(f(`W)/F(V)) is identified withits imageunder T(W/ V). PROOF.The firstpart is well known,and the secondpart follows from the definitionsapplied to theexpression (0.1). Q REMARK 1.1. There can be certainnotational problems in identifying Aut(W/V, F) and G(F(W)/F(V)), especiallywhen we give these groups thle structureof permutationsgroups. For example, the group of 144 MICHAELFRIED automorphismsof W as a cover of V (defined over F; Aut(W/ V, F)) is canonically identified with Cens (G(F(W)/F(V))) (the centralizerin S, of the image of G(F(W)/F(V)) under T(W/V)). However, the group of automorphismsof F(W)/F(V) is canonically identified with the quotient group: NG(F(w-)IF( V))( G ( ?)F ( W))) / G ( )F ( W)) Of course, these two groupsare isomorphic;the isomorphismis just a simple fact of group theory[Fr, 1, ?2].