Shimura Varieties and Moduli - James Milne

1 Shimura Varieties and MilneApril 30, 2011, Shimura Varieties are the quotients of hermitian symmetric domainsby discrete groups defined by congruence conditions. We examine their relation withmoduli ..41 Elliptic modular curves5 Definition of elliptic modular curves ..5 Elliptic modular curves as Moduli Varieties ..62 Hermitian symmetric domains9 Preliminaries on Cartan involutions and polarizations ..9 Definition of hermitian symmetric domains ..10 Classification in terms of real groups ..11 Classification in terms of root systems ..12 Example: the Siegel upper half space ..133 Discrete subgroups of Lie groups14 Lattices in Lie groups ..14 Arithmetic subgroups of algebraic groups ..15 Arithmetic lattices in Lie groups ..16 Congruence subgroups of algebraic groups ..184 Locally symmetric varieties18 Quotients of hermitian symmetric domains ..18 The algebraic structure on the quotient ..19 Chow s theorem.

2 19 The Baily-Borel theorem ..19 Borel s theorem ..19 Locally symmetric Varieties ..20 Example: Siegel modular Varieties ..201 CONTENTS25 Variations of Hodge structures21 The Deligne torus ..21 Real Hodge structures ..22 Rational Hodge structures ..23 Polarizations ..23 Local systems and vector sheaves with connection ..23 Variations of Hodge structures ..246 Mumford-Tate groups and their variation in families24 The conditions (SV) ..24 Definition of Mumford-Tate groups ..25 Special Hodge structures ..27 The generic Mumford-Tate group ..28 Variation of Mumford-Tate groups in families ..29 Proof of (a) of Theorem ..30 Proof of the first statement of (b) of Theorem ..31 Proof of the second statement of (b) of Theorem ..32 Variation of Mumford-Tate groups in algebraic families ..337 Period subdomains33 Flag manifolds ..33 Period domains ..34 Period subdomains ..35 Why Moduli Varieties are (sometimes) locally symmetric.

3 37 Application: Riemann s theorem in families ..388 Variations of Hodge structures on locally symmetric varieties39 Existence of Hodge structures of CM-type in a family ..39 Description of the variations of Hodge structures onD. /..40 Existence of variations of Hodge structures ..429 Absolute Hodge classes and motives43 The standard cohomology theories ..44 Absolute Hodge classes ..44 Proof of Deligne s theorem ..47 Motives for absolute Hodge classes ..49 Abelian motives ..50CM motives ..51 Special motives ..51 Families of abelian motives ..5110 Symplectic Representations52 Preliminaries ..53 The real case ..53 TypeAn..57 TypeBn..57 TypeCn..57 TypeDn..58 CONTENTS3 TypeE6..58 TypeE7..59 The rational case .. ;xh/for which there do not exist symplectic representations .. ;xh/for which there exist symplectic representations ..61 Conclusion ..6311 Moduli64 Mumford-Tate groups ..64 Families of abelian Varieties and motives.

4 66 Shimura Varieties ..67 Shimura Varieties as Moduli Varieties ..68 Remarks ..70 References71 Index of definitions and symbols75 IntroductionThe hermitian symmetric domains are the complex manifolds isomorphic to bounded sym-metric domains. The Griffiths period domains are the parameter spaces for polarized ratio-nal Hodge structures. A period domain is a hermitian symmetric domain if the universalfamily of Hodge structures on it is a variation of Hodge structures, , satisfies Griffithstransversality. This rarely happens, but, as Deligne showed, every hermitian symmetric do-main can be realized as the subdomain of a period domain on which certain tensors for theuniversal family are of ;p/( , are Hodge tensors).In particular, every hermitian symmetric domain can be realized as a Moduli space forHodge structures plus tensors. This all takes place in the analytic realm, because hermitiansymmetric domains are not algebraic Varieties .

5 To obtain an algebraic variety, we mustpass to the quotient by an arithmetic group. In fact, in order to obtain a Moduli variety, weshould assume that the arithmetic group is defined by congruence conditions. The algebraicvarieties obtained in this way are the connected Shimura arithmetic subgroup lives in a semisimple algebraic group overQ, and the varia-tions of Hodge structures on the connected Shimura variety are classified in terms of aux-iliary reductive algebraic groups. In order to realize the connected Shimura variety as amoduli variety, we must choose the additional data so that the variation of Hodge struc-tures is of geometric origin. The main result of the article classifies the connected Shimuravarieties for which this is known to be possible. Briefly, in a small number of cases, theconnected Shimura variety is a Moduli variety for abelian Varieties with polarization, en-domorphism, and level structure (the PEL case); for a much larger class, the variety is amoduli variety for abelian Varieties with polarization, Hodge class, and level structure (thePHL case); for all connected Shimura Varieties except those of typeE6,E7, and certaintypesD, the variety is a Moduli variety for abelianmotiveswith additional structure.

6 In theremaining cases, the connected Shimura variety is not a Moduli variety for abelian motives,and it is not known whether it is a Moduli variety at now summarize the contents of the 1. As an introduction to the general theory, we review the case of elliptic modularcurves. In particular, we prove that the modular curve constructed analytically coincideswith the modular curve constructed algebraically using geometric invariant theory. 2. We briefly review the theory of hermitian symmetric domains. To give a hermitiansymmetric domain amounts to giving a real semisimple Lie groupHwith trivial centre anda homomorphismufrom the circle group toHsatisfying certain conditions. This leads toa classification of hermitian symmetric domains in terms of Dynkin diagrams and specialnodes. 3. The group of holomorphic automorphisms of a hermitian symmetric domain is a realLie group, and we are interested in quotients of the domain by certain discrete subgroupsof this Lie group.

7 In this section we review the fundamental theorems of Borel, Harish-Chandra, Margulis, Mostow, Selberg, Tamagawa, and others concerning discrete subgroupsof Lie groups. 4. The arithmetic locally symmetric Varieties (resp. connected Shimura Varieties ) arethe quotients of hermitian symmetric domains by arithmetic (resp. congruence) groups. Weexplain the fundamental theorems of Baily and Borel on the algebraicity of these varietiesand of the maps into them. 5. We review the definition of Hodge structures and of their variations, and state thefundamental theorem of Griffiths that motivated their definition. 6. We define the Mumford-Tate group of a rational Hodge structure, and we prove thebasic results concerning their behaviour in families. 7. We review the theory of period domains, and explain Deligne s interpretation ofhermitian symmetric domains as period subdomains. 8. We classify certain variations of Hodge structures on locally symmetric Varieties interms of group-theoretic data.

8 Order to be able to realize all but a handful of locally symmetric Varieties asmoduli Varieties , we shall need to replace algebraic Varieties and algebraic classes by moregeneral objects. In this section, we prove Deligne s theorem that all Hodge classes onabelian Varieties are absolutely Hodge, and have algebraic meaning, and we define abelianmotives. 10. Following Satake and Deligne, we classify the symplectic embeddings of an alge-braic group that give rise to an embedding of the associated hermitian symmetric domaininto a Siegel upper half space. 11. We use the results of the preceding sections to determine which Shimura varietiescan be realized as Moduli Varieties for abelian Varieties (or abelian motives) plus the expert will find little that is new in this article, there is much that is notwell explained in the literature. As far as possible, complete proofs have been usekto denote the base field (always of characteristic zero), andkalto denote an alge-braic closure ofk.

9 Algebraic group means affine algebraic group scheme and algebraicvariety means geometrically reduced scheme of finite type over a field . For a smooth al-gebraic varietyXoverC, we letXandenote the with its natural structureof a complex manifold. The tangent space at a pointpof spaceXis denoted spaces and representations are finite dimensional unless indicated otherwise. The1 ELLIPTIC MODULAR CURVES5linear dual of a vector spaceVis denoted byV_. For ak-vector spaceVand commutativek-algebraR,VRDR kV. For a topological spaceS, we letVSdenote the constant localsystem of vector spaces onSdefined byV. By a lattice in a real vector space, we mean afull lattice, , theZ-module generated by a basis for the vector sheafon a complex manifold (or scheme)Sis a locally free sheaf ofOS-modules of finite rank. In order forWto be a vector subsheaf of a vector sheafV, werequire that the maps on the fibresWs!Vsbe injective. With these definitions, vectorsheaves correspond to vector bundles and vector subsheaves to vector quotient of a Lie group or algebraic groupGby its denoted Lie group or algebraic group is said to beadjointif it is semisimple (in particular, con-nected) with trivial centre.

10 An algebraic group issimple( simple) if it connectednoncommutative and every proper normal subgroup is trivial (resp. finite). Anisogenyofalgebraic groups is a surjective homomorphism with finite kernel. An algebraic groupGissimply connectedif it is semisimple and every isogenyG0!GwithG0connected is anisomorphism. The inner automorphism ofGdefined by an elementgis denoted by adWG!Gadbe the quotient map. There is an action ofGadonGsuch that For an algebraic groupGoverR, the identity com-ponent the real topology. For a finite extension of fieldsL=kand an algebraicgroupGoverL, we algebraic group overkobtained by (Weil) restrictionof scalars. As usual,GmDGL1and Nis the kernel ofGmN ! a number fieldkis a prime ideal inOk(a finite prime), an embedding ofkintoR(a real prime), or a conjugate pair of embeddings ofkintoC(a complex prime).The ring of finite ad`eles ofQisAfDQ QpZp .We use orz7!xzto denote complex conjugation onCor on a subfield ofC, andwe useX'Yto mean thatXandYisomorphic with a specific isomorphism whichisomorphism should always be clear from the algebraic groups we use the language of modern algebraic geometry, not the moreusual language, which is based on Weil s Foundations.