Introduction to Shimura Varieties J.S. Milne October 23, 2004; revised September 16, 2017 Abstract This is an introduction to the theory of Shimura varieties, or, in other words, to the arithmetic theory of automorphic functions and holomorphic automorphic forms. In this revised version, the numbering is unchanged from the original published version except for displays. Contents 1 Hermitian symmetric domains..................5 2 Hodge structures and their classifying spaces............. 22 3 Locally symmetric varieties................... 32 4 Connected Shimura varieties................... 42 5 Shimura varieties....................... 52 6 The Siegel modular variety................... 67 7 Shimura varieties of Hodge type................. 76 8 PEL Shimura varieties..................... 79 9 General Shimura varieties.................... 90 10 Complex multiplication: the Shimura–Taniyama formula........ 96 11 Complex multiplication: the main theorem.............. 106 12 Definition of canonical models.................. 110 13 Uniqueness of canonical models................. 117 14 Existence of canonical models.................. 120 15 Abelian varieties over finite fields................. 130 16 The good reduction of Shimura varieties.............. 141 17 A formula for the number of points................ 148 A Appendix Complements.................... 151 B Appendix List of Shimura varieties of abelian type........... 157 C Appendix Review of Shimura’s Collected Papers........... 160 References........................... 163 Index............................. 170 These are my notes for a series of fifteen lectures at the Clay Summer School, Fields Institute, Toronto, June 2 – June 27, 2003. The original version was published as: “Introduction to Shimura varieties, In Harmonic analysis, the trace formula, and Shimura varieties, 265–378, Clay Math. Proc., 4, Amer. Math. Soc., Providence, RI, 2005.” The notes were revised in 2017. Copyright c 2004, 2017 J.S. Milne. 1 Introduction The arithmetic properties of elliptic modular functions and forms were extensively studied in the 1800s, culminating in the beautiful Kronecker Jugendtraum. Hilbert emphasized the importance of extending this theory to functions of several variables in the twelfth of his famous problems at the International Congress in 1900. The first tentative steps in this direction were taken by Hilbert himself and his students Blumenthal and Hecke in their study of what are now called Hilbert (or Hilbert–Blumenthal) modular varieties. As the theory of complex functions of several variables matured, other quotients of bounded symmetric domains by arithmetic groups were studied (Siegel, Braun, and others). However, the modern theory of Shimura varieties1 only really began with the development of the theory of abelian varieties with complex multiplication by Shimura, Taniyama, and Weil in the mid-1950s, and with the subsequent proof by Shimura and his students of the existence of canonical models for certain families of Shimura varieties. In two fundamental articles, Deligne recast the theory in the language of abstract reductive groups and extended Shimura’s results on canonical models. Langlands made Shimura varieties a central part of his program, both as a source of representations of Galois groups and as tests for his conjecture that all motivic L-functions are automorphic. These notes are an introduction to the theory of Shimura varieties from the point of view of Deligne and Langlands. Because of their brevity, many proofs have been omitted or only sketched. The first nine sections study Shimura varieties over the complex numbers, the next five study them over number fields of characteristic zero (the theory of canonical models), and the final three study them in mixed characteristic and over finite fields. INTRODUCTION TO THE REVISED VERSION (2017) On looking at these notes thirteen years after they were written, I found that they read too closely as being my personal notes for the lectures. In particular, they lacked the motivation and historical background that (I hope) the lectures provided. In revising them, I have added this background, and I have fixed all the errors and instances of careless writing that have been pointed out to me. Unnumbered asides are new, and this version includes three appendices not in the published version. One point I should emphasize is that this is an introduction to the theory of general Shimura varieties. Although Shimura varieties of PEL-type form a very important class — they are the moduli varieties of abelian varieties with polarization, endomorphism, and level structure — they make up only a small class in the totality of Shimura varieties.2 The simplest Shimura varieties are the elliptic modular curves. My notes Modular Functions and Modular Forms emphasize the arithmetic and the geometry of these curves, and so provide an elementary preview of some of the theory discussed in these notes. The entire foundations of the theory of Shimura varieties need to be reworked. Once that has been accomplished, perhaps I will write a definitive version of the notes. 1Ihara (1968) introduced the term “Shimura curve” for the “algebraic curves uniformized by automorphic functions attached to quaternion algebras over totally real fields, whose beautiful arithmetic properties have been discovered by Shimura (Annals 1967).” Langlands (1976) introduced the term “Shimura variety” for “certain varieties” studied “very deeply” by Shimura. His definition is that of Deligne 1971b. 2“Dans un petit nombre de cas, X= peut s’interpreter´ comme l’ensemble des classes d’isomorphie des variet´ es´ abeliennes´ complexes, muni de quelques structures algebriques´ additionelles (polarisations, endomor- phismes, structures sur les points d’ordre n).” Deligne 1971b 2 NOTATION AND CONVENTIONS Throughout, k is a field. Unless indicated otherwise, vector spaces are assumed to be finite-dimensional, and free Z-modules are assumed to be of finite rank. The linear dual Hom.V;k/ of a k-vector space (or module) V is denoted by V _. For a k-vector space V and a commutative k-algebra R, V .R/ denotes V k R (and similarly for Z-modules). By ˝ a lattice, we always mean a full lattice. For example, a lattice in an R-vector space V is a Z-submodule such that R V — throughout denotes a canonical isomorphism. ˝Z ' ' The symbol ka denotes an algebraic closure of the field k and ks the separable closure of k in ka. The transpose of a matrix C is denoted by C t . An algebraic group over a field k is a group scheme of finite type over k. As k is always of characteristic zero, such groups are smooth, and hence are not essentially different from the algebraic groups in Borel 1991 or Springer 1998. Let G be an algebraic group over a field k of characteristic zero. If G is connected or, more generally, if every connected component of G has a k-point, then G.k/ is dense in G for the Zariski topology (Milne 2017, 17.93). This implies that a connected algebraic subgroup of an algebraic group over k is determined by its k-points, and that a homomorphism from a connected algebraic group is determined by its action on the k-points. Semisimple and reductive groups, whether algebraic or Lie, are required to be connected. A simple algebraic or Lie group is a semisimple group whose only proper normal subgroups are finite (sometimes such a group is said to be almost-simple). For example, SLn is simple. For a torus T over k, X .T / denotes the character group of Tka . The derived group of a reductive group G is denoted by Gder (it is a semisimple group), and the adjoint group (quotient of G by its centre) is denoted by Gad. Let g G.k/; then g acts on G by the inner def 2 automorphism ad.g/ .x gxg 1/ and hence on Lie.G/ by an automorphism Ad.g/. For D 7! more notation concerning reductive groups, see 5. For a finite extension of fields L F , the algebraic group over F obtained by restriction of scalars from an algebraic group G over L is denoted by .G/L=F . A superscript C (resp. ı) denotes a connected component relative to a real topology (resp. a Zariski topology). For an algebraic group, it means the identity connected component. For example, .On/ı SOn, .GLn/ı GLn, and GLn.R/C consists of the n n matrices with D D det > 0. For an algebraic group G over Q, G.Q/C G.Q/ G.R/C. Following Bourbaki, D \ I require compact topological spaces to be Hausdorff. Throughout, I use the notation standard in algebraic geometry, which sometimes conflicts with that used in other areas. For example, if G and G0 are algebraic groups over a field k, then a homomorphism G G means a homomorphism defined over k; if K is a field ! 0 containing k, then GK is the algebraic group over K obtained by extension of the base field and G.K/ is the group of points of G with coordinates in K. If k , K is a W ! homomorphism of fields and V is an algebraic variety (or other algebro-geometric object) over k, then V has its only possible meaning: apply to the coefficients of the equations defining V . Let A and B be sets and let be an equivalence relation on A. If there exists a canonical surjection A B whose fibres are the equivalence classes, then I say that B classifies the ! elements of A modulo or that it classifies the -classes of elements of A. The cardinality of a set S is denoted by S . Throughout, I write A B C=D for the double coset space j j n A .B C /=D (apply before and =). n n A functor F A B is fully faithful if the maps Hom .a;a / Hom .F a;F a / are W ! A 0 ! B 0 bijective. The essential image of such a functor is the full subcategory of B whose objects are 3 4 isomorphic to an object of the form F a.