Introduction to Shimura Varieties - James Milne

1 Introduction to Shimura MilneOctober 23, 2004; revised September 16, 2017 AbstractThis is an Introduction to the theory of Shimura Varieties , or, in other words, to thearithmetic theory of automorphic functions and holomorphic automorphic forms. Inthis revised version, the numbering is unchanged from the original published versionexcept for symmetric domains ..52 Hodge structures and their classifying spaces ..223 Locally symmetric Varieties ..324 Connected Shimura Varieties ..425 Shimura Varieties ..526 The Siegel modular variety ..677 Shimura Varieties of Hodge type ..768 PEL Shimura Varieties ..799 General Shimura Varieties ..9010 Complex multiplication: the Shimura Taniyama formula ..9611 Complex multiplication: the main theorem .. 10612 Definition of canonical models.

2 11013 Uniqueness of canonical models .. 11714 Existence of canonical models .. 12015 Abelian Varieties over finite fields .. 13016 The good reduction of Shimura Varieties .. 14117 A formula for the number of points .. 148 AAppendix Complements .. 151 BAppendix List of Shimura Varieties of abelian type .. 157 CAppendix 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 , InHarmonicanalysis, the trace formula, and Shimura Varieties ,265 378, Clay Math. Proc., 4, Amer. Math. Soc., Providence,RI, 2005.

3 The notes were revised in 2017. Copyrightc 2004, 2017 arithmetic properties of elliptic modular functions and forms were extensively studiedin the 1800s, culminating in the beautiful Kronecker Jugendtraum. Hilbert emphasizedthe importance of extending this theory to functions of several variables in the twelfth ofhis famous problems at the International Congress in 1900. The first tentative steps in thisdirection were taken by Hilbert himself and his students Blumenthal and Hecke in their studyof what are now called Hilbert (or Hilbert Blumenthal) modular Varieties . As the theoryof complex functions of several variables matured, other quotients of bounded symmetricdomains by arithmetic groups were studied (Siegel, Braun, and others). However, the moderntheory of Shimura varieties1only really began with the development of the theory of abelianvarieties 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 canonicalmodels for certain families of Shimura Varieties .

4 In two fundamental articles, Deligne recastthe theory in the language of abstract reductive groups and extended Shimura s results oncanonical models. Langlands made Shimura Varieties a central part of his program, both asa source of representations of Galois groups and as tests for his conjecture that all motivicL-functions are automorphic. These notes are an Introduction to the theory of Shimuravarieties from the point of view of Deligne and Langlands. Because of their brevity, manyproofs have been omitted or only first nine sections study Shimura Varieties over the complex numbers, the next fivestudy them over number fields of characteristic zero (the theory of canonical models), andthe final three study them in mixed characteristic and over finite TO THE REVISED VERSION(2017)On looking at these notes thirteen years after they were written, I found that they read tooclosely as being my personal notes for the lectures.

5 In particular, they lacked the motivationand historical background that (I hope) the lectures provided. In revising them, I haveadded this background, and I have fixed all the errors and instances of careless writing thathave been pointed out to me. Unnumbered asides are new, and this version includes threeappendices not in the published point I should emphasize is that this is an Introduction to the theory ofgeneralShimura Varieties . Although Shimura Varieties of PEL-type form a very important class they are the moduli Varieties of abelian Varieties with polarization, endomorphism, and levelstructure they make up only a small class in the totality of Shimura simplest Shimura Varieties are the elliptic modular curves. My notesModularFunctions and modular Formsemphasize the arithmetic and the geometry of these curves,and so provide an elementary preview of some of the theory discussed in these entire foundations of the theory of Shimura Varieties need to be reworked.

6 Once thathas been accomplished, perhaps I will write a definitive version of the (1968) introduced the term Shimura curve for the algebraic curves uniformized by automorphicfunctions attached to quaternion algebras over totally real fields, whose beautiful arithmetic properties have beendiscovered by Shimura (Annals 1967). Langlands (1976) introduced the term Shimura variety for certainvarieties studied very deeply by Shimura . His definition is that of Deligne Dans un petit nombre de cas,X= peut s interpr eter comme l ensemble des classes d isomorphie desvari et es ab eliennes complexes, muni de quelques structures alg ebriques additionelles (polarisations, endomor-phismes, structures sur les points d ordren). Deligne 1971b2 NOTATION AND CONVENTIONST hroughout,kis a field.

7 Unless indicated otherwise, vector spaces are assumed to befinite-dimensional, and freeZ-modules are assumed to be of finite rank. The linear ;k/of ak-vector space (or module)Vis denoted byV_. For ak-vector spaceVand a commutativek-algebraR, kR(and similarly forZ-modules). Bya lattice, we always mean a full lattice. For example, a lattice in anR-vector spaceVis aZ-submodule such that ZR'V throughout'denotes symbolkadenotes an algebraic closure of the fieldkandksthe separable closure ofkinka. The transpose of a matrixCis denoted algebraic group over a fieldkis a group scheme of finite type overk. Askis alwaysof characteristic zero, such groups are smooth, and hence are not essentially different fromthe algebraic groups in Borel 1991 or Springer 1998.

8 LetGbe an algebraic group overa fieldkof characteristic zero. IfGis connected or, more generally, if every connectedcomponent ofGhas ak-point, dense inGfor the Zariski topology (Milne2017, ). This implies that a connected algebraic subgroup of an algebraic group overkis determined by itsk-points, and that a homomorphism from a connected algebraic group isdetermined by its action on and reductive groups, whether algebraic or Lie, are required to be simple algebraic or Lie group is a semisimple group whose only proper normal subgroupsare finite (sometimes such a group is said to be almost-simple). For example,SLnis a torusToverk,X .T/denotes the character group ofTka. The derived group ofa reductive groupGis denoted byGder(it is a semisimple group), and the adjoint group(quotient ofGby its centre) is denoted byGad.

9 ; thengacts onGby the !gxg 1/and hence an Formore notation concerning reductive groups, see 5. For a finite extension of fieldsL F,the algebraic group overFobtained by restriction of scalars from an algebraic groupGoverLis denoted superscriptC(resp. ) denotes a connected component relative to a real topology ( Zariski topology). For an algebraic group, it means the identity connected component. Forexample,.On/ DSOn,.GLn/ DGLn, of then nmatrices withdet> 0. For an algebraic groupGoverQ, \ Following Bourbaki,I require compact topological spaces to be , I use the notation standard in algebraic geometry, which sometimes conflictswith that used in other areas. For example, ifGandG0are algebraic groups over a fieldk, then a homomorphismG!G0means a homomorphism defined overk; ifKis a fieldcontainingk, thenGKis the algebraic group overKobtained by extension of the basefield the group of points ofGwith coordinates inK.

10 If Wk ,!Kis ahomomorphism of fields andVis an algebraic variety (or other algebro-geometric object)overk, then Vhas its only possible meaning: apply to the coefficients of the sets and let be an equivalence relation onA. If there exists a canonicalsurjectionA!Bwhose fibres are the equivalence classes, then I say thatBclassifies theelements ofAmodulo or that it classifies the -classes of elements ofA. The cardinalityof a setSis denoted byjSj. Throughout, I writeAnB C=Dfor the double coset C/=D(apply beforenand=).A functorFWA!Bis fully faithful if the ;a0/! ;Fa0/arebijective. The essential image of such a functor is the full subcategory ofBwhose objects are34isomorphic to an object of the formFa. An equivalence is a fully faithful functorFWA!Bwhose essential image addition to the references listed at the end, I refer to the following of my course notes(available at ).