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Shimura Varieties Shimura varieties Spring 2015 These are my notes from the working group on Shimura varieties http://www.math.u-bordeaux1.fr/~jtong/GAGA.html Please send any remarks to [email protected] Contents 1 Linear algebraic groups (review of results)1 1.1 Basic definitions and examples..........................................1 1.2 Tori..........................................................2 1.3 K /k-forms......................................................3 1.4 Structure of algebraic groups...........................................3 1.5 Root systems.....................................................5 1.6 Classification over an algebraically closed field................................ 11 1.7 Classification over a perfect field......................................... 11 2 Hermitian symmetric spaces: examples and basic properties 12 2.1 Almost complex manifolds............................................ 12 2.2 Hermitian manifolds................................................ 13 2.3 Symmetric spaces.................................................. 14 2.4 Bergman metric................................................... 15 2.5 A higher-dimensional example: Siegel upper half-space........................... 17 2.6 Isometries of Hermitian symmetric domains.................................. 18 3 Classification of symmetric Hermitian domains 20 3.1 Preliminaries: Cartan involutions........................................ 20 8 January 2015 (Alan Hertgen) 1 Linear algebraic groups (review of results) The used reference for this is Satake, Classification theory of semi-simple algebraic groups. A modern reference for algebraic groups is http://jmilne.org/math/CourseNotes/ala.html 1.1 Basic definitions and examples Let G be an algebraic group over k, by which we mean a smooth group scheme over k. (REMARK: usually we also assume that G is of finite type.) If the underlying scheme is affine, then we say that G is an affine algebraic group. Example 1.1. • Any finite group may be seen as an algebraic group (take the constant functor of points). 1 • The additive group G corresponds to the functor R (R, ), which is represented by Speck[T ]. a Å 1 • The additive group G corresponds to the functor R (R£, ), which is represented by Speck[T,T ¡ ] m ¢ Æ Speck[T ,T ]/(T T 1). 1 2 1 2 ¡ • The special linear group SLn corresponds to the functor R SLn(R), which is represented by Speck[T ,...,T ]/(det(T ,...,T ) 1), 11 nn 11 nn ¡ where det(T11,...,Tnn) is the determinant X sgn(σ)T T . 1,σ1 ¢¢¢ n,σn σ Sn 2 • The general linear group GLn corresponds to the functor R GLn(R), which is represented by Speck[T ,...,T ,T ]/(det(T ,...,T )T 1). 11 nn 11 nn ¡ More generally, if V is a finite dimensional vector space over k, then we may define GLV as a representable functor R Aut(V R). ­k • Similarly, we have SOn and Spn (for chark , 2). N Definition 1.2. An algebraic subgroup of GLV is called a linear algebraic group. In particular, a linear algebraic group is affine. Conversely, every affine algebraic group is isomorphic to a linear algebraic group. A theorem of Chevalley asserts that an algebraic group G is built up from a linear algebraic group and an abelian variety. Precisely, we have Theorem 1.3 (Chevalley). If G is a connected algebraic group over a perfect field k, then there exists a unique short exact sequence of algebraic groups 1 H G A 1 ! ! ! ! where H is a linear algebraic group and A is an abelian variety. Proposition 1.4. Let G be an algebraic group over k. There is a unique short exact sequence 1 G± G ¼ (G) 1 ! ! ! 0 ! where G± is connected, and ¼0(G) is finite étale over k. 1.2 Tori d Definition 1.5. An algebraic group is called a split torus (tore déployé) if it is isomorphic to Gm. An algebraic group T is called a torus if Tk is a split torus. Example 1.6. The algebraic group D GL formed by diagonal matrices is a split torus. n ½ n 0 1 ¤ B C B ¤ C B . C @ .. A ¤ In fact, any subtorus T in GLn /k is split iff it lies in Dn after some basis change; that is, iff there is a matrix P 1 2 GL (k) such that PTP ¡ D . N n ½ n 2 Definition 1.7. The group of characters of a torus T is given by X ¤(T ) : Hom(T ,(G ) ). Æ k m k It is a free Z-module of finite rank, which is the dimension of T . Further, X ¤(T ) is equipped with an action of Gal(k/k). Theorem 1.8. There is an equivalence of categories tori over k free modules of finite rank with Gal(k/k)-action, $ T X ¤(T ). 1.3 K /k-forms Let G be an algebraic group over k and let K /k be a Galois extension. Definition 1.9. A K /k-form of G is an algebraic group H/k such that there exists an isomorphism f : H G . K ! K For K k we will simply say that H/k is a k-form of G. Æ Let ¡ : Gal(K /k). For each σ ¡ we have f σ : H G , and we put Æ 2 K ! K σ 1 Á : f f ¡ Aut(G ). σ Æ ± 2 K Such maps Áσ satisfy the cocycle condition 1 Á¡ Á Á . σ ± ¿ Æ στ One shows that two K /k-forms are isomorphic iff the corresponding cocycles are cohomological: Theorem 1.10. There is a natural bijection {K /k-forms of G}/ H 1(¡,Aut(G )). isomorphism $ K d Example 1.11. A torus of dimension d is by definition a k-form of Gm. 1 Gd 1 Z Z H (¡,Aut(( m)k )) H (¡,GLn( )) with the trivial action Homcont(¡,GLn( ))/conjugation N 1.4 Structure of algebraic groups Definition 1.12. Let G be an algebraic group over k. An element u G(k) is called unipotent if for an embedding 2 G , (GL ) its image in GL (k) is unipotent, i.e. (u 1)k 0 for some k 1,2,3,.... k ! n,k n ¡ Æ Æ An algebraic group G is called unipotent if all elements of G(k) are unipotent. Example 1.13. Consider the algebraic group U GL consisting of upper triangular matrices with 1 on the diag- n ½ n onal 01 1 ¤ ¤ ¢¢¢ ¤ B 1 C B C B ¤ ¢¢¢ ¤C B 1 C B ¢¢¢ ¤. C B .. C @ . A 1 Un is unipotent. N Definition 1.14. We say that an algebraic group G is solvable if G(k) is solvable. 3 Proposition 1.15. Assume that k k and let G be a connected solvable algebraic group over k. Æ 1. By the Lie–Kolchin theorem G is isomorphic to an algebraic subgroup of the group Tn of upper triangular matrices. 2. There is a unique maximal unipotent algebraic subgroup Gu. It is a connected normal subgroup with k-points G (k) {u G(k) u is unipotent}. u Æ 2 j 3. Let T be the maximal subtorus of G. Then G G o T. Æ u 4. The maximal tori are conjugate by an element of G(k). Definition 1.16. Let G be an algebraic group over k. G has a unique maximal connected solvable normal subgroup R(G), called the radical of G. G has a unique maximal connected unipotent normal subgroup Ru(G), called the unipotent radical. We say that G is semisimple if R(G ) 0 and that G is reductive if R (G ) 0. k Æ u k Æ G Example 1.17. SLn is semisimple. GLn is reductive, with radical R((GLn)k ) ( m)k . N We say that G is simple if it has no normal algebraic subgroups of dimension 0. Recall that a morphism of È algebraic groups f : G H is called an isogeny if it is surjective and has finite kernel. ! Proposition 1.18. 1. A semisimple algebraic group G is isogenous to a direct product of simple algebraic groups. 2. A reductive algebraic group G is isogenous to a direct product of a semisimple group and a torus. Definition 1.19. A Borel subgroup in G is a maximal solvable connected algebraic subgroup. Proposition 1.20. Assume k k. Let G be a connected algebraic group over k. Æ 1. The Borel subgroups in G are conjugate. 2. If B is a Borel subgroup, then G/B is projective. 3. Any Borel subgroup equals its normalizer. S 1 4.G g G(k) g ¡ B g. Æ 2 The proof is based on the Borel fixed-point theorem. Proposition 1.21 (Borel). Let G be a connected solvable algebraic group acting regularly on a non-empty, complete algebraic variety W over k k. Then there exists a fixed point. Æ Borel’s result generalizes Lie–Kolchin theorem. If G GL is a connected solvable algebraic group, then G acts ½ V on the complete flags of V 0 V ( V ( ( V V Æ 0 1 ¢¢¢ n Æ and such flags form a projective variety, so by the Borel’s theorem there is a fixed flag. The corresponding choice of basis gives an embedding G , T . ! n Definition 1.22. The rank of G is the dimension of a maximal torus in G. The k-rank of G is the dimension of a maximal split torus. We say that G is anisotropic if its k-rank is 0. 4 1.5 Root systems Let G be a connected semisimple algebraic group over k and let T be a maximal subtorus. Definition 1.23. We say that a character ® X ¤(T ) is a root if there exists an isomorphism u : G U to a sub- 2 a ¡! ® group U G such that for all t T and s G ® ½ 2 2 a 1 t u(s)t ¡ u(®(t)s).
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