The Siegel Modular Variety Eran Assaf Dartmouth College Shimura Varieties Reading Seminar, April 2020 Eran Assaf The Siegel Modular Variety 1 Symplectic Spaces 2 Siegel Modular Variety 3 Complex Abelian Varieties 4 Modular Description Structure of the talk Eran Assaf The Siegel Modular Variety 2 Siegel Modular Variety 3 Complex Abelian Varieties 4 Modular Description Structure of the talk 1 Symplectic Spaces Eran Assaf The Siegel Modular Variety 3 Complex Abelian Varieties 4 Modular Description Structure of the talk 1 Symplectic Spaces 2 Siegel Modular Variety Eran Assaf The Siegel Modular Variety 4 Modular Description Structure of the talk 1 Symplectic Spaces 2 Siegel Modular Variety 3 Complex Abelian Varieties Eran Assaf The Siegel Modular Variety Structure of the talk 1 Symplectic Spaces 2 Siegel Modular Variety 3 Complex Abelian Varieties 4 Modular Description Eran Assaf The Siegel Modular Variety Symplectic Spaces Definition Symplectic space is a pair pV ; q with V a k-vector space and : V ˆ V Ñ k a symplectic form. bilinear. alternating - pv; vq “ 0. nondegenerate - pu; V q “ 0 ñ u “ 0. Remark Definition dimpV q “ 2n W Ď V is totally isotropic if pW ; W q “ 0. Definition Symplectic basis is a basis B such that 0 ´1n r sB “ 1n 0 ˆ ˙ Eran Assaf The Siegel Modular Variety Symplectic Spaces Lemma (Milne, 6.1) W Ď V totally isotropic, BW a basis of W . Then there is a symplectic basis BV of V extending BW . Proof. By induction on n. Identify W _ with the complement of W K via v ÞÑ pv; ´q. The dual basis to BW gives a symplectic basis of W ` W _. By the induction hypothesis, pW ` W _qK also has a symplectic basis. Corollary Any two symplectic spaces of the same dimension are isomorphic. V has a symplectic basis. Eran Assaf The Siegel Modular Variety Symplectic Groups Definition The symplectic group Spp q Spp qpkq “ tg P GLpV q| pgu; gvq “ pu; vqu The group of symplectic similitudes GSpp q GSpp qpkq “ tg P GLpV q| pgu; gvq “ νpgq¨ pu; vq; νpgq P kˆu The group of projective symplectic similitudes PGSpp q PGSpp qpkq “ GSpp qpkq{kˆ Spp q ad & Gm / GSpp q / PGSpp q ν # Gm Eran Assaf The Siegel Modular Variety Shimura Datum Reductive group Let Gp q “ GSpp q. Note that Gp qad “ PGSpp q and Gp qder “ Spp q. GpRq-conjugacy class of homomorphisms h : S Ñ GR For J a complex structure on V pRq s.t. RpJu; Jvq “ Rpu; vq, i.e. 2 J P Spp qpRq such that J “ ´1, let J pu; vq :“ Rpu; Jvq. Write ` X p q “ tJ P Spp qpRq | J ¡ 0u, ´ ` ´ X p q “ tJ P Spp qpRq | J ă 0u and X p q “ X p q X p q . Lemma Ů ˆ The map J ÞÑ hJ : X p q Ñ HompC ; GpRqq defined by hJ pa ` biq “ a ` bJ identifies X with a GpRq-conjugacy class in HompCˆ; GpRqq. Eran Assaf The Siegel Modular Variety Shimura Datum Proof. For z “ a ` bi, we have phJ pzqu; hJ pzqvq “ pau ` bJu; av ` bJvq “ “ a2 pu; vq ` b2 pJu; Jvq ` abp pu; Jvq ` pJu; vqq But pJu; Jvq “ pu; vq and pJu; vq “ pJ2u; Jvq “ p´u; Jvq “ ´ pu; Jvq hence 2 2 phJ pzqu; hJ pzqvq “ pa ` b q pu; vq “ zz¯ ¨ pu; vq In particular, hJ pzq P GpRq. For g P GpRq ´1 ´1 ´1 gJg ´1 pu; vq “ pu; gJg vq “ νpgq pg u; Jpg vqq “ ´1 ´1 “ νpgq J pg u; g vq so that for J P X p q, gJg ´1 P X p q. (Cont...) Eran Assaf The Siegel Modular Variety Shimura Datum Proof. Also, for z “ a ` bi ´1 ´1 ´1 hgJg ´1 pzq “ a ` bgJg “ gpa ` bJqg “ ghJ pzqg It remains to show transitivity. For that, let Bp q be the set of symplectic bases of pV ; q. We ` 2n have Bp q Ñ X p q : B ÞÑ JB defined for B “ pei qi“1 by ei`n 1 ¤ i ¤ n JB pei q “ #´ei´n n ` 1 ¤ i ¤ 2n 2 Indeed, JB “ ´1, and B is orthonormal for J . This map is surjective (orthonormal basis for J ) and equivariant: ´1 JgB pgei q “ gJB pei q “ gJB g pgei q Spp qpRq acts transitively on Bp q, hence on X p q`. Finally, the map g P GpRq swapping ei with ei`n has νpgq “ ´1 and swaps X p q` with X p q´. Eran Assaf The Siegel Modular Variety Shimura Datum Proposition The pair pGp q; X p qq is a Shimura datum. It satisfies SV1-SV6. (SV1) For all h P X , the Hodge structure on LiepGRq defined by Ad ˝h is of type tp´1; 1q; p0; 0q; p1; ´1qu. Proof. We have LiepGRp qq Ď LiepGLRpV qq “ EndpV q and the action defined by Ad ˝h on EndpV q is pzf qpvq “ phpzq ˝ f ˝ hpzq´1qpvq Let V pCq “ V ` ` V ´ hence hpzqv “ zv for v P V ` and hpzqv “ zv¯ for v P V ´. We have ` ´ ` ´ ´ ` EndpV pCqq “ EndpV q`EndpV q`HompV ; V q`HompV ; V q with actions by 1; 1; z¯{z; z{z¯ respectively. Eran Assaf The Siegel Modular Variety Shimura Datum (SV2) For all h X , ad h i is a Cartan involution of G ad. P p p qq R Proof. 1 J “ hpiq. Let : V pCq ˆ V pCq Ñ C be the sesquilinear form 1 defined by pu; vq “ Cpu; v¯q. Then for g P GpCq 1 ´1 1 1 pgu; JpJ gJ¯ qvq “ pgu; gJv¯ q “ Cpgu; gJvq “ pu; Jvq 1 1 1 so that J pgu; adpJqpg¯qvq “ J pu; vq, and J is invariant under pad Jq 1 G . Since J is symmetric and positive(negative)-definite, J pad Jq 1 is Hermitian and positive(negative)-definite. Then G “ Up J q is a definite unitary group, hence compact. Eran Assaf The Siegel Modular Variety Shimura Datum (SV3) ad G has no Q-factor on which the projection of h is trivial. Proof. ad The root system of Spp q is irreducible, hence G is Q-simple. Finally, PGSppRq is not compact (SV4) The weight homomorphism wX : Gm Ñ GR is defined over Q. Proof. ˆ ` ´ r P R acts as r on both V and V , so wX prq “ r. Eran Assaf The Siegel Modular Variety Shimura Datum (SV5) The group ZpQq is discrete in ZpAf q. Proof. ˆ ˆ Z “ Gm, and Q is discrete in Af . (SV6) The torus Z ˝ splits over a CM-field. Proof. Z “ Gm is already split over Q. Eran Assaf The Siegel Modular Variety The Siegel Modular Variety Moduli space K Ď GpAf q compact open. HK - set of triples ppW ; hq; s; ηKq s.t. pW ; hq is a rational Hodge structure of type p´1; 0q; p0; ´1q. s or ´s is a polarization for pW ; hq. ηK is a K-orbit in HomAf pV pAf q; W pAf qq with ˆ sAf pηpuq; ηpvqq “ νpηq ¨ Af pu; vq for some νpηq P Af . An isomorphism ppW ; hq; s; ηKq Ñ ppW 1; h1q; s1; η1Kq is an isomorphism b : pW ; hq Ñ pW 1; h1q s.t. s1pbpuq; bpvqq “ µpbqspu; vq for some µpbq P Qˆ. b ˝ ηK “ η1K. Proposition (Milne, 6.3) The set ShK pGp q; X p qqpCq classifies the elements of HK modulo isomorphism. Eran Assaf The Siegel Modular Variety The Siegel Modular Variety Proof. Let a : W Ñ V be an isomorphism s.t. papuq; apvqq “ µpaqspu; vq for some µpaq P Qˆ. Then ´1 ´1 ´1 ´1 RpaJa u; aJa vq “ µpaqsRpJa u; Ja vq “ ´1 ´1 “ µpaqsRpa u; a vq “ Rpu; vq and ´1 ´1 ´1 aJa´1 pu; uq “ Rpu; aJa uq “ µpaqsRpa u; Ja uq “ ´1 ´1 “ µpaqsJ pa u; a uq so that pahqpzq :“ a ˝ hpzq ˝ a´1 P X . From Af ppa ˝ ηqpuq; pa ˝ ηqpvqq “ µpaqsAf pηpuq; ηpvqq “ “ µpaqνpηq Af pu; vq we see that a ˝ η P GpAf q. Define a map HK Ñ GpQqzX ˆ GpAf q{K : ppW ; hq; s; ηKq ÞÑ rah; a ˝ ηsK Eran Assaf The Siegel Modular Variety The Siegel Modular Variety Proof. 1 If a1 : W Ñ V is another such isomorphism, then a1 ˝ a´ P GpQq, 1 1 hence rah; a ˝ ηsK “ ra h; a ˝ ηsK , so the map is independent of a. If b : ppW ; hq; s; ηKq Ñ ppW 1; h1q; s1; η1Kq is an isomorphism, and a1 : W 1 Ñ V , then a ˝ b : W Ñ V , and ppa ˝ bqhqpzq “ a ˝ b ˝ hpzq ˝ b´1 ˝ a´1 “ a ˝ h1pzq ˝ a´1 “ pah1qpzq 1 1 so that rpa ˝ bqh; a ˝ b ˝ ηsK “ rah ; a ˝ η sK . Thus, the map factors through the equivalence relation. If ppW ; hq; s; ηKq and ppW 1; h1q; s1; η1Kq map to the same element, take a : W Ñ V and a1 : W 1 Ñ V . Then pah; a ˝ ηq “ pa1h1; a1 ˝ η1 ˝ kq. Take b “ a´1 ˝ a1 : W 1 Ñ W . Surjectivity - rh; gsK is the image of ppV ; hq; ; gKq. Eran Assaf The Siegel Modular Variety Abelian Varieties Definition An Abelian variety over k is a proper connected group variety. Remark group variety is smooth - translate the smooth locus. connected group variety is geometrically connected - 1 P Gpkq. Lemma (Rigidity Lemma) α : X ˆ Y Ñ Z with X proper and X ˆ Y geom. irreducible. αpX ˆ tyuq “ tzu “ αptxu ˆ Y q Then αpX ˆ Y q “ tzu. Observations May assume k “ k¯.