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The Arithmetic Volume of Shimura Varieties of Orthogonal Type Outline Algebraic geometry and Arakelov geometry Orthogonal Shimura varieties Main result Integral models and compactifications of Shimura varieties The arithmetic volume of Shimura varieties of orthogonal type Fritz H¨ormann Department of Mathematics and Statistics McGill University Montr´eal,September 4, 2010 Fritz H¨ormann Department of Mathematics and Statistics McGill University The arithmetic volume of Shimura varieties of orthogonal type Outline Algebraic geometry and Arakelov geometry Orthogonal Shimura varieties Main result Integral models and compactifications of Shimura varieties 1 Algebraic geometry and Arakelov geometry 2 Orthogonal Shimura varieties 3 Main result 4 Integral models and compactifications of Shimura varieties Fritz H¨ormann Department of Mathematics and Statistics McGill University The arithmetic volume of Shimura varieties of orthogonal type Outline Algebraic geometry and Arakelov geometry Orthogonal Shimura varieties Main result Integral models and compactifications of Shimura varieties Algebraic geometry X smooth projective algebraic variety of dim d =C. Classical intersection theory Chow groups: CHi (X ) := fZ j Z lin. comb. of irr. subvarieties of codim. ig=rat. equiv. Class map: i 2i cl : CH (X ) ! H (X ; Z) Intersection pairing: i j i+j CH (X ) × CH (X ) ! CH (X )Q compatible (via class map) with cup product. Degree map: d deg : CH (X ) ! Z Fritz H¨ormann Department of Mathematics and Statistics McGill University The arithmetic volume of Shimura varieties of orthogonal type Outline Algebraic geometry and Arakelov geometry Orthogonal Shimura varieties Main result Integral models and compactifications of Shimura varieties Algebraic geometry L very ample line bundle on X . 1 c1(L) := [div(s)] 2 CH (X ) for any (meromorphic) global section s of L. Definition of geometric volume d volL(X ) := deg(c1(L) ) 2 Z L defines an embedding N X ,! P (C) X \ H1 \···\ Hd consists of volL(X ) points. (Hi 's generic hyperplanes). Fritz H¨ormann Department of Mathematics and Statistics McGill University The arithmetic volume of Shimura varieties of orthogonal type Outline Algebraic geometry and Arakelov geometry Orthogonal Shimura varieties Main result Integral models and compactifications of Shimura varieties Algebraic geometry Explanation of terminology: If L is equipped with a Hermitian metric h, c c1(L; h) := d d log h(s) (which is independent of s) is a 2-form, with cl(c1(L)) = [c1(L; h)]. Hence Z d volL(X ) = c1(L; h) : X Define a `geometric degree' map: i CH (X ) ! Z Z d−i d−i Z 7! deg(Z · c1(L) ) = c1(L; h) Z Fritz H¨ormann Department of Mathematics and Statistics McGill University The arithmetic volume of Shimura varieties of orthogonal type Outline Algebraic geometry and Arakelov geometry Orthogonal Shimura varieties Main result Integral models and compactifications of Shimura varieties Arakelov geometry Let X be a model of X , i.e. a regular projective scheme, flat over spec(Z) with fixed isomorphism XC ' X . Arakelov intersection theory Arithmetic Chow groups: i Z lin. comb. of irreducible subschemes of codim. i rat. equiv. CHc (X) := (Z; g) g a Greens current for Z on X = C trivial Greens cur. Arithmetic intersection pairing: i j i+j CHc (X) × CHc (X) ! CHc (X)Q Arithmetic degree map: d+1 degd : CHc (X) ! R c ( g \Greens current" means that d d [g] + δZ = [!] holds true, for ! a smooth form.) C Fritz H¨ormann Department of Mathematics and Statistics McGill University The arithmetic volume of Shimura varieties of orthogonal type Outline Algebraic geometry and Arakelov geometry Orthogonal Shimura varieties Main result Integral models and compactifications of Shimura varieties Arakelov geometry L very ample line bundle on X and h a smooth Hermitian metric on LC. 1 c (L; h) := [div(s); log h(s )] 2 CH (X) b1 C c for any (meromorphic) global section s of L. Definition of arithmetic volume d+1 volc L;h(X) := deg(d bc1(L; h) ) 2 R Define linear `height' maps: i Z (XQ) ! R Z d+1−i d+1−i Z 7! deg((d Z; g) · bc1(L; h) ) − g c1(L; h) (any g) X d+1−i = deg(d bc1(LjZ ; hjZ ) ) (if Z is a reasonable model) For i = d, we recover the naive height (up to a bounded function)! Fritz H¨ormann Department of Mathematics and Statistics McGill University The arithmetic volume of Shimura varieties of orthogonal type Outline Algebraic geometry and Arakelov geometry Orthogonal Shimura varieties Main result Integral models and compactifications of Shimura varieties 1 Algebraic geometry and Arakelov geometry 2 Orthogonal Shimura varieties 3 Main result 4 Integral models and compactifications of Shimura varieties Fritz H¨ormann Department of Mathematics and Statistics McGill University The arithmetic volume of Shimura varieties of orthogonal type Outline Algebraic geometry and Arakelov geometry Orthogonal Shimura varieties Main result Integral models and compactifications of Shimura varieties Orthogonal Shimura varieties ... Z-Lattice L with orthogonal quadratic form Shimura variety /Q of signature (m − 2; 2) (dim m − 2) O ? auxiliary Z-lattice M special cycles with pos. def. quadratic form consisting of smaller of dimension n orthogonal Shimura varieties (dim m − 2 − n) Structure of the special cycles reflects representations of M by L, i.e. fisometries M ,! Lg. Fritz H¨ormann Department of Mathematics and Statistics McGill University The arithmetic volume of Shimura varieties of orthogonal type Outline Algebraic geometry and Arakelov geometry Orthogonal Shimura varieties Main result Integral models and compactifications of Shimura varieties ... and their geometric and arithmetic volume orthogonal Shimura varieties special cycles geometric Q −1 Q −1 volume ν λν (L) ν λν (L)µν (L; M; κ; s = 0) calculated in a lot of A relation to d Q arithmetic special cases d s ν µν (L; M; κ; s) s=0 volume conjectured (Kuhn,¨ Burgos, Bruinier, Kudla, ::: ) (Kudla, Rapoport, Yang) Q Here ν µν (L; M; κ; s) is a Fourier coefficient of a certain Eisenstein m series E(τ; s) of weight 2 and genus n. λp(L) is the volume of SO(LZp ) for almost all p. Fritz H¨ormann Department of Mathematics and Statistics McGill University The arithmetic volume of Shimura varieties of orthogonal type Outline Algebraic geometry and Arakelov geometry Orthogonal Shimura varieties Main result Integral models and compactifications of Shimura varieties Orthogonal Shimura varieties Let L be a Z-lattice with quadratic form QL of signature (m − 2; 2). Definition: Orthogonal symmetric space D(L) := f oriented maximal negative definite subspaces of LRg D(L) is a complex analytic space, homogenous under SO(LR). Fritz H¨ormann Department of Mathematics and Statistics McGill University The arithmetic volume of Shimura varieties of orthogonal type Outline Algebraic geometry and Arakelov geometry Orthogonal Shimura varieties Main result Integral models and compactifications of Shimura varieties The corresponding orthogonal Shimura datum There is an open (Borel) embedding: _ D(L) ,! Sh (L)(C) := f< v >2 P(LC) j QL(v) = 0g N 7! < z > determined by NC =< z; z > (taking the orientation of N into account). Associate with N the Hodge filtration ? 0 ⊆< z >⊆< z > ⊆ LC which corresponds to the Hodge structure of type (−1; 1); (0; 0); (1; −1) L =< z > ⊕N?⊕ < z > C C with associated morphism h : S ,! SO(LR). This identifies D(L) with a conjugacy class of these morphisms and O(L) := (SO(L); D(L)) is a Shimura datum. It is of Abelian type. Fritz H¨ormann Department of Mathematics and Statistics McGill University The arithmetic volume of Shimura varieties of orthogonal type Outline Algebraic geometry and Arakelov geometry Orthogonal Shimura varieties Main result Integral models and compactifications of Shimura varieties The orthogonal Shimura variety Definition: (Analytic) orthogonal Shimura variety Choose a compact open subgroup K ⊆ SO(L (1) ). A an K [ Sh ( O(L)) := [SO(L )n (L) × (SO(L (1) )=K)] = [Γ n (L)] ; Q D A i D i where Γi ⊂ SO(LQ) are certain arithmetic subgroups. Mostly K = SO0(L ) ⊆ SO(L ) (discriminant kernel). Zb Zb Lemma The reflex field is Q (if m = dim(L) ≥ 3). Fritz H¨ormann Department of Mathematics and Statistics McGill University The arithmetic volume of Shimura varieties of orthogonal type Outline Algebraic geometry and Arakelov geometry Orthogonal Shimura varieties Main result Integral models and compactifications of Shimura varieties Special cycles Let M be another Z-lattice, with positive definite quadratic form QM . For any ring R, denote: I(M; L)(R) := fα : MR ! LR j α is an isometry g: Definition: Special cycle Let κ ⊂ Hom(M (1) ; L (1) ) be compact open and K-stable. Then A A X 0 Zan(L; M; κ) := Shan(K O(α?)) (1) Kα⊆I(M;L)(A )\κ is a cycle on Shan(K O(L)). Mostly κ will be a coset in (L∗ =L ) ⊗ M∗ and K the discriminant kernel. Zb Zb Fritz H¨ormann Department of Mathematics and Statistics McGill University The arithmetic volume of Shimura varieties of orthogonal type Outline Algebraic geometry and Arakelov geometry Orthogonal Shimura varieties Main result Integral models and compactifications of Shimura varieties Examples dim(L) Witt index Sh(K O(L)) Z(L; M) (dim(M) = 1) 2 0 CM points ; 3 0 Shimura curves Heegner points 3 1 modular curves Heegner points 4 1 Hilbert modular surfaces Hirzebruch-Zagier cycles 5 2 Siegel threefolds Humbert surfaces . Fritz H¨ormann Department of Mathematics and Statistics McGill University The arithmetic volume of Shimura varieties of orthogonal type Outline Algebraic geometry and Arakelov geometry Orthogonal Shimura varieties Main result Integral models and compactifications of Shimura varieties The Weil representation and associated Eisenstein series Assume for simplicity that m = dim(L) is even. Let M be a lattice and ∗ ∗ s choose some (reference) bilinear form γ0 2 (M ⊗ M ) . The Weil representation ρ is a certain representation of Sp(M ) on Zb C[(L∗=L) ⊗ M∗], where M = M∗ ⊕ M with standard symplectic form.
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