Eric Hofmann. AKLS Seminar Aachen, March 3, TU Darmstadt. Borcherds Products on Unitary Groups. Eric Hofmann. Setup.

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1 TU Darmstadt AKLS Seminar Aachen, March 3, 2010

2 Outline 1 Hermitian lattices and unitary on U(L) 2 Sketch of proof: Pullback from O(V ) 3

3 Let F be an imaginary quadratic number field, F = Q( d), d < 0, with discriminant D F, different δ and ring of integers O F. Let L be a hermitian lattice of signature (1, q) over O F, with, a nondegenerate hermitian form av, bw = a b v, w = bw, av, extended to the F-vectorspace V = L F and to the complex space V R = V F R. Assume L to be integral and even, that is tr F/Q λ, µ Z and tr F/Q λ, λ 2Z for all λ, µ L. Denote by L the Z-dual of L, L = { v V ; tr F/Q λ, v Z, for all λ L }.

4 Denote by U(V ) the unitary group of V,, and by U(V )(R) its real points. The isometries of L, form an arithmetic subgroup, U(L). Symmetric attached to U(V ) can be obtained as the Grassmannian Gr U = U(V )(R)/C, with C U(V )(R), C a maximal compact subgroup isomorphic to U(1) U(q) under V R C 1,q. Gr U is isomorphic to the projective cone K U = {[z] P(V R ) ; z, z > 0}. We will give an affine model for this later on.

5 Modified Witt basis We assume that L splits off a hyperbolic plane over O F, so that L = H D, with H, D = 0, H of signature (1, 1) and D negative definite. Choose lattice vectors l, l with l L primitive isotropic and l L also isotropic, such that l, l 0. Then, with a,b (fractional) O F -ideals. L = ( al + bl ) D, We call an F-basis of V of the form l, l, v 1,..., v q 1 L with l, l as above a modified Witt basis.

6 A Siegel model Now, for the elements of K U, consider representatives z V of the form z = l τ l, l δl + σ, with σ D F R. The condition z, z > 0 yields the following description. Siegel The following subset of C q is an affine model for Gr U : H U = { (τ, σ) C (D F R) ; l, l 2 Iτ > σ, σ }. Note that H U is not a tube, unless q = 1, in which case H U H, the upper half plane in C.

7 A Siegel model Now, for the elements of K U, consider representatives z V of the form z = l τ l, l δl + σ, with σ D F R. The condition z, z > 0 yields the following description. Siegel The following subset of C q is an affine model for Gr U : H U = { (τ, σ) C (D F R) ; l, l 2 Iτ > σ, σ }. Note that H U is not a tube, unless q = 1, in which case H U H, the upper half plane in C.

8 Let Γ U(L) be an arithmetic subgroup. Then, Γ acts on H U with an automorphy factor j : Γ H U C. Definition An automorphic form of weight k on Γ is a meromorphic function f : H U C satisfying f (γ(τ, σ)) = j(γ; τ, σ) k f (τ, σ) for all γ Γ. If in addition f is holomorphic and regular at the cusps, f is called a modular form.

9 Fourier-Jacobi expansion A holomorphic automorphic form can be partially expanded as a Fourier series: f (τ, σ) = n Z a n (σ)e ( nτ N + b), with N Z, b Q. The coefficients a n (σ) transform as Jacobi-. a 0 is constant, we have lim τ i f (τ, σ) = a 0. For q > 1 the Koecher principle holds: If f is holomorphic, a n = 0 for n < 0, so f is regular at the cusps. If q = 1, f is an elliptic modular form and all the a n are constant.

10 The Heisenberg Group The arithmetic Heisenberg group in in U(L) consists of pairs [h, t], with h 1 N Z for some integer N, t from a sublattice of D. The group law is given by [h, t] [h, t ] = [h + h + I t, t, t + t ]. δ On H U, the Heisenberg group acts according to [0, t](τ, σ) = [h, 0](τ, σ) = (τ + h, σ) ( τ + σ, t δ l, l t, t, σ + l, l t δ ).

11 V as a quadratic space The hermitian space V also carries a structure as a quadratic space over Q: The following is a non-degenerate symmetric form, bilinear of signature (2, 2q) on V as a Q-vectorspace (, ) = tr F/Q, = 2R,. Attached to this is the quadratic form q ( ) =,. By extension of scalars, (, ) is a real bilinear form on V R as a vector space over R.

12 Weil- The double cover of SL 2 (R), the metaplectic group Mp 2 (R) consists of elements ( ) ( M, φ(τ), M = a b ) c d, φ(τ) 2 = cτ + d. The subgroup Mp 2 (Z) has two generators, T = (( ), 1), S = (( ), τ ). It has a ρ L on the group algebra C[L/L ]. ρ L (T )e γ = e ( q (γ) ) q 1 i e γ, ρ L (S)e γ = L /L e ( (γ, δ) ) e δ. δ L /L

13 Vector-values modular There is also a dual of Mp 2 (Z) on the vector-valued functions H C[L /L], ( f κ (M, φ) ) (τ) = φ(τ) 2κ ρ 1 L (M, φ) f (Mτ). Definition The space M! κ(ρ L ) of nearly holomorphic modular consists of functions f : H C[L /L] satisfying f κ (M, φ) = f, for all (M, φ) Mp 2 (Z). f is holomorphic on H. f is meromorphic at the cusp i. Such a modular form has a Fourier expansion f (τ) = c(γ, n)e(nτ)e γ. γ L /L n Z q(γ) n

14 products for Theorem Given f M! q 1 (ρ L) with Fourier coefficients satisfying c(γ, n) Z for n < 1, there is a meromorphic function Ξ f : H U C with the following properties: Ξ f is an automorphic form of weight c(0, 0)/2. Its poles and zeros lie on generalized Heegner divisors. On each Weyl chamber W of H U, Ξ f has an absolutely converging infinite product expansion around the cusps, Ξ W f (z) = Ce ρ λ K λ L + (W ) ( 1 e ( )) z, λ c(λ, λ,λ ) l., l

15 The product converges absolutely when z, z 0. Heegner dvisors in K U respectively H U are given by unions of primitive Heegner divisors, sets of the form H λ = {[z] K U ; z, λ = 0}, for a λ L, λ, λ < 0. For c(γ, n) with n < 0, the Heegner divisors H γ dissect K U respectively H U into components called Weyl chambers. Considering L as a lattice over Z, K is a Z-sublattice of rank 2q of the form K = (Za + Zb) D, with a, b H. The positivity condition λ L + (W ) means that [ I z, λ l, l ] 1 > 0 for all z W K U. C is a constant, while ρ = 2πi z, ρ f W l, l 1, where ρ f W is the Weyl vector attached to f and W.

16 Sketch of the proof Consider V R with the form (, ) as a real quadratic space. The orthogonal group O(V )(R) is isomorphic to O(2, 2q). Since (, ) = 2R,, we have an embedding of U(V )(R) O(V )(R), This induces an embedding of the attached symmetric s α : H U H O, permitting us to pull back the automorphic products constructed by to the unitary side, Ξ f (z(τ, σ)) = α (Ψ(f ; Z))(τ, σ).

17 Some remarks on the embedding: Since End C (V R ) End R (V R ), a complex scalar defines an endomorphism of the real space V R, (, ). U(L) is embedded into the orthogonal group O(L) of L (as a Z-lattice). The discriminant kernel is mapped to the discriminant kernel. Sub of finite index remain sub of finite index. (Similarly for commensurable sub.) The Heisenberg group in U(V ) is mapped into the Heisenberg group of O(V ).

18 A symmetric for O(V ) for O(V ) can be obtained as a Grassmannian Gr O = O(V )(R)/K, where K O(2) O(2q) is a maximal compact subgroup. The points of Gr O are 2-dimensional positive definite subspaces of V R. Choose a continuously varying orientation on these and for each v Gr O an orthogonal basis X L, Y L, with X 2 L = Y 2 L, X L Y L. With this, we introduce a complex structure on Gr O.

19 A positive cone in V C. The real quadratic space V R is complexified to V C = V R R C and (, ) is extended to a complex bilinear form. Now, consider the maps X L, Y L Z L = X L + iy L V C, and Gr O v [Z L ] P(V C ). This sends Gr O to either of the two components of the set K O = {[Z L ] P(V C ) ; (Z L, Z L ) = 0, ( Z L, Z L ) > 0}. Choose one component, K + O. It is isomorphic to Gr O.

20 Construction of the tube For the construction of the tube, we require a primitive vector e 1 L and an e 2 L satisfying e 2 1 = 0, (e 1, e 2 ) = 1. Normalize X L and Y L as follows (X L, e 1 ) = 1, (Y L, e 1 ) = 0. Set K = L e1 e 2 and denote by X and Y the projections of X L and Y L to the subspace K R. The tube model H O is defined as the image of K + O under [Z L ] Z K C: H O { Z = X + iy K C ; Y 2 > 0 }.

21 Matching basis We already have a C-basis l,l consisting of lattice vectors for the hyperbolic space H F R,,. To fix the embedding H U H O induced by α, we need to choose an R-basis of H Q R, (, ), as well. Since l corresponds to the cusp it should be part of this new basis. Set e 1 = l, and choose lattice vectors e 2, e 3, e 4 H with the following properties (e 1, e 2 ) = (e 3, e 4 ) = 1 and (e i, e j ) = 0 otherwise. Clearly, e 1,..., e 4 span H R as a real quadratic space.

22 Matching basis We already have a C-basis l,l consisting of lattice vectors for the hyperbolic space H F R,,. To fix the embedding H U H O induced by α, we need to choose an R-basis of H Q R, (, ), as well. Since l corresponds to the cusp it should be part of this new basis. Set e 1 = l, and choose lattice vectors e 2, e 3, e 4 H with the following properties (e 1, e 2 ) = (e 3, e 4 ) = 1 and (e i, e j ) = 0 otherwise. Clearly, e 1,..., e 4 span H R as a real quadratic space.

23 The image of [z] K U in Gr O is given by [z] RX L + RY L, X L = z 2 l, l, Y L = iz 2 l, l. It is quickly checked that the following statements hold: 1 The two complex structures are compatible. For example, we have iz Y L + ix L = iz L. 2 We have (X L, l) = 1, (Y L, l) = 0. 3 Further X L Y L, and X 2 L = Y 2 L > 0.

24 Example: Choice of basis Assume D F to be even. Consider a lattice L of the form L = O F δ 1 O F. Clearly L = L. Fix an isometry L C to C 1,1. Set l = (1, 0), l = (0, δ 1 ). Then, l, l = δ 1 and L = O F l O F l. For L as the Z-lattice (Z + δ/2z)l (Z + δ/2z)l, a basis of the required form is given by l, e 2 = δ 2 l, e 3 = δ 2 l, e 4 = l. With this, the representative z = l τl maps to v = RX L + RY L, where X L = δ 2 z, Y L = 2 δ z. Finally, τ H U maps to Z = δ/2e 4 + τe 3 in H O.

25 Example: Liftings from M! 0 (ρ L) Assume D F 1(2), L = O F δ 1 O F. Denote ω = 1 2 (1 + δ). Consider the function J(τ) M! 0 (ρ L) given by J(τ) = j(τ) 744 = q q +..., where j(τ) is the usual j-function. We find that there are two Weyl chambers, W > and W <, defined by Iτ > δ and Iτ < δ. On the W >, the Weyl vector is found to be ρ J W > = e 4 and the constant C equals 1. Thus, the infinite product expansion of the lift Ξ J takes the form Ξ W> J (τ) = e( τ) ( ) c(mn). 1 e (mτ n ω) m,n Z m>0