Download e-book for iPad: Borcherds Products on O(2,l) and Chern Classes of Heegner by Jan H. Bruinier

By Jan H. Bruinier

ISBN-10: 3540433201

ISBN-13: 9783540433200

Round 1994 R. Borcherds found a brand new kind of meromorphic modular shape at the orthogonal team $O(2,n)$. those "Borcherds items" have limitless product expansions analogous to the Dedekind eta-function. They come up as multiplicative liftings of elliptic modular types on $(SL)_2(R)$. the truth that the zeros and poles of Borcherds items are explicitly given by way of Heegner divisors makes them fascinating for geometric and mathematics purposes. within the current textual content the Borcherds' development is prolonged to Maass wave types and is used to review the Chern sessions of Heegner divisors. A communicate theorem for the lifting is proved.

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Extra resources for Borcherds Products on O(2,l) and Chern Classes of Heegner Divisors

Sample text

Here condition (iii) means that f has a Fourier expansion of the form c(γ, n)eγ (nτ ). f (τ ) = γ∈L /L n∈Z−q(γ) n≥0 Moreover, if all c(γ, n) with n = 0 vanish, then f is called a cusp form. The C-vector space of modular forms of weight κ with respect to ∗L and Mp2 (Z) is denoted by Mκ,L , the subspace of cusp forms by Sκ,L . It is easily seen that Mκ,L is finite dimensional. 3. Fix a positive integer t. Let L be the 1-dimensional lattice Z, equipped with the positive definite quadratic form q(a) = ta2 .

We find that θL+γ (τ, v) is given by 1 2yzv2 e (τ q((λK + cz )w ) + τ¯q((λK + cz )w⊥ )) c,d∈Z λK ∈K+p(γ−cz ) c≡(γ,z) (N ) ×e − |(λK + cz , z)τ + d|2 d(λK + cz + f (c, γ)z, zv − zv⊥ ) − 4iyzv2 2zv2 , where we have used zw = zw⊥ = 0 and q(z) = 0. 1) of ΘK (τ, w; dµ, −cµ), we see that we have to prove − d(λK + cz + f (c, γ)z, zv − zv⊥ ) ≡ −(λK − cµ/2, dµ) − d(γ, z ) + cdq(z ) 2zv2 modulo 1. Since (z, zv − zv⊥ )/2zv2 = 1, we find −d (f (c, γ)z, zv − zv⊥ ) = −df (c, γ) 2zv2 (γ, z) − c dB = −dbγ + N 2 = −d(γ, z ) + d(γ, z)z + From (ζ, z ) ∈ Z it follows that B ≡ −N z −d 2 (γ, z) − c dB.

We write L = {x ∈ L ⊗ Q; (x, y) ∈ Z for all y ∈ L} for the dual lattice. Then the quotient L /L is a finite Abelian group, the socalled discriminant group. The modulo 1 reduction of q(x) is a Q/Z-valued quadratic form on L /L whose associated bilinear form is the modulo 1 reduction of the bilinear form (·, ·) on L . Recall that there is a unitary representation L of Mp2 (Z) on the group algebra C[L /L]. If we denote the standard basis of C[L /L] by (eγ )γ∈L /L , then L can be defined by the action of the generators S, T ∈ Mp2 (Z) as follows (cp.

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Borcherds Products on O(2,l) and Chern Classes of Heegner Divisors by Jan H. Bruinier


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