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

By Jan H. Bruinier

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

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**Example text**

6) of L . Obviously L is contained in L0 . Beside the orthogonal projection from L0 to K there is another natural projection L0 → K : If we combine the linear map L0 → L ∩ z ⊥ , λ → λ − (λ,z) N ζ with the orthogonal projection from L ∩ z ⊥ to K , we get a projection p : L0 −→ K , p(λ) = λK − (λ, z) ζK . 7) This map has the property that p(L) = K. In fact, if λ = k + aζ + bz ∈ L with k ∈ K and a, b ∈ Z, then it is easily seen that p(λ) = k. Thus p induces a surjective map L0 /L → K /K which will also be denoted by p.

Note that k − 1 = 1 − κ. For γ ∈ L /L and n ∈ Z − q(γ) with n > 0 let aγ,n : Sκ,L → C denote the ∗ of Sκ,L which maps a cusp form f to its functional in the dual space Sκ,L (γ, n)-th Fourier coeﬃcient aγ,n (f ). 5, aγ,n can be described by means of the Petersson scalar product as aγ,n = (4πn)κ−1 ( · , Pγ,n ). 41) The following theorem can also be proved using Serre duality (see [Bo3] Thm. 1). 16. 3 Non-holomorphic Poincar´e series of negative weight 37 ! 17. There exists a nearly holomorphic modular form f ∈ Mk,L with prescribed principal part c(β, m)eβ (mτ ) β∈L /L m∈Z+q(β) m<0 (c(β, m) ∈ C with c(β, m) = c(−β, m)), if and only if the functional c(β, m)aβ,−m , β∈L /L m∈Z+q(β) m<0 ∗ equals 0 in Sκ,L .

Then as a set of representatives for λ ∈ L0 /L with p(λ) = γ +K we may take γ −(γ, ζ)z/N +bz/N where b runs modulo N . A set of representatives for L0 /L is given by λ = γ − (γ, ζ)z/N + bz/N , where γ runs through a set of representatives for K /K and b runs modulo N . 6. We use the same notation as above. Let (M, φ) ∈ Mp2 (Z), M = ac db . Then FK (τ ; r, t) satisﬁes FK (M τ ; ar + bt, cr + dt) = φ(τ )2k Proof. 3. K (M, φ)FK (τ ; r, t). 12) by Fβ,m (τ, s; r, t). 7. Let β ∈ L /L and m ∈ Z + q(β) with m < 0.