By Jan H. Bruinier

Round 1994 R. Borcherds came across a brand new form 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 varieties 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 functions. within the current textual content the Borcherds' building is prolonged to Maass wave varieties and is used to review the Chern periods of Heegner divisors. A communicate theorem for the lifting is proved.

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31) Using the standard integral representation ∞ −z/2 µ+1/2 Γ (1/2 − ν + µ)Wν, µ (z) = e e−tz t−1/2−ν+µ (1 + t)−1/2+ν+µ dt z 0 ( (µ − ν) > −1/2, (z) > 0) of the W -Whittaker function ([E1] Vol. I p. 274 (18)), we ﬁnd for y < 0: W1−k/2 (y) = |y|−k/2 W−k/2, 1/2−k/2 (|y|) ∞ −|y|/2 =e 1−k |y| e−t|y| (1 + t)−k dt 0 ∞ = e|y|/2 |y|1−k e−t|y| t−k dt. 1 If we insert the deﬁnition of the incomplete Gamma function (cf. [AbSt] p. 32) x we obtain for y < 0 the identity W1−k/2 (y) = e−y/2 Γ (1 − k, |y|). The usual Laplace operator of weight k (cf.

2) γ∈L /L If r and t are both 0 we will omit them and simply write θγ (τ, v) resp. ΘL (τ, v). Let Λ be the lattice Z2 equipped with the standard symplectic form and W = Λ ⊗ R (such that Sp(Λ) = SL2 (Z)). Then W ⊗ V is a symplectic vector space. The usual theta function attached to the lattice Λ ⊗ L ⊂ W ⊗ V restricted to the symmetric subspace H × Gr(L) ⊂ Sp(W ⊗ V )/K (where K denotes a maximal compact subgroup) equals the Siegel theta function θL (τ, v). H. Bruinier: LNM 1780, pp. 39–61, 2002. c Springer-Verlag Berlin Heidelberg 2002 40 2 The regularized theta lift It is well known that ΘL (τ, v; r, t) is a real analytic function in (τ, v) ∈ H × Gr(L).

Then we have z/N ∈ L . Denote by K the lattice ⊥ Over Q we have K = L ∩ z⊥ ∩ z . 3) L ⊗ Q = (K ⊗ Q) ⊕ (Qz ⊕ Qz ). 1 Siegel theta functions 41 Hence K has signature (b+ − 1, b− − 1). If n ∈ V = L ⊗ R then we write nK for the orthogonal projection of n to K ⊗ R. It can be easily checked that nK can be computed as nK = n − (n, z)z + (n, z)(z , z )z − (n, z )z. If n ∈ L then nK lies in the dual lattice K of K. (Warning: Notice that K is not necessarily contained in L . ) Let ζ ∈ L be a lattice vector with (ζ, z) = N .