Integrals of Borcherds Forms

By S. Kudla. Compositio Mathematica, 2001

131 citations recorded by the source corpus.

In his Inventiones papers in 1995 and 1998, Borcherds constructed holomorphic automorphic forms Ψ(f) with product expansions on bounded domains D associated to rational quadratic spaces V of signature (n2), starting from vector valued modular forms f of weight 1 − n2 for SL2(Z) which are allowed to have poles at the cusp and whose nonpositive Fourier coefficients are integers cμ(−m), m ≥ 0. In this paper, we use the Siegel–Weil formula to give an explicit formula for the integral κ(Ψ(f)) of − log||Ψ(f)||2 over X = Γ\D, where || ||2 is the Petersson norm. This integral is given by a sum for m0 of quantities cμ(−m)κμ(m), where κμ(m) is the limit as Im(τ) → ∞ of the mth Fourier coefficient of the second term in the Laurent expansion at s = n2 of a certain Eisenstein series E(τs) of weight (n2) + 1 attached to V. The possible role played by the quantity κ(Ψ(f)) in the Arakelov theory of the divisors Zμ(m) on X is explained in the last section.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1fd1b2df8f4b48c47371340e65c09951b8c9c53e

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Citations in the corpus

CM-values of Hilbert modular functions (2006, methodology): “The period integral of the Borcherds product over the 2-cycle (i.e., the volume form) was studied by Kudla [ Ku2 ] and Bruinier, Burgos, and Kuhn [BBK,BK] independently.”
L-functions and theta correspondence for classical groups (2014, background): “This is a kind of the matching phenomenon discussed by Kudla in [24].”
On the converse theorem for Borcherds products (2012, methodology): “attice L, then we brieﬂy write ρL for ρL′/L. Note that ρL can be identiﬁed with a sub-representation of the usual Weil representation of Mp2(Z) on the space of Schwartz-Bruhat functions on L ⊗Qˆ, see [Ku2]. 2.2. Vector valued modular forms. Let k ∈ 1 2 Z. A holomorphic function f : H→ C[A] is called”
Faltings heights of CM cycles and derivatives of L-functions (2008, methodology): “We use a similar setup as in [26].”
Borcherds products and arithmetic intersection theory on Hilbert modular surfaces (2003, background): “(z1,z2,s) are constructed as regularized theta lifts of non-holomorphic Hejhal-Poincar´e series of weight 0. Here, for brevity, we have preferred to argue as above. Observe that (v) also follows from [Ku5]. Deﬁnition 4.4. Hilbert modular forms that arise as lifts via Theorem 4.3 are called Borcherds”