Integrals of Borcherds Forms

Abstract

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($\open Z$) which are allowed to have poles at the cusp and whose nonpositive Fourier coefficients are integers c_μ(−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 c_μ(−m)κ_μ(m), where κ_μ(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 Z_μ(m) on X is explained in the last section.