2001/10/21 by Stephen S. Kudla, Kudla, Stephen S. · 5 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT
paper · pdf · doi:10.48550/arxiv.math/0110236
arxiv created 2001/10/21 · openalex publication_date 2001/10/21 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (n,2). The input F for his construction is a vector valued modular form of weight 1-n/2 for SL2(Z) which is allowed to have a pole at the cusp and whose non-positive Fourier coefficients are integers cμ(-m), m≥0. For example, the divisor of Ψ(F) is the sum over m>0 and the coset parameter μ of cμ(-m) Zμ(m) for certain rational quadratic divisors Zμ(m) on the arithmetic quotient X = ΓD. In this paper, we give an explicit formula for the integral κ(Ψ(F)) of -log||Ψ(F)||2 over X, where ||.||2 is the Petersson norm. More precisely, this integral is given by a sum over μ and m>0 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= n/2 of a certain Eisenstein series E(τ,s) of weight n/2 + 1 attached to V. It is also shown, via the Siegel--Weil formula, that the value E(τ, n/2) of the Eisenstein series at this point is the generating function of the volumes of the divisors Zμ(m) with respect to a suitable Kähler form. 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.