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29 March 2024
 
  » arxiv » math.NT/0110236

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Integrals of Borcherds forms
Stephen S. Kudla ;
Date 21 Oct 2001
Subject Number Theory; Algebraic Geometry | math.NT math.AG
AbstractIn his Inventiones papers in 1995 and 1998, Borcherds constructed holomorphic automorphic forms $Psi(F)$ 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 $SL_2(Z)$ which is allowed to have a pole at the cusp and whose non-positive Fourier coefficients are integers $c_mu(-m)$, $mge0$. For example, the divisor of $Psi(F)$ is the sum over $m>0$ and the coset parameter $mu$ of $c_mu(-m) Z_mu(m)$ for certain rational quadratic divisors $Z_mu(m)$ on the arithmetic quotient $X = Gamma D$. In this paper, we give an explicit formula for the integral $kappa(Psi(F))$ of $-log||Psi(F)||^2$ over $X$, where $||.||^2$ is the Petersson norm. More precisely, this integral is given by a sum over $mu$ and $m>0$ of quantities $c_mu(-m) kappa_mu(m)$, where $kappa_mu(m)$ is the limit as $Im( au) -> infty$ of the $m$th Fourier coefficient of the second term in the Laurent expansion at $s= n/2$ of a certain Eisenstein series $E( au,s)$ of weight $n/2 + 1$ attached to $V$. It is also shown, via the Siegel--Weil formula, that the value $E( au, n/2)$ of the Eisenstein series at this point is the generating function of the volumes of the divisors $Z_mu(m)$ with respect to a suitable Kähler form. The possible role played by the quantity $kappa(Psi(F))$ in the Arakelov theory of the divisors $Z_mu(m)$ on $X$ is explained in the last section.
Source arXiv, math.NT/0110236
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