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We prove a relation between a generating series for the heights of Heegner cycles on the arithmetic surface associated with a Shimura curve and the second term in the Laurent expansion at s = ½ of an Eisenstein series of weight $\frac32$ for SL(2). On the geometric side, a typical coefficient of the generating series involves the Faltings heights of abelian surfaces isogenous to a product of CM elliptic curves, an archimedean contribution, and contributions from vertical components in the fibers of bad reduction. On the analytic side, these terms arise via the derivatives of local Whittaker functions. It should be noted that s = ½ is not the central point for the functional equation of the Eisenstein series in question. Moreover, the first term of the Laurent expansion at s = ½ coincides with the generating function for the degrees of the Heegner cycles on the generic fiber and, in particular, does not vanish.

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 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.