Derivatives of Eisenstein series and Faltings heights

In a series of papers, [25], [30], [28], [29], [31], [26], we showed that certain quantities from the arithmetic geometry of Shimura varieties associated to orthogonal groups occur in the Fourier coefficients of the derivative of suitable Siegel-Eisenstein series. It was essential in these examples that this derivative was the second term in the Laurent expansion of a Siegel-Eisenstein series at the center of symmetry, and that the first term in this Laurent expansion vanished (incoherent case). In the present paper we prove a relation between a generating function for the heights of Heegner cycles on the arithmetic surface associated to a Shimura curve and the second term in the Laurent expansion at s = 12 of an Eisenstein series of weight 32 for SL2. It is remarkable that s = 1 2 is not the center of symmetry and that the first term of the Laurent expansion is non-zero. In fact, this nonzero value has a geometric interpretation in terms of the Shimura curve over the field of complex numbers. Considering the fact that the Eisenstein series is a rather familiar classical object, it is surprising that this interpretation of its Laurent expansion at s = 12 has not been noticed before. As we will argue below in this introduction, we believe that our result is part of a general pattern involving the heights of divisors on arithmetic models of Shimura varieties associated to orthogonal groups.