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Derivatives of Eisenstein series and Faltings heights

Published online by Cambridge University Press:  04 December 2007

Stephen S. Kudla
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, [email protected]
Michael Rapoport
Affiliation:
Mathematisches Institut der Universität Bonn, Beringstr. 1, 53115 Bonn, [email protected]
Tonghai Yang
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, WI 53706, [email protected]
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Abstract

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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.

Type
Research Article
Copyright
Foundation Compositio Mathematica 2004