Derivatives of Eisenstein series and Faltings heights

By S. Kudla, Michael Rapoport, Tonghai Yang. Compositio Mathematica, 2001

90 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4b74b978b569aee8caa319f71feb68f8854e5904

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Citations in the corpus

Heegner Divisors and Nonholomorphic Modular Forms (2002, result): “However, recent results of Kudla, Rapoport and Yang interpret these coefficients as the (geometric) degree of certain 0-cycles supported in the cusps, see [16].”
Traces of CM values of modular functions (2004, background): “For D ∈ Z and v > 0, Kudla, Rapoport and Yang [20, 23] construct cycles Ẑ(D, v) in the extended arithmetic Chow group of M with real coefficients ĈH1R(M), see [3, 14, 6, 22].”
Integrals of Borcherds Forms (2001, methodology): “The group G0A acts in SðVðAÞÞ via the Weil representation o (determined by the standard additive character c of A”
Modular Forms and Special Cycles on Shimura Curves. (2006, methodology): “The normalized Eisenstein series of genus one satisfies a functional equation [9], Section 15, (5.”