Derivatives of Eisenstein series and Faltings heights

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.

[1]  Helmut Hasse,et al.  Number Theory , 2020, An Introduction to Probabilistic Number Theory.

[2]  S. Kudla.  Modular forms and arithmetic geometry , 2003, math/0308296.

[3]  Jens Funke.  Heegner Divisors and Nonholomorphic Modular Forms , 2002, Compositio Mathematica.

[4]  S. Kudla.  Integrals of Borcherds Forms , 2001, Compositio Mathematica.

[5]  S. Kudla,et al.  Cycles on Siegel threefolds and derivatives of Eisenstein series , 2000 .

[6]  R. Hain.  Moduli of Riemann Surfaces, Transcendental Aspects , 2000, math/0003144.

[7]  J. Bost.  Potential theory and Lefschetz theorems for arithmetic surfaces , 1999 .

[8]  S. Kudla,et al.  Arithmetic Hirzebruch Zagier cycles , 1999, math/9904083.

[9]  Tonghai Yang,et al.  An Explicit Formula for Local Densities of Quadratic Forms , 1998 .

[10]  S. Kudla,et al.  Height pairings on Shimura curves and p-adic uniformization , 1998, math/9809149.

[11]  U. Kuehn.  Generalized arithmetic intersection numbers , 1998 .

[12]  S. Kudla.  Central Derivatives of Eisenstein Series and Height Pairings , 1997 .

[13]  S. Kudla.  Splitting metaplectic covers of dual reductive pairs , 1994 .

[14]  Jean-Benoît Bost,et al.  Heights of projective varieties and positive Green forms , 1994 .

[15]  P. Colmez.  Périodes des variétés abéliennes à multiplication complexe , 1993 .

[16]  R. Ranga Rao,et al.  On some explicit formulas in the theory of Weil representation , 1993 .

[17]  R. Kottwitz.  Points on some Shimura varieties over finite fields , 1992 .

[18]  C. Soulé,et al.  Arithmetic intersection theory , 1990 .

[19]  Angelo Vistoli.  Intersection theory on algebraic stacks and on their moduli spaces , 1989 .

[20]  Benedict H. Gross,et al.  On canonical and quasi-canonical liftings , 1986 .

[21]  G. Faltings.  Endlichkeitssätze für abelsche Varietäten über Zahlkörpern , 1983 .

[22]  G. Shimura,et al.  Confluent hypergeometric functions on tube domains , 1982 .

[23]  Helmut Hasse.  Number Theory , 1979 .

[24]  F. Hirzebruch,et al.  Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus , 1976 .

[25]  V. Drinfel'd,et al.  Coverings of p-adic symmetric regions , 1976 .

[26]  Ihrer Grenzgebiete,et al.  Ergebnisse der Mathematik und ihrer Grenzgebiete , 1975, Sums of Independent Random Variables.

[27]  Henri Cohen,et al.  Sums involving the values at negative integers of L-functions of quadratic characters , 1975 .

[28]  R. A. Silverman,et al.  Special functions and their applications , 1966 .

[29]  A. Weil.  Sur certains groupes d'opérateurs unitaires , 1964 .

[30]  H. Rauch,et al.  ON THE MODULI OF RIEMANN SURFACES. , 1955, Proceedings of the National Academy of Sciences of the United States of America.

[31]  Tonghai Yang.  THE SECOND TERM OF AN EISENSTEIN SERIES , 2002 .

[32]  S. Kudla.  Derivatives of Eisenstein series and generating functions for arithmetic cycles , 2000 .

[33]  Tonghai Yang,et al.  On the Derivative of an Eisenstein Series of Weight One , 1999 .

[34]  S. Kudla.  Algebraic cycles on Shimura varieties of orthogonal type , 1997 .

[35]  Y. Nakkajima,et al.  A generalization of the Chowla-Selberg formula. , 1991 .

[36]  J. Bost.  Théorie de l'intersection et théorème de Riemann-Roch arithmétiques , 1991 .

[37]  H. Carayol,et al.  Uniformisation p-adique des courbes de Shimura : les théorèmes de Čerednik et de Drinfeld , 1991 .

[38]  G. Faltings,et al.  Degeneration of Abelian varieties , 1990 .

[39]  S. Gelbart.  Weil's Representation and the Spectrum of the Metaplectic Group , 1976 .

[40]  Don Zagier.  Nombres de classes et formes modulaires de poids 3/2. , 1975 .

[41]  P. Deligne,et al.  Les Schémas de Modules de Courbes Elliptiques , 1973 .

[42]  D. Mumford,et al.  The irreducibility of the space of curves of given genus , 1969 .

[43]  S. Kudla.  Institute for Mathematical Physics Special Cycles and Derivatives of Eisenstein Series , 2022 .