Arithmetic Special Cycles and Derivatives of L-functions

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Abstract Stephen Kudla University of Maryland 9970506 This project involves the construction of modular generating series associated to algebraic cycles on certain Shimura varieties. The fact that such generating series should be the q-expansions of modular forms reflects some sort of hidden symmetry in the distribution of algebraic subvarieties of these varieties. In addition, in certain cases, there is a more delicate type of generating series whose coefficients lie in the arithmetic Chow groups of an integral model of a Shimura variety. Such series appear to be related to the derivatives of Siegel-Eisenstein series and hence, in turn, to the central derivatives of automorphic L-functions. Thus, it is hoped that one may ultimately obtain information about higher dimensional analogues of the Gross-Zagier formula and the Birch-Swinnerton Dyer conjecture. The powerful techniques which are now emerging in modern number theoretic geometry have made it possible to manage previously intractable problems centered about solving equations with integers. Strong connections have been found between the geometric properties of graphs associated with an equation and the integer solutions to that equation. For example, the celebrated Mordell conjecture, proved by Faltings in 1983, asserts an equation whose graph is a surface with at least two holes can have only a finite number of integer solutions. The present project involves more complicated versions of this connection in which the equations and their graphs require extra dimensions and the `integer solutions' may actually be other kinds of geometric objects (curves, surfaces, etc.)