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This paper is about the arithmetic of Kudla-Rapoport divisors on Shimura varieties of type GU(n - 1,1). In the first part of the paper we construct a toroidal compactification of N. Krämer's integral model of the Shimura variety. This extends work of K.-W. Lan, who constructed a compactification at unramified primes. In the second, and main, part of the paper we use ideas of Kudla to construct Green functions for the Kudla-Rapoport divisors on the open Shimura variety, and analyze the behavior of these functions near the boundary of the compactification. The Green functions turn out to have logarithmic singularities along certain components of the boundary, up to log-log error terms. Thus, by adding a prescribed linear combination of boundary components to a Kudla-Rapoport divisor one obtains a class in the arithmetic Chow group of Burgos-Kramer-Kühn. In the third and final part of the paper we compute the arithmetic intersection of each of these divisors with a cycle of complex multiplication points. The computation is quickly reduced to the calculations of the author's earlier work Complex multiplication cycles and Kudla-Rapoport divisors. The arithmetic intersection multiplicities are shown to appear as Fourier coefficients of the diagonal restriction of the central derivative of a Hilbert modular Eisenstein series. [ABSTRACT FROM AUTHOR]

If E is an elliptic curve defined over a number field and p is a prime of good ordinary reduction for E, a theorem of Rubin relates the p-adic height pairing on the p-power Selmer group of E to the first derivative of a cohomologically defined p-adic L-function attached to E. Bertolini and Darmon have defined a sequence of "derived" p-adic heights. In this paper we give an alternative definition of the p-adic height pairing and prove a generalization of Rubin's result, relating the derived heights to higher derivatives of p-adic L-functions. We also relate degeneracies in the derived heights to the failure of the Selmer group of F over a Zp-extension to be "semi-simple" as an Iwasawa module, generalizing results of Perrin-Riou. [ABSTRACT FROM AUTHOR]