After the work of Kisin, there is a good theory of canonical integral models of Shimura varieties of Hodge type at primes of good reduction. The first part of this paper develops a theory of Hodge type Rapoport-Zink formal schemes, which uniformize certain formal completions of such integral models. In the second part, the general theory is applied to the special case of Shimura varieties associated with groups of spinor similitudes, and the reduced scheme underlying the Rapoport-Zink space is determined explicitly.
Let M be the Shimura variety associated with the group of spinor similitudes of a quadratic space over Q of signature (n, 2). We prove a conjecture of Bruinier-Kudla-Yang, relating the arithmetic intersection multiplicities of special divisors and big CM points on M to the central derivatives of certain L-functions.As an application of this result, we prove an averaged version of Colmez's conjecture on the Faltings heights of CM abelian varieties.
Abstract. Given a weight two modular form f with associated p-adic Galois representation V f , for certain quadratic imaginary fields K one can construct canonical classes in the Galois cohomology of V f by taking the Kummer images of Heegner points on the modular abelian variety attached to f . We show that these classes can be interpolated as f varies in a Hida family and construct an Euler system of big Heegner points for Hida's universal ordinary deformation of V f . We show that the specialization of this big Euler system to any form in the Hida family is nontrivial, extending results of Cornut and Vatsal from modular forms of weight two and trivial character to all ordinary modular forms, and propose a horizontal nonvanishing conjecture for these cohomology classes. The horizontal nonvanishing conjecture implies, via the theory of Euler systems, a conjecture of Greenberg on the generic ranks of Selmer groups in Hida families.
In Bull. Soc. Math. France 115 (1987), 399-456, Perrin-Riou formulates a form of the Iwasawa main conjecture which relates Heegner points to the Selmer group of an elliptic curve defined over Q, as one goes up the anticyclotomic Z p -extension of a quadratic imaginary field K. Building on the earlier work of Bertolini on this conjecture, and making use of the recent work of Mazur and Rubin on Kolyvagin's theory of Euler systems, we prove one divisibility of Perrin-Riou's conjectured equality. As a consequence, one obtains an upper bound on the rank of the Mordell-Weil group E(K) in terms of Heegner points.
We study the intersections of special cycles on a unitary Shimura variety of signature (n − 1, 1) and show that the intersection multiplicities of these cycles agree with Fourier coefficients of Eisenstein series. The results are new cases of conjectures of Kudla and suggest a Gross-Zagier theorem for unitary Shimura varieties.
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