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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. [ABSTRACT FROM AUTHOR]

Abstract: Building on ideas of Vatsal [Uniform distribution of Heegner points, Invent. Math. 148(1) (2002) 1–46], Cornut [Mazur''s conjecture on higher Heegner points, Invent. Math. 148(3) (2002) 495–523] proved a conjecture of Mazur asserting the generic nonvanishing of Heegner points on an elliptic curve as one ascends the anticyclotomic -extension of a quadratic imaginary extension . In the present article, Cornut''s result is extended by replacing the elliptic curve E with the Galois cohomology of Deligne''s two-dimensional -adic representation attached to a modular form of weight , and replacing the family of Heegner points with an analogous family of special cohomology classes. [Copyright &y& Elsevier]

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]