Generalized Heegner cycles at Eisenstein primes and the Katz p-adic L-function

We consider normalized newforms f∈Sk(Γ0 (N),εf)whose nonconstant term Fourier coefficients are congruent to those of an Eisenstein series modulo some prime ideal above a rational prime p. In this situation, we establish a congruence between the anticyclotomic p-adic L-function of Bertolini, Darmon, and Prasanna and the Katz two-variable p-adic L-function. From this we derive congruences between images under the p-adic Abel-Jacobi map of certain generalized Heegner cycles attached to f and special values of the Katz p-adic L-function. Our results apply to newforms associated with elliptic curves E=Q whose mod-p Galois representations ETpU are reducible at a good prime p. As a consequence, we show the following: if K is an imaginary quadratic field satisfying the Heegner hypothesis with respect to E and in which p splits, and if the bad primes of E satisfy certain congruence conditions mod p and p does not divide certain Bernoulli numbers, then the Heegner point PE .K/ is nontorsion, implying, in particular, that rankℤ E.K/ D 1. From this we show that if E is semistable with reducible mod-3 Galois representation, then a positive proportion of real quadratic twists of E have rank 1 and a positive proportion of imaginary quadratic twists of E have rank 0.