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