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

In this paper, we consider normalized newforms $f\in S_k(\Gamma_0(N),\varepsilon_f)$ whose non-constant 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-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. In particular, our results apply to newforms associated with elliptic curves $E/\mathbb{Q}$ whose mod $p$ Galois representations $E[p]$ 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 $P_{E}(K)$ is non-torsion, in particular implying that $\text{rank}_{\mathbb{Z}}E(K) = 1$. From this, we show that when $E$ 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.