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

Abstract

We consider normalized newforms $f\in S_k\left({\Gamma }_0\left(N\right),{\epsilon }_f\right)$ 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 $E∕\mathbb{Q}$ whose mod-$p$ Galois representations $E\left[p\right]$ 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 $modp$ and $p$ does not divide certain Bernoulli numbers, then the Heegner point $P_E\left(K\right)$ is nontorsion, implying, in particular, that ${rank}_{\mathbb{Z}}E\left(K\right)=1$. 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.