#### Vol. 10, No. 2, 2016

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

### Daniel Kriz

Vol. 10 (2016), No. 2, 309–374
##### Abstract

We consider normalized newforms $f\in {S}_{k}\left({\Gamma }_{0}\left(N\right),{\epsilon }_{\phantom{\rule{0.3em}{0ex}}f}\right)$ 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 $E∕ℚ$ 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 $\phantom{\rule{0.3em}{0ex}}mod\phantom{\rule{0.3em}{0ex}}p$ and $p$ does not divide certain Bernoulli numbers, then the Heegner point ${P}_{E}\left(K\right)$ is nontorsion, implying, in particular, that ${rank}_{ℤ}E\left(K\right)=1$. 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.

##### Keywords
Heegner cycles, $p$-adic Abel–Jacobi map, Katz $p$-adic $L$-function, Beilinson–Bloch conjecture, Goldfeld's conjecture
##### Mathematical Subject Classification 2010
Primary: 11G40
Secondary: 11G05, 11G15, 11G35