## Algebra & Number Theory

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

Daniel Kriz

#### Abstract

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 $E[p]$ 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 $modp$ and $p$ does not divide certain Bernoulli numbers, then the Heegner point $PE(K)$ is nontorsion, implying, in particular, that $rankℤE(K) = 1$. 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.

#### Article information

Source
Algebra Number Theory, Volume 10, Number 2 (2016), 309-374.

Dates
Received: 10 December 2014
Revised: 12 December 2015
Accepted: 15 December 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1510842481

Digital Object Identifier
doi:10.2140/ant.2016.10.309

Mathematical Reviews number (MathSciNet)
MR3477744

Zentralblatt MATH identifier
06561467

#### Citation

Kriz, Daniel. Generalized Heegner cycles at Eisenstein primes and the Katz $p$-adic $L$-function. Algebra Number Theory 10 (2016), no. 2, 309--374. doi:10.2140/ant.2016.10.309. https://projecteuclid.org/euclid.ant/1510842481

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