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

We consider normalized newforms $f \in S_{k} (Γ_{0} (N), ε_{f})$ 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 $mod p$ and $p$ does not divide certain Bernoulli numbers, then the Heegner point $P_{E} (K)$ is nontorsion, implying, in particular, that ${rank}_{ℤ} E (K) = 1$ . 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.

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

Milestones

Received: 10 December 2014

Revised: 12 December 2015

Accepted: 15 December 2015

Published: 16 March 2016

Authors

Daniel Kriz
Department of Mathematics Princeton University Fine Hall, Washington Rd Princeton, NJ 08544 United States

Vol. 10, No. 2, 2016

Daniel Kriz

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors