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

Kriz, Daniel

doi:10.2140/ant.2016.10.309

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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

DOI: 10.2140/ant.2016.10.309

Abstract

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