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

Abstract

In this paper, we consider normalized newforms $f\in S_k({\Gamma }_0(N),{\epsilon }_f)$ whose non-constant 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-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. In particular, our results apply to newforms associated with elliptic curves $E/\mathbb{Q}$ 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 non-torsion, in particular implying that ${\text{rank}}_{\mathbb{Z}}E(K)=1$. From this, we show that when $E$ 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.

53 pages, accepted for publication in Algebra and Number Theory

References in corpus (2)

• The Parity of Analytic Ranks among Quadratic Twists of Elliptic Curves over Number Fields
• Quadratic Twists of Elliptic Curves with 3-Selmer Rank 1

Cited by in corpus (5)

• On the anticyclotomic Iwasawa theory of rational elliptic curves at Eisenstein primes
• Three-isogeny Selmer groups and ranks of abelian varieties in quadratic twist families over a number field
• A New $p$-adic Maass-Shimura operator and Supersingular Rankin-Selberg $p$-adic $L$-functions
• Heegner points at Eisenstein primes and twists of elliptic curves
• Supersingular main conjectures, Sylvester's conjecture and Goldfeld's conjecture