Algebra & Number Theory

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

Daniel Kriz

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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 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 rankE(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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10]
Secondary: 11G05: Elliptic curves over global fields [See also 14H52] 11G15: Complex multiplication and moduli of abelian varieties [See also 14K22] 11G35: Varieties over global fields [See also 14G25]

Heegner cycles $p$-adic Abel–Jacobi map Katz $p$-adic $L$-function Beilinson–Bloch conjecture Goldfeld's conjecture


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.

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