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2016 Generalized Heegner cycles at Eisenstein primes and the Katz $p$-adic $L$-function
Daniel Kriz
Algebra Number Theory 10(2): 309-374 (2016). DOI: 10.2140/ant.2016.10.309

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 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.

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Daniel Kriz. "Generalized Heegner cycles at Eisenstein primes and the Katz $p$-adic $L$-function." Algebra Number Theory 10 (2) 309 - 374, 2016. https://doi.org/10.2140/ant.2016.10.309

Information

Received: 10 December 2014; Revised: 12 December 2015; Accepted: 15 December 2015; Published: 2016
First available in Project Euclid: 16 November 2017

zbMATH: 06561467
MathSciNet: MR3477744
Digital Object Identifier: 10.2140/ant.2016.10.309

Subjects:
Primary: 11G40
Secondary: 11G05 , 11G15 , 11G35

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

Rights: Copyright © 2016 Mathematical Sciences Publishers

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