Vol. 10, No. 2, 2016

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Generalized Heegner cycles at Eisenstein primes and the Katz $p$-adic $L$-function

Daniel Kriz

Vol. 10 (2016), No. 2, 309–374

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.

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
Received: 10 December 2014
Revised: 12 December 2015
Accepted: 15 December 2015
Published: 16 March 2016
Daniel Kriz
Department of Mathematics
Princeton University
Fine Hall, Washington Rd
Princeton, NJ 08544
United States