Algebra & Number Theory
- Algebra Number Theory
- Volume 10, Number 2 (2016), 309-374.
Generalized Heegner cycles at Eisenstein primes and the Katz $p$-adic $L$-function
We consider normalized newforms whose nonconstant term Fourier coefficients are congruent to those of an Eisenstein series modulo some prime ideal above a rational prime . In this situation, we establish a congruence between the anticyclotomic -adic -function of Bertolini, Darmon, and Prasanna and the Katz two-variable -adic -function. From this we derive congruences between images under the -adic Abel–Jacobi map of certain generalized Heegner cycles attached to and special values of the Katz -adic -function.
Our results apply to newforms associated with elliptic curves whose mod- Galois representations are reducible at a good prime . As a consequence, we show the following: if is an imaginary quadratic field satisfying the Heegner hypothesis with respect to and in which splits, and if the bad primes of satisfy certain congruence conditions and does not divide certain Bernoulli numbers, then the Heegner point is nontorsion, implying, in particular, that . From this we show that if is semistable with reducible mod- Galois representation, then a positive proportion of real quadratic twists of have rank 1 and a positive proportion of imaginary quadratic twists of have rank 0.
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
Permanent link to this document
Digital Object Identifier
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]
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. https://projecteuclid.org/euclid.ant/1510842481