Generalized Heegner cycles at Eisenstein primes and the Katz p-adic L-function Article Swipe
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· 2016
· Open Access
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· DOI: https://doi.org/10.2140/ant.2016.10.309
· OA: W2200623020
We consider normalized newforms [math] whose nonconstant term Fourier coefficients are congruent to those of an Eisenstein series modulo some prime ideal above a rational prime [math] . In this situation, we establish a congruence between the anticyclotomic [math] -adic [math] -function of Bertolini, Darmon, and Prasanna and the Katz two-variable [math] -adic [math] -function. From this we derive congruences between images under the [math] -adic Abel–Jacobi map of certain generalized Heegner cycles attached to [math] and special values of the Katz [math] -adic [math] -function. ¶ Our results apply to newforms associated with elliptic curves [math] whose mod- [math] Galois representations [math] are reducible at a good prime [math] . As a consequence, we show the following: if [math] is an imaginary quadratic field satisfying the Heegner hypothesis with respect to [math] and in which [math] splits, and if the bad primes of [math] satisfy certain congruence conditions [math] and [math] does not divide certain Bernoulli numbers, then the Heegner point [math] is nontorsion, implying, in particular, that [math] . From this we show that if [math] is semistable with reducible mod- [math] Galois representation, then a positive proportion of real quadratic twists of [math] have rank 1 and a positive proportion of imaginary quadratic twists of [math] have rank 0.
