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.
"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