Kolyvagin systems and Iwasawa theory of generalized Heegner cycles

Abstract

Iwasawa theory of Heegner points on abelian varieties of GL2 type has been studied by, among others, Mazur, Perrin-Riou, Bertolini and Howard. The purpose of this paper is to describe extensions of some of their results in which abelian varieties are replaced by the Galois cohomology of Deligne's p-adic representation attached to a modular form of even weight > 2. In this setting, the role of Heegner points is played by higher-dimensional Heegner-type cycles that have been recently defined by Bertolini, Darmon and Prasanna. Our results should be compared with those obtained, via deformation-theoretic techniques, by Fouquet in the context of Hida families of modular forms.2010 Mathematics Subject Classification. 11R23, 11F11.

Comparing anticyclotomic Selmer groups of positive co-ranks for congruent modular forms

“…Let f ∈ H be a modular form of weight k = 2r and tame level Γ 0 (N ) as in Section 1.1. Following [LV18], we denote by T the associated G Q -representation constructed in [Nek92]; it is a free rank-2 O-module. Then V = T ⊗ L f,P is the r-twist of the Galois representation constructed by Deligne.…”

“…Let h K denote the class number of K, and let φ denote the Euler totient function. Following [LV18], we assume (admiss. )…”

“…inert) in K. We will assumeN − = 1 and that p = pp splits in K. The assumption thatN − = 1 seems important to our method, as we use it to guarantee the global-to-local maps defining certain Selmer group are surjective; see the discussion following Lemma 3.3 as well as the proofs of Lemma 3.10 and Proposition 5.1. It is also present in the assumptions of [LV18], which we cite in Section 3.3. It is not immediately clear to the authors whether this assumption can be relaxed to N − having an even number of prime divisors.…”

“…One suite of hypotheses, labeled (admiss. ), is introduced in Section 3.3 in order to apply results of [LV18]. There is also a mild technical hypothesis (H.div) which is introduced in Section 3.5; see Remark 3.15.…”

p‐adic heights of Heegner points and Beilinson–Flach classes

“…This appears to be an important technical point, since it allows us to dispense with the twist of E by a rather undesirable p-distinguished character. † After the first version of this paper was released, this has largely been carried out by the combined works of Büyükboduk-Lei [6] and Longo-Vigni [35].…”