Iwasawa theory for Rankin-Selberg products of p-nonordinary eigenforms

Abstract

Our main goal in this article is to prove a divisibility statement in the Iwasawa main conjectures for symmetric squares of non-p-ordinary eigenforms (twisted by an auxiliary Dirichlet character). This task is carried out with the aid of Beilinson-Flach elements, which need to be suitably modified to obtain their integral counterparts. The key technical novelty is a significant improvement of the signed factorization procedure employed in the semi-ordinary Rankin-Selberg products, dwelling on ideas of Perrin-Riou on higher rank Euler systems.2010 Mathematics Subject Classification. 11R23 (primary); 11F11, 11R20 (secondary) .

Rank-two Euler systems for symmetric squares

“…The hypotheses (Ψ 1 ), (Ψ 2 ) and (Im) of Theorem A were already present in [BLV18]. The hypothesis (SD) allows us to show that the corestriction map…”

“…One expects that these classes are shadows of a (bounded) rank-two Euler system; in more precise terms, one expects that these unbounded classes arise from a rank-two Euler system via appropriate Perrin-Riou functionals. Evidence towards this was given in [BLLV18,BLV18].…”

“…Proof. All the references in this proof are to [BLV18], unless otherwise stated. The assertion is essentially Theorem 2.3.20, after minor modifications.…”

“…The main goal of this article is to construct a rank-two Euler system in the setting of [BLV18]. We fix forever a prime number p ≥ 7 as well as embeddings ι ∞ : Q → C and ι p : Q → C p .…”

“…In this article, with the aid of "signed" Beilinson-Flach elements constructed in [BLV18], we give a rather explicit construction of a rank-two Euler system for T under the following hypotheses.…”