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

Abstract

Let $f$ and $g$ be two modular forms which are nonordinary at $p$. The theory of Beilinson–Flach elements gives rise to four rank-one nonintegral Euler systems for the Rankin–Selberg convolution $f\otimes g$, one for each choice of $p$-stabilisations of $f$ and $g$. We prove (modulo a hypothesis on nonvanishing of $p$-adic $L$-functions) that the $p$-parts of these four objects arise as the images under appropriate projection maps of a single class in the wedge square of Iwasawa cohomology, confirming a conjecture of Lei–Loeffler–Zerbes.

Furthermore, we define an explicit logarithmic matrix using the theory of Wach modules, and show that this describes the growth of the Euler systems and $p$-adic $L$-functions associated to $f\otimes g$ in the cyclotomic tower. This allows us to formulate “signed” Iwasawa main conjectures for $f\otimes g$ in the spirit of Kobayashi’s $\pm$-Iwasawa theory for supersingular elliptic curves; and we prove one inclusion in these conjectures under our running hypotheses.