Let and be two modular forms which are nonordinary at . The theory of Beilinson–Flach elements gives rise to four rank-one nonintegral Euler systems for the Rankin–Selberg convolution , one for each choice of -stabilisations of and . We prove (modulo a hypothesis on nonvanishing of -adic -functions) that the -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 -adic -functions associated to in the cyclotomic tower. This allows us to formulate “signed” Iwasawa main conjectures for in the spirit of Kobayashi’s -Iwasawa theory for supersingular elliptic curves; and we prove one inclusion in these conjectures under our running hypotheses.
"Iwasawa theory for Rankin-Selberg products of $p$-nonordinary eigenforms." Algebra Number Theory 13 (4) 901 - 941, 2019. https://doi.org/10.2140/ant.2019.13.901