#### Vol. 13, No. 4, 2019

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

### Kâzım Büyükboduk, Antonio Lei, David Loeffler and Guhan Venkat

Vol. 13 (2019), No. 4, 901–941
##### 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 $±$-Iwasawa theory for supersingular elliptic curves; and we prove one inclusion in these conjectures under our running hypotheses.

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##### Keywords
Iwasawa theory, elliptic modular forms, nonordinary primes
##### Mathematical Subject Classification 2010
Primary: 11R23
Secondary: 11F11, 11R20
##### Milestones
Received: 12 February 2018
Revised: 13 September 2018
Accepted: 10 February 2019
Published: 6 May 2019
##### Authors
 Kâzım Büyükboduk UCD School of Mathematics and Statistics University College Dublin Ireland Antonio Lei Département de Mathématiques et de Statistique Université Laval Pavillion Alexandre-Vachon Quebec City, QC Canada David Loeffler Mathematics Institute University of Warwick Zeeman Building Coventry United Kingdom Guhan Venkat Département de Mathématiques et de Statistique Laval University Pavillion Alexandre-Vachon Quebec City, QC Canada