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

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 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 g in the cyclotomic tower. This allows us to formulate “signed” Iwasawa main conjectures for f 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.

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