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

Büyükboduk, Kâzım; Lei, Antonio; Loeffler, David; Venkat, Guhan

doi:10.2140/ant.2019.13.901

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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

DOI: 10.2140/ant.2019.13.901

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.

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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

Vol. 13, No. 4, 2019