Vol. 13, No. 4, 2019

Download this article
Download this article For screen
For printing
Recent Issues

Volume 13
Issue 6, 1243–1507
Issue 5, 995–1242
Issue 4, 749–993
Issue 3, 531–747
Issue 2, 251–530
Issue 1, 1–249

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Subscriptions
Editorial Board
Editors' Interests
Submission Guidelines
Submission Form
Editorial Login
Ethics Statement
Author Index
To Appear
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Other MSP Journals
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 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.

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