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Iwasawa theory for
Rankin-Selberg products of $p$-nonordinary eigenforms
Kâzım Büyükboduk, Antonio Lei, David Loeffler and Guhan Venkat |
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Vol. 13 (2019), No. 4, 901–941
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Abstract |
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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. |
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Keywords
Iwasawa theory, elliptic modular forms, nonordinary primes
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Mathematical Subject Classification 2010
Primary: 11R23
Secondary: 11F11, 11R20
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Milestones
Received: 12 February 2018
Revised: 13 September 2018
Accepted: 10 February 2019
Published: 6 May 2019
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Authors |
| Kâzım Büyükboduk | |
| UCD School of Mathematics and
Statistics University College Dublin Ireland |
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| Antonio Lei | |
| Département de Mathématiques et de
Statistique Université Laval Pavillion Alexandre-Vachon Quebec City, QC Canada |
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| David Loeffler | |
| Mathematics Institute University of Warwick Zeeman Building Coventry United Kingdom |
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| Guhan Venkat | |
| Département de Mathématiques et de
Statistique Laval University Pavillion Alexandre-Vachon Quebec City, QC Canada |
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