Algebra & Number Theory

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

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

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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 fg, 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 fg in the cyclotomic tower. This allows us to formulate “signed” Iwasawa main conjectures for fg in the spirit of Kobayashi’s ±-Iwasawa theory for supersingular elliptic curves; and we prove one inclusion in these conjectures under our running hypotheses.

Article information

Algebra Number Theory, Volume 13, Number 4 (2019), 901-941.

Received: 12 February 2018
Revised: 13 September 2018
Accepted: 10 February 2019
First available in Project Euclid: 18 May 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11R23: Iwasawa theory
Secondary: 11F11: Holomorphic modular forms of integral weight 11R20: Other abelian and metabelian extensions

Iwasawa theory elliptic modular forms nonordinary primes


Büyükboduk, Kâzım; Lei, Antonio; Loeffler, David; Venkat, Guhan. Iwasawa theory for Rankin-Selberg products of $p$-nonordinary eigenforms. Algebra Number Theory 13 (2019), no. 4, 901--941. doi:10.2140/ant.2019.13.901.

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