Algebra & Number Theory
- Algebra Number Theory
- Volume 13, Number 4 (2019), 901-941.
Iwasawa theory for Rankin-Selberg products of $p$-nonordinary eigenforms
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.
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
Permanent link to this document
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
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. https://projecteuclid.org/euclid.ant/1558144824