Iwasawa theory of elliptic modular forms over imaginary quadratic fields at non-ordinary primes

TL;DR

This paper formulates and investigates Iwasawa main conjectures for non-ordinary modular forms over imaginary quadratic fields, introducing signed Selmer groups and Beilinson-Flach elements to relate p-adic L-functions and Selmer groups.

Contribution

It introduces signed Coleman maps and Beilinson-Flach elements for non-ordinary forms over imaginary quadratic fields, advancing the formulation of Iwasawa main conjectures in this setting.

Findings

Defined doubly-signed Selmer groups and p-adic L-functions.

Constructed Euler systems from Beilinson-Flach elements.

Established one inclusion in the main conjectures under technical assumptions.

Abstract

We formulate integral Iwasawa main conjectures for suitable twists of a newform $f$ that is non-ordinary at $p$, over the cyclotomic $\mathbb{Z}_p$-extension, the anticyclotomic $\mathbb{Z}_p$-extensions (in both the definite and the indefinite cases) as well as the $\mathbb{Z}_p^2$-extension of an imaginary quadratic field $K$ where $p$ splits. In order to do so, we define Kobayashi-Sprung-style signed Coleman maps, which we use to introduce doubly-signed Selmer groups. In the same spirit, we construct signed (integral) Beilinson-Flach elements (out of the collection of unbounded Beilinson-Flach elements of Loeffler-Zerbes), which we use to define doubly-signed $p$-adic $L$-functions. The main conjecture then relates these two set of objects. Furthermore, we show that the integral Beilinson-Flach elements form a locally restricted Euler system, that in turn allows us to deduce (under certain technical assumptions) one inclusion in each one of the four main conjectures we formulate here (which may be turned into equalities under favourable circumstances).