Anticyclotomic Iwasawa theory of elliptic modular forms at non-ordinary primes

This article is a continuation of our previous work on the Iwasawa theory of an elliptic modular form over an imaginary quadratic field $K$, where the modular form in question was assumed to be ordinary at a fixed odd prime $p$. We formulate the Iwasawa main conjecture 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 \emph{definite} and the \emph{indefinite} cases) as well as the maximal $\mathbb{Z}_p^2$-tower of an imaginary quadratic field $K$ where $p$ splits. In order to do so, we define Kobayashi-Sprung-style doubly-signed Coleman maps, which we use to define doubly signed Selmer groups. In the same spirit, we construct doubly-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 allows us to deduce (under certain technical assumptions) one inclusion in each one of the four main conjectures we formulate here.