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

We will report on our joint work with A. Lei towards the main conjectures (in one or two-variables) for the
Rankin-Selberg convolutions of the base change of a $p$-non-ordinary modular form to an imaginary quadratic
field $K$, with ray class characters of $K$. The crucial ingredient is a signed-splitting procedure for two families
of $p$-stabilized (unbounded) Beilinson-Flach classes, much in the spirit of Kobayashi and Pollack, which yields
a pair of Euler systems (collections of bounded cohomology classes) for the associated to motive. In the indefinite
anticyclotomic set up (where we show that the main conjectures themselves reduce to $0=0$), our methods also
yield a divisibility in a $\Lambda$-adic Birch and Swinnerton-Dyer formula. (These circle of ideas partially
extend to allow the treatment more general $p$-non-ordinary Rankin-Selberg products and symmetric squares;
this is joint work in progress with with A. Lei, D. Loeffler and G. Venkat.)