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.)

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