Iwasawa theory for $$\mathrm {GL}_2\times \mathrm {Res}_{K/\mathbb {Q}}\mathrm {GL}_1$$

Let K be an imaginary quadratic field where the prime p splits. Our goal in this article is to prove results towards Iwasawa main conjectures for p-nearly-ordinary families associated $$\mathrm {GL}_2\times \mathrm {Res}_{K/\mathbb {Q}}\mathrm {GL}_1$$ with a minimal set of assumptions. The technical input improvement on locally restricted Euler system machinery that allows treatment residually reducible cases, which we apply Beilinson–Flach system.