On the Iwasawa main conjectures for modular forms at non-ordinary primes

In this paper, we prove under mild hypotheses the Iwasawa main conjectures of Lei–Loeffler–Zerbes for modular forms of weight $2$ at non-ordinary primes. Our proof is based on the study of the two-variable analogues of these conjectures formulated by B"uy"ukboduk–Lei for imaginary quadratic fields in which $p$ splits, and on anticyclotomic Iwasawa theory. As application of our results, we deduce the $p$-part of the Birch and Swinnerton-Dyer formula in analytic ranks $0$ or $1$ for abelian varieties over $\mathbb{Q}$ of ${\rm GL}_2$-typefornon-ordinaryprimes$p>2$.