Kato's main conjecture for nonordinary modular forms

We prove Kato's main conjecture for modular forms at nonordinary primes for weights in the Fontaine-Laffaille range. The key ingredients in the proof are a reformulation of the conjecture in terms of signed Selmer groups due to Lei-Loeffler-Zerbes, certain $p$-adic families of Rankin-Eisenstein classes arising from the work of Lei-Loeffler-Zerbes and Kings-Loeffler-Zerbes, and the lower bound divisibility in an Iwasawa-Greenberg main conjecture for Rankin-Selberg $p$-adic $L$-functions obtained in our earlier work \cite{CLW}.