On the BDP Iwasawa main conjecture for modular forms

Abstract

Let K be an imaginary quadratic field where p splits, $$p\ge 5$$ a prime number and f an eigen-newform of even weight and level $$N>3$$ that is coprime to p. Under the Heegner hypothesis, Kobayashi–Ota showed that one inclusion of the Iwasawa main conjecture of f involving the Bertolini–Darmon–Prasanna p-adic L-function holds after tensoring by $$\mathbb {Q}_p$$ . Under certain hypotheses, we improve upon Kobayahsi–Ota’s result and show that the same inclusion holds integrally. Our result implies the vanishing of the Iwasawa $$\mu $$ -invariants of several anticyclotomic Selmer groups.