The anticyclotomic main conjectures for elliptic curves

Let E/Q be an elliptic curve of conductor N and let f be the cuspidal eigenform on Γ0(N) associated to E by the modularity theorem. Denote by K∞ the anticyclotomic Zp-extension of an imaginary quadratic field K, where p is a prime number unramified in K. Under appropriate arithmetic assumptions we prove the main conjectures of Iwasawa theory for E over K∞. Our results cover both the cases where p is good ordinary and supersingular for E, and both the definite and indefinite settings. This leaves out a single case, which we term exceptional, for which we establish one of the two expected divisibilities.