Anticyclotomic main conjectures for CM modular forms

Abstract

Generalizing work of Agboola-Howard on CM elliptic curves, we prove a version of the main conjecture of Iwasawa theory over the anticyclotomic ℤp-extension of a quadratic imaginary field K for Selmer groups attached to modular forms with complex multiplication by K. We deduce a statement to the effect that if the L-function of a CM modular form f of even weight has odd order of vanishing at its central point, then a Selmer group attached to f over ℚ is infinite.