Recent work of Vatsal yields an exact formula for the μ-invariant of the anticyclotomic p-adic L-function of a weight two modular form in terms of certain congruence numbers. In particular, unlike the cyclotomic case, these μ-invariants can be non-zero even when the p-torsion of the curve is irreducible.
Using Vatsal's formula, along with results of Bertolini and Darmon, we derive an exact formula for the μ-invariant of the anticyclotomic Selmer group of an elliptic curve in terms of certain local invariants attached to the Galois representation of this modular form. Moreover, we show that the formulas for the analytic and algebraic μ-invariants agree, thus establishing the "μ-part" of the main conjecture for these modular forms.
This is a joint project with Tom Weston.