Main conjecture for elliptic curves

Question

Suppose $E$ is an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at a prime $p$. Then one can define nonnegative integers $ \lambda_{E}^{alg} $, $ \mu_{E}^{alg} $, $ \lambda_{E}^{an} $ and $ \mu_{E}^{an} $.

The "algebraic" Iwasawa invariants $ \lambda_{E}^{alg} $ and $ \mu_{E}^{alg} $ are defined in terms of the structure of the $p$-primary subgroup $ Sel_{E}(\mathbb{Q}_{\infty})_{p} $ of the Selmer group for $E$ over the cyclotomic $ \mathbb{Z}_{p} $-extension $ \mathbb{Q}_{\infty} $ of $\mathbb{Q}$. The definition of the "analytic" invariants $ \lambda_{E}^{an} $ and $ \mu_{E}^{an} $ is in terms of the $p$-adic $L$-function for $E$ constructed by Mazur and Swinnerton-Dyer.

Now the Main Conjecture (Mazur) implies that $ \mu_{E}^{alg}=\mu_{E}^{an} $ and $ \lambda_{E}^{alg}=\lambda_{E}^{an} $. I want to know

1) What are the results proved till now towards proving the Main Conjecture $?$

2) For a particular elliptic curve over $\mathbb{Q}$ having good ordinary reduction at a prime $ p $, are there any methods to check that it satisfies the Main Conjecture $?$

EDIT: Prof. D Loeffler has mentioned that the main conjecture is now a theorem if the image of the mod $p$ Galois representation of $E$ is the whole of $GL_2(\mathbf{F}_p)$. But what happens if the residual representation of $E$ is not irreducible and $E$ has a $p$-isogeny $?$

Answer 1

The main conjecture is a theorem if the image of the mod $p$ Galois representation of E is the whole of $GL_2(\mathbf{F}_p)$. The full statement of the conjecture, which implies what you wrote about lambda and mu invariants but is quite a bit stronger, is the claim that the p-adic L-function generates the characteristic ideal of the Selmer group. This was proved in two steps.

The statement that the p-adic L-function lies in the char ideal of Selmer group (the Selmer group is not too large) was proved by Kato in 2004, using his Euler system; see his paper in Asterisque 295. This implies that $\mu^{alg} \le \mu^{an}$ and $\lambda^{alg} \le \lambda^{an}$.

The reverse inclusion, that the p-adic L-function divides the characteristic ideal of Selmer (the Selmer group is not too small) was proved by Skinner and Urban in this 2014 paper.

Answer 2

A small additional note: the main conjecture was also completely proven by Rubin for elliptic curves with complex multiplication. (You have to assume that $p$ is odd but that is the only assumption.)

This is Theorem 12.3 in Rubin's Inventiones paper "The main conjectures of Iwasawa theory for imaginary quadratic fields".