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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} $ at $p$.

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} $.

Let $E_1$ and $E_2$ are two elliptic curves over $\mathbb{Q}$ such that there exists a $\mathbb{Q}$-isogeny from $E_1$ to $E_2$. Is the Main Conjecture invariant under a $\mathbb{Q}$-isogeny i.e, if the conjecture is true for any one of the curves then it is also true for the other one $?$

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Yes, the main conjecture is isogeny-invariant. See here:

B. Perrin-Riou, Variation de la fonction $L$ $p$-adique par isogénie, Algebraic number theory, Adv. Stud. Pure Math. 17 (1989), pp. 347-358.

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    $\begingroup$ Oh ! I didn't know this. $\endgroup$ – Chris Wuthrich May 12 '14 at 8:47
  • $\begingroup$ I got this via Google, from the references of a more recent article by Ochiai (proving the same result for 2-variable main conjectures over a Hida family). Of course there are lots of other related results, e.g. the isogeny-invariance statement for the Bloch--Kato conjecture proved in the B-K paper in the Grothendieck Festschrift. $\endgroup$ – David Loeffler May 12 '14 at 8:58
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The full main conjecture is invariant under isogenies defined over $\mathbb{Q}$, not just the statement about $\mu$ and $\lambda$-invariants.

The only thing that changes on the analytic side is the period: Recall the $p$-adic $L$-function interpolates values of the form Gaussum*L-value/Period. The complex $L$-function is invariant under isogeny, so only the Néron periods of the curves enter into the change.

On the algebraic side, one has to determine the change of the Selmer group under isogeny. Over $\mathbb{Q}$ this is what Cassels had to do to show that the Birch and Swinnerton-Dyer conjecture is invariant. Now over the $\mathbb{Z}_p$-extension, one can do it in a similar way. That the changes agree is proven in the appendix of a paper by Perrin-Riou called "Fonctions $L$ $p$-adiques, théorie d’Iwasawa et points de Heegner", Bull. Soc. Math. France 115 (1987), no. 4, 399–456. this link on numdam

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