Congruent elliptic curves and the $\tau $-component in the Iwasawa main conjecture

Abstract Let p be an odd prime, and suppose that $E_1$ and $E_2$ are two elliptic curves which are congruent modulo p . Fix an Artin representation $\tau\,{:}\,G_{F}\rightarrow \mathrm{GL}_2(\mathbb{C})$ over a totally real field F , induced from a Hecke character over a CM-extension $K/F$ . Assuming $E_1$ and $E_2$ are ordinary at p , we compute the variation in the $\mu$ - and $\lambda$ -invariants for the $\tau$ -part of the Iwasawa Main Conjecture, as one switches from $E_1$ to $E_2$ . Provided an Euler system exists, it will follow directly that IMC $(E_1,\tau)$ is true if and only if IMC $(E_2,\tau)$ is true.