This question is based off [these notes][1] by Preston Wake about Iwasawa invariants and Hida Families. In the notes, the author asks "why" the elliptic curve $11A3$ has $\mu$-invariant equal to $2$ for the prime $p=5$. He offers a heuristic justification by saying that for all primes $\ell \neq 11$, we have the congruence

$$a_{\ell} \equiv \chi(\ell) + \ell \chi^{-1}(\ell) \mod 5^2$$

where $\chi: (\mathbf{Z}/11\mathbf{Z})^{\times} \to (\mathbf{Z}/11\mathbf{Z})^{\times}$ is the character that sends $2$ to $6$. Here, as usual, $a_{\ell}$ is defined by $a_{\ell} = |E(\mathbf{F}_{\ell})| - (\ell+1)$. According to the notes, the first term $\chi(\ell)$ in the above sum "comes from Eisensteiness" and the second term $\ell \chi^{-1}(\ell)$ in the sum "comes from extra reducibility".

*My question is:* where does the equation above come from? And what does it have to do with the $\mu$-invariant of $11A3$? Because a priori, it is simply a congruence between the various $a_{\ell}$'s; it seems unrelated to $\mu$-invariants. **More generally, I'd like to ask**: what is the concrete link between the $\mu$-invariants of elliptic curves, and congruences between Fourier coefficients of modular forms like the one shown above? Is there a reference that explains this connection?

To be clear, I've read about how Hida families of modular forms can shed light on the $\mu$-invariants of *modular forms*. (As an example, in the paper of Pollack, Wake, and Weston.) My question, however, is about how Hida families of modular forms give us info about the $\mu$-invariants of *elliptic curves*. The equation above, for example, seems to suggest that we can gleam information about the $\mu$-invariants of *elliptic curves* by examining congruences between their Fourier coefficients. I'm not too clear on the link between those two sides of the story. 

So if anyone knew of a reference specifically about congruences between modular forms and $\mu$-invariants of *elliptic curves*, I'd be immensely grateful. Thanks for the help! 
 

 


  [1]: http://www.math.pitt.edu/~caw203/pdfs/notes/Wake2018.pdf