Let $\Lambda := \Bbb Z_p[[T]]$ be the usual Iwasawa algebra. In [Jha and Sujatha - On the Hida deformations of fine Selmer groups](https://doi.org/10.1016/j.jalgebra.2011.02.042) on page 181, the authors refer to the quotient $\Bbb H^{\text{ord}}_{\mathcal{F}}$ of the universal ordinary Hecke algebra $\Bbb H^{\text{ord}}_{Np^\infty}$ corresponding to an ordinary $\Lambda$-adic eigenform $\mathcal{F}$. I'm somewhat new to the subject and have been asked to give this paper a go by my masters supervisor, and while I "know" what all of these objects are, Hida's original 1986 paper doesn't explicitly say "the quotient of blah corresponding to blah" and I don't know how to make this connection. Could someone explain this to me like I'm five?


  [1]: https://reader.elsevier.com/reader/sd/pii/S0021869311001499