I have been reading Hida's book "p-Adic automorphism forms on Shimura varieties" and I don't understand a point. He first describes p-adic modular forms of tame level N as functions on the Igusa tower which can be interpreted as functions on triples $(E,\phi_{p^\alpha},\phi_N)$ where $\phi_{p^\alpha}: \mu_{p^\alpha}\hookrightarrow E$ is the level structure Cartier dual to the identification $E[p^\alpha]^{et}\cong \mathbb{Z}/p^\alpha\mathbb{Z}$. Then he claims that since $\mu_{p^\alpha}$ has a canonical differential $\frac{dt}{t}$ we can consider p-adic modular forms as functions on triples $(E,\phi_{p^\alpha},\phi_N, \phi_{p^\alpha *}\frac{dt}{t})$.

I guess the idea is that such a differential is supposed to trivialize the pushforward of the sheaf of differential of the elliptic curve over the base, when we think about modular forms as section of that sheaf for the universal elliptic curve. However, I cannot make sense of the pushforward of the differential as a differential of the elliptic curve.

Does anyone know how to understand all this?

Thank you in advance.