Definition of p-adic modular forms

Question

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.

Answer 1

It's not a pushforward of a differential (or maybe it can be described as one, but it doesn't have to be).

Rather there is a pullback map from differentials on the elliptic curve to differentials on $\mu_{p^\alpha}$. However, this map is an isomorphism mod $p^\alpha$, so you can apply the inverse to reverse the process.

The fact that it is an isomorphism follows from:

The differentials on an elliptic curve over a ring $R$ are a rank $1$ free module over $R$.

The differentials on $\mu_{p^\alpha}$ over $R$ are $R/ p^\alpha$, because it's giving by the equation $x^{p^\alpha}=1$, hence $p^\alpha x^{p^\alpha-1} dx =0$, and $x$ is invertible so $p^\alpha dx=0$.

The map is a unit mod $p$ - after specializing to a field of characteristic $p$, it's sufficient to check that it's nonzero. But if it's zero, then the map $\mu_p \to E$ is trivial on differentials, so by exponentiating (and stopping before $p$th powers) it is zero, which contradicts the fact that it comes from Cartier duality.