Let $N\geq4$ be a positive integer and p be a prime such that $(p,N)=1$, and $X=X_1(N)$ be the modular curve parameterizing (generalized) elliptic curves with $\Gamma_1(N)$-level structure. Base change X to $\mathbb{Z}_p$, and complete along the special fiber, we get a formal scheme $\hat{X}$. Let $\pi: E\to\hat{X}$ be the universal elliptic curve, and $\omega$ be the modular sheaf $\pi_*\Omega^1_{E/\hat{X}}$. Let $Spf(R)\subset{\hat{X}}$ be an open affine formal sub-scheme such that $\omega$ is trivialized, so that we can view the Hasse invariant as an element of $R/pR$ (up to units), if H is any lifting of the Hasse invariant in R, do we have $\mathbb{Z}_p\langle{H}\rangle\subset{R}$ is etale? This is claimed in the proof of Lem.3.3 of the paper Triple product p-adic L-function associated to finite slope p-adic families of modular forms, by saying that Hasse invariant has simple zeroes. But I could understand the logical relation, would anyone please explain a little bit to me? or give another proof? Thanks very much.
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1$\begingroup$ I think the correct statement is that it is etale in a neighborhood of the zeroes of the Hasse invariant. Presumably one can immediately reduce Lemma 3.3 to a neighborhood of the supersingular points. $\endgroup$– Will SawinCommented Jun 29, 2018 at 14:37
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$\begingroup$ Thanks very much. This seems much more plausible. $\endgroup$– GRHCommented Jun 29, 2018 at 14:44
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