I was reading Gouvea's book on $p$-adic modular forms, and I found something strange at the very beginning of the book. When he introduces test objects, he defines an elliptic curve over a ring $R$ with arithmetic $\Gamma_1(Np^{\nu})$ level structure as an elliptic curve $E$ over $R$ plus an injection $\mu_{Np^{\nu}}\hookrightarrow E[Np^{\nu}]$. After this definition, he compares this notion of level structure with the notion of Katz Mazur's book, which says that an elliptic curve $E/R$ has level $Np^{\nu}$ when there exists a point $P\in E[Np^{\nu}](R)$ with exact order $Np^{\nu}$, in the sense of Drinfeld's level structure. He then says that, for $\nu=0$ the two objects coincide, while for $\nu>0$, the first notion gives rise to a modular curve which is an open subscheme of the modular curve defined by Katz and Mazur. I really don't see the connection between these two situations. Any suggestion?


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