In the definition of level structure of level $n$ for an elliptic curve $A$, there are two versions:
- an isomorphism of group schemes $(\mathbf Z/n\mathbf Z)^2 \to A[n]$.
- an isomorhpism of group schemes $(\mathbf Z/n\mathbf Z)^2 \to A[n]$ after passing to some etale cover of the base.
There is a natural map from the moduli problem of 1 to the moduli problem of 2. This map appears to be etale.
Question: for more general moduli problems (when the level is given by subgroups $H\subset GL(2,n)$), normally 2. is used. What is the reason for this?