level structures and moduli of abelian varieties

Hello,

In the definition of level structure of level $n$ for an elliptic curve $A$, there are two versions:

1. an isomorphism of group schemes $(\mathbf Z/n\mathbf Z)^2 \to A[n]$.
2. 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?

Thanks

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I think you're a bit muddled. A full level $n$ structure is 1. I am not sure what you mean by 2 -- is the cover part of the data for example? -- but my guess is that there's been a misunderstanding here. The reason that one has to be a bit more careful when doing the general $H$ case is because you want to allow the situation that the curve have a level $H$ structure but have no full level $n$ structures at all -- e.g. if $H=\Gamma_0(n)$ then you want an $H$-structure to mean a subgroup of order $n$, but a curve can have such a subgroup without having a full level $n$ structure. – Kevin Buzzard Jun 11 '11 at 16:17
So one way of doing a level $H$ structure (assuming $n$ invertible on the base) is to look at the isom scheme and then quotient out, and then observe that even if the isom scheme doesn't have points, the quotient still might. If you want to make this "explicit" you might start allowing base changes so you can see the points moving around under $H$. – Kevin Buzzard Jun 11 '11 at 16:19
[s/subgroup/cyclic subgroup/ in the above] – Kevin Buzzard Jun 11 '11 at 16:20