This is a very soft question.

Let $n\geq 1$ and let $Y(n)$ be the (open) modular curve associated to $\Gamma(n)\subset SL_2(\mathbb{Z})$. Interpreted correctly, $Y(n)\to Y(d)$ is finite etale, whenever $d$ is a positive integer dividing $n$.

Now, it is a priori possible that $Y(n)\to Y(d)$ is a "trivial" finite etale cover, i.e., it has a section.

But, of course, it isn't. With a view towards higher-dimensional analogues of the modular curves I would like to know:

Why is $Y(n)\to Y(d)$ a non-trivial finite etale cover?

I'm looking for an argument which is as "soft" as possible, and could be applied to higher-dimensional situations (or even situations different from the moduli of abelian varieties).

For instance, the genus of $Y(n)$ grows with $n$, so these covers can't be trivial. But is there a "softer", slightly more direct, argument?

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    $\begingroup$ Maybe I am missing something: a morphism of irreducible curves cannot have a section unless it's an isomorphism. $\endgroup$ – Pop Jul 26 '18 at 12:02
  • $\begingroup$ Oops, I should have said "birational" instead of "isomorphism". (But the point remains the same.) $\endgroup$ – Pop Jul 26 '18 at 12:27
  • $\begingroup$ Yes, but let's not use that $Y(n)$ is irreducible. (In fact, if one defines level $n$ structure without demanding compatibility with the Weil pairing, then the resulting modular curve is not necessarily irreducible.) Basically, I am looking for an argument which only uses the moduli-interpretation somehow. $\endgroup$ – Dasz Jul 26 '18 at 13:05

Since you are looking for moduli interpretations, we may either work with stacks, or assume $d$ is large enough that $Y(d)$ is representable (i.e., at least 3).

Perhaps the easiest answer is that in a small neighborhood of the infinite cusp, $Y(n)$ is ramified over $Y(1)$ with degree $n$, while $Y(d)$ is ramified with degree $d$. In other words, there is a $\mathbb{C}((q))$-point of $Y(n)$ whose image in $Y(d)$ factors through a map from the subfield $\mathbb{C}((q^{n/d}))$, and the point from this subfield does not lift to $Y(n)$. If there were a section, all points of $Y(d)$ would have lifts.

A moduli-theoretic interpretation of this ramification is that if we are given an elliptic curve with level $n$ structure $(E,a,b)$, monodromy around the infinite cusp induces a shearing automorphism $(E,a,b) \mapsto (E,a,a+b)$ of order $n$, and this automorphism induces an automorphism $(E,n\frac{a}{d},n\frac{b}{d}) \mapsto (E,n\frac{a}{d},n\frac{a+b}{d})$ of order $d$ on the corresponding curve with level $d$ structure.

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