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Assume that $X/S$ is a family of smooth projective curves say in characteristic zero. Can the relative Jacobian of $X/S$ always be endowed with a level structure? In other words does a map $S\to A_{g,n}$ exists for $n>3$? I would appreciate any reference for this question.

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This is not always possible.

Consider an elliptic curve $E$ over $S= Spec \mathbb{Q}$ with trivial Mordell-Weil group, for instance. Can you take it from here?

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  • $\begingroup$ Good point, I now understand. thank you very much! $\endgroup$
    – Jack
    Oct 26, 2016 at 15:51
  • $\begingroup$ @Jack By the way, you can also argue similarly over function fields over $\mathbb C$. That's enough to also show that there are elliptic curves over positive-dimensional varieties with no level structure. Basically: the existence of a level $n$ structure implies the existence of non-trivial $n$-torsion points. If there aren't any of the latter, then there is no level $n$ structure. $\endgroup$ Oct 26, 2016 at 16:01
  • $\begingroup$ OK, so the same argument even works in higher dimensional cases (not just Elliptic curves), right? $\endgroup$
    – Jack
    Oct 26, 2016 at 16:56
  • $\begingroup$ @Jack Yes, but you don't have to redo any of the arguments. If $E$ is an elliptic curve over $S$ with no level $n$ structure, then $E^g$ (self-product of $E$ over $S$) has no level $n$ structure either. $\endgroup$ Oct 26, 2016 at 17:03

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