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.
2026-03-26 05:56:24.1774504584
Level structure on relative Jacobian
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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?