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David Loeffler
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User0829
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Global section of horizontalvertical differential 1 forms on universal elliptic curve

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User0829
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Global section of horizontal differential 1 forms on universal elliptic curve

Let $B$ be a modular curve (of some level) over a number field $K$ (here, we implicitly assume that $K$ is large enough to make sense the phrase "$B$ is a $K$-variety"). Let $E\to B$ the universal elliptic curve. For a given geometric point $b\to B$, the space $\Gamma(E_b,\Omega^1_{E_b/b})$ is a 1 dimension vector space (over some separably closed field defining $b$) where $E_b$ is, as usual, the fibre over $b$. Let $\{\omega_b\}_{b\to B}$ be a collection basis of $\Gamma(E_b,\Omega^1_{E_b/b})$ for all geometric points $b\to B$. Can we find a "global" $\omega$ giving $\omega_b$ for each geometric point $b\to B$? Probably this is not a good question to be answered. So, I try to make it in a less stupid way: I am wondering if there is an element $\omega$ of $\Gamma(E,\Omega^1_{E/B})$ which maps down to $\omega_b\in\Omega^1_{E/B,b}$. One may think that this is still very far from being well-posed. So, any suggestion for improving the statement of the question itself would also be very appreciated.