Global section of vertical differential 1 forms on universal elliptic curve

Question

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.

Answer 1

What you're looking for is a section of the sheaf $\omega = \pi_* \Omega^1_{E/B}$, where $\pi: E \to B$ is the structure map, which is a line bundle on $B$. This line bundle $\omega$ has a canonical extension to the compactification $\bar{B}$, and global sections of this line bundle over $\bar{B}$ are exactly weight 1 modular forms (and more generally $H^0(\bar{B}, \omega^{\otimes k})$ is the space of weight $k$ modular forms).