Let S be a scheme, $p:E\rightarrow S$ be an elliptic curve with zero section $e:S\rightarrow E$. My question is: Are $p_*(\Omega_{E/S}^1)$ and $e^*(\Omega_{E/S}^1)$ isomorphic? I think it's right because I saw both definitions in different notes and they seem to represent the same thing.
Thank you!
Yes, this is true.
Namely, note that [1, §4.2 Proposition 2] one has an isomorphism of $\mathcal{O}_E$-modules
$$p^\ast e^\ast\Omega^1_{E/S}\cong \Omega^1_{E/S}$$
Applying $p_\ast$ to both sides gives an isomorphism
$$p_\ast(p^\ast e^\ast \Omega^1_{E/S})\cong p_\ast \Omega^1_{E/S}$$
But, by the projection formula
$$\begin{aligned} p_\ast(p^\ast e^\ast\Omega^1_{E/S}) &=p_\ast(p^\ast e^\ast\Omega^1_{E/S}\otimes_{\mathcal{O}_E}\mathcal{O}_E)\\ &=e^\ast\Omega^1_{E/S}\otimes_{\mathcal{O}_S} p_\ast(\mathcal{O}_E)\\ &= e^\ast\Omega^1_{E/S}\otimes_{\mathcal{O}_S}\mathcal{O}_S\\ &= e^\ast \Omega^1_{E/S}\end{aligned}$$
where we have used the fact that since $p:E\to S$ is a smooth proper curve with geometrically connected fibers that $p_\ast\mathcal{O}_E\cong\mathcal{O}_S$.
[1] Bosch, S., Lütkebohmert, W. and Raynaud, M., 2012. Néron models (Vol. 21). Springer Science & Business Media.