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I have seen two equivalent definitions of the modular sheaf $\omega$. Let $S$ be some base scheme. If $p \colon \mathcal{E} \to X$ is the universal generalized elliptic curve over the modular curve $X$, and $e \colon X \to \mathcal{E}$ is the zero section (so that $e(x)$ is the identity in the fiber $p^{-1}(x)$), then $$ \omega = p_*\Omega^1_{\mathcal{E}/S} = e^*\Omega^1_{\mathcal{E}/S}. $$ Here $\Omega^1_{\mathcal{E}/S}$ is the sheaf of relative 1-forms on $\mathcal{E}$ with logarithmic poles at the cusps.

I believe that "the reason" that these are the same is because $\Omega^1_{E/S}$ is a trivial line bundle for any generalized elliptic curve $E = p^{-1}(x)$, and its sections are all constant. This gives a natural way to identify the sections of $e^*\Omega^1_{\mathcal{E}/S}$ with the sections of $p_*\Omega^1_{\mathcal{E}/S}$: just extend any section of $e^*\Omega^1_{\mathcal{E}/S}$ to all of $\Omega^1_{\mathcal{E}/S}$ by making it constant along the fibers, which we can do because it's a trivial bundle. The property above says that any section of $p_*\Omega^1_{\mathcal{E}/S}$ will be constant along the fibers anyway, so we get all of them.

My question is, in how much generality can we expect a natural isomorphism $e^*\mathcal{F} \cong p_*\mathcal{F}$ where $p \colon T \to X$ is a fiber bundle, $e \colon X \to T$ is a section, and $\mathcal{F}$ is a sheaf on $T$? Does my reasoning from the previous paragraph have to be true, or can we weaken it? (I know this is supposed to be true for, for example, Hilbert and Siegel modular forms, but I think the reasoning above applies in those cases.)

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The condition that $e^*(\mathcal{F})$ be naturally identified with $p^*(\mathcal{F})$ can be tautologically interpreted as saying that for any open set $U$ in $X$, any section of $\mathcal{F}$ on $p^{-1}U$ is uniquely determined by its restriction to the subset $e(U)$.

For example, if $\mathcal{F}$ is a line bundle and $p$ has connected fibers, then it is necessary and sufficient that $p_*(\mathcal{F})$ be a line bundle. For another example, if $\mathcal{F}$ is a sheaf of relative differentials, then we need all global differentials to be uniquely determined by their restriction to $e$. When $p$ is a family of principally polarized abelian varieties, this gives the Siegel modular forms that you mention.

A slightly more exotic class of examples is given by sheaves $\mathcal{F}$ that are scheme-theoretically supported on the image of $e$.

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    $\begingroup$ Maybe one can add that there is always a natural map $p_* \mathcal F \to e^*\mathcal F$, and one wants to check if this is an isomorphism. This isomorphism can be checked locally, and thus your answer. Otherwise it’s slightly ambiguous how to make precise the notion of natural isomorphism here. $\endgroup$
    – Will Sawin
    Oct 5 '19 at 16:24
  • $\begingroup$ Thank you @WillSawin, that is a very clarifying observation. $\endgroup$
    – S. Carnahan
    Oct 5 '19 at 16:28
  • $\begingroup$ So with the map Will Sawin mentioned, and the fact that the isomorphism can be checked locally, we just need that both sheaves are line bundles and the map $p_*\mathcal{F} \to e^*\mathcal{F}$ is nonzero on each stalk. In the case that we get a vector bundle or another sheaf, checking the isomorphism is more complicated; when you say it can be "checked locally" you mean we can check on small opens, can we check on each stalk? $\endgroup$
    – Jon Aycock
    Oct 6 '19 at 19:49

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