Let $M$ be a compactified moduli stack of elliptic curves which is considered as a stack over a field $L$. Then modular forms (which are homolorphic at the cusp) of weight $2k$ are nothing but sections of the $k$-th power of the Hodge bundle which comes from the universal family. I have two questions.
is there any good intuition or a modular interpretation of sections of $\Omega^{1}_{M/L}$?
If $L$ is a finite field or, more generally, a field of positive characteristic. Is it still a reasonable thing to consider sections of the Hodge bundle and think about them as of "modular forms in char. p"? Especially it is interesting if the characteristic is $2$ or $3$.