  We prove that the space of holomorphic $p$-forms on the moduli space
$\overline{\mathcal{M}}_{g,n}$ of stable curves of genus $g$ with $n$ marked
points vanishes for $p=14, 16, 18$ unconditionally and also for $p=20$ under a
natural assumption in the case $g=3$. This result is consistent with the
Langlands program and it is obtained by applying the Arbarello-Cornalba
inductive approach to the cohomology of moduli spaces.
