Holomorphic differential forms on moduli spaces of stable curves

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.