$\overline{M}_{1,n}$ is usually not uniruled in characteristic $p$
Épijournal de Géométrie Algébrique, Tome 3 (2019)
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Using etale cohomology, we define a birational invariant for varieties in characteristic $p$ that serves as an obstruction to uniruledness - a variant on an obstruction to unirationality due to Ekedahl. We apply this to $\overline{M}_{1,n}$ and show that $\overline{M}_{1,n}$ is not uniruled in characteristic $p$ as long as $n \geq p \geq 11$. To do this, we use Deligne's description of the etale cohomology of $\overline{M}_{1,n}$ and apply the theory of congruences between modular forms.
@article{10_46298_epiga_2019_volume3_4134,
author = {Sawin, Will},
title = {$\overline{M}_{1,n}$ is usually not uniruled in characteristic $p$},
journal = {\'Epijournal de G\'eom\'etrie Alg\'ebrique},
publisher = {mathdoc},
volume = {3},
year = {2019},
doi = {10.46298/epiga.2019.volume3.4134},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.46298/epiga.2019.volume3.4134/}
}
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Sawin, Will. $\overline{M}_{1,n}$ is usually not uniruled in characteristic $p$. Épijournal de Géométrie Algébrique, Tome 3 (2019). doi: 10.46298/epiga.2019.volume3.4134
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