${\overline{M}}_{1,n}$ is usually not uniruled in characteristic $p$

Abstract

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 $n\ge p\ge 11$. 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.

arXiv comment10 pages, published version

References in corpus (1)

• Varieties over a finite field with trivial Chow group of 0-cycles have a rational point