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

## TL;DR

The paper introduces a new birational invariant in characteristic p that obstructs uniruledness, and demonstrates that the moduli space ar{M}_{1,n} is not uniruled for certain n and p using etale cohomology and modular forms.

## Contribution

It defines a novel birational invariant in characteristic p and applies it to show ar{M}_{1,n} is not uniruled when n nd p nd p 1, extending understanding of moduli spaces in positive characteristic.

## Key findings

- ar{M}_{1,n} is not uniruled for n nd p nd p 1.
- A birational invariant obstructing uniruledness is constructed using etale cohomology.
- Application of modular forms theory to algebraic geometry 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 $n \geq p \geq 11$. 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.