We show that the Deligne-Mumford moduli space of genus 1 curves with n marked points is not unirational in characteristic p, for n sufficiently large with respect to p, when p>7. This statement, as well as the non-unirationality of many other Deligne-Mumford moduli spaces, was well-known in characteristic zero, but is more difficult in characteristic p. Our method uses the action of Frobenius on the etale cohomology of M_{1,n}. Following Deligne, we can compute this cohomology in terms of modular forms. The unirationality then follows from classical results on modular forms modulo p.
Speaker(s): Will Sawin (Princeton/ETH)