Obstructions to unirationality for product-quotient surfaces over $\overline{\mathbb{F}}_p$

TL;DR

This paper constructs a supersingular surface over an algebraic closure of a finite field with trivial étale fundamental group that is not unirational, providing a counterexample to Shioda's 1977 conjecture and introducing new obstructions to unirationality.

Contribution

It presents the first known counterexample to Shioda's conjecture by constructing a non-unirational, supersingular surface with trivial fundamental group over _p, and develops new obstructions to unirationality for product-quotient surfaces.

Findings

Constructed a supersingular surface with trivial _p fundamental group that is not unirational.

Provided counterexample to Shioda's 1977 conjecture.

Introduced new obstructions to unirationality for product-quotient surfaces.

Abstract

We construct a surface over $\overline{\mathbb{F}}_p$ with $\pi_1^{\'{e}t}(X) = 1$ that is supersingular -- in the sense that $H^2_{\'{e}t}(X, \mathbb{Q}_{\ell}(1))$ is spanned by algebraic cycles -- but is not unirational. This provides a counterexample to a 1977 conjecture of Shioda. To achieve this, we produce new obstructions to unirationality for product-quotient surfaces.