Finite projective planes, Fermat curves, and Gaussian periods

Abstract

One of the oldest and most fundamental problems in the theory of finite projective planes is to classify those having a group which acts transitively on the incident point-line pairs (flags). The conjecture is that the only ones are the Desarguesian projective planes (over a finite field). In this paper, we show that non-Desarguesian finite flag-transitive projective planes exist if and only if certain Fermat surfaces have no nontrivial rational points, and formulate several other equivalences involving Fermat curves and Gaussian periods. In particular, we show that a non-Desarguesian flag-transitive projective plane of order $n$ exists if and only if $n>8$, the number $p=n^2+n+1$ is prime, and the square of the absolute value of the Gaussian period ${\sum }_{a\in {\mathcal{D}}_n}{\zeta }^a$ ($\zeta =$ primitive $p$th root of unity, ${\mathcal{D}}_n=$ group of $n$th powers in ${\mathbb{F}}_p^{\times }$) belongs to $\mathbb{Z}$. We also formulate a conjectural classification of all pairs $(p,n)$ with $p$ prime and $n\mid p-1$ having this latter property, and give an application to the construction of symmetric designs with flag-transitive automorphism groups. Numerical computations are described verifying the first conjecture for $p<4\times 10^{22}$ and the second for $p<10^7$.