A translation generalized quadrangle in characteristic $\ne 0$ is linear

Abstract

It is a long-standing conjecture from the 1970s that every translation generalized quadrangle is linear, that is, has an endomorphism ring which is a division ring (or, in geometric terms, that has a projective representation). We show that any translation generalized quadrangle $\Gamma$ is ideally embedded in a translation quadrangle which is linear. This allows us to weakly represent any such $\Gamma$ in projective space, and moreover, to have a well-defined notion of "characteristic" for these objects. We then show that each translation quadrangle in positive characteristic indeed is linear.