  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.
