On the number of even latin squares of even order

We show that for each reduced odd latin square of even order there exists at least one map such that its image is a reduced even latin square of the same order. We prove that this map is injective. As a consequence, we can show that the number of even latin squares of even order is bounded from below by the number of odd latin squares of the same order. This gives a positive answer to the Alon-Tarsi conjecture on even latin squares