Erratum to “ Model Theory and Diophantine Geometry ”

Our informal description of the modularity of a definable set X of Morley dimension and degree 1 should have been: “There is no n-dimensional definable family of definable subsets of X × X with n ≥ 2, each of Morley dimension and Morley degree 1 and with pairwise intersections finite” in place of “There should be no infinite definable family of definable subsets of X×X , each of Morley dimension and Morley degree 1 and with pairwise intersections finite.” (See page 415.) Thanks to Gregory Cherlin for pointing out the mistake. I also mistakenly omitted some important historical and mathematical references concerning the Mordell-Lang conjecture in positive characteristic. Firstly, the Mordell-Lang conjecture for function fields in characteristic p > 0, as proved by Hrushovski (the positive characteristic case of Theorem 2.1 in my paper), was first raised in this precise form by Abramovich and Voloch [1] (where some special cases were proved). In an earlier paper [2] Voloch proved the special case where X is a projective variety embedded in its Jacobian A where A is ordinary. It is also in this paper of Voloch’s that the idea of considering only the prime-to-p division points of a finitely generated subgroup of A appears. This restriction to the prime-to-p division points makes the problem tractable, even for Hrushovski. On the other hand, Voloch has pointed out to me that there is no evidence for not including the p-division points in Γ too. This more general case (where Γ is the group of all division points of some finitely generated subgroup of the semi-abelian variety A) remains open.