On the Manin–Mumford and Mordell–Lang conjectures in positive characteristic

Abstract

We prove that in positive characteristic, the Manin-Mumford conjecture implies the Mordell-Lang conjecture, in the situation where the ambient variety is an abelian variety defined over the function field of a smooth curve over a finite field and the relevant group is a finitely generated group. In particular, in the setting of the last sentence, we provide a proof of the Mordell-Lang conjecture, which does not depend on tools coming from model theory.

Sparsity of p-divisible unramified liftings for subvarieties of abelian varieties with trivial stabilizer

“…Th. 4.1 in [Rös13]). Theorem 1.1 makes Rössler's result effective, showing that if the stabilizer of X is trivial, then it is sufficient to consider the map (2) for m = 1.…”

Infinitely p-Divisible Points on Abelian Varieties Defined over Function Fields of Characteristic p>0

“…Lemma 2.4 below (which plays a key role in the proof of Conjecture 1.1) implies that the infinitely p-divisible points defined over a certain separable (but transcendental) field extension of K 0 are annulled by a fixed Weil polynomial (in particular it has no roots of unity among its roots) applied to a lifting of the Frobenius automorphism. This fact, combined with the existence of arc schemes (see for instance [23, before Lemma 2.3])) as well as Proposition 6.1 in [19] can be used to give a quick proof of Theorem 4.1 in [23]. This last theorem is the main tool in the proof of the Mordell-Lang conjecture given in [23]; more precisely, the Mordell-Lang conjecture follows quickly from it (see par.…”

“…This fact, combined with the existence of arc schemes (see for instance [23, before Lemma 2.3])) as well as Proposition 6.1 in [19] can be used to give a quick proof of Theorem 4.1 in [23]. This last theorem is the main tool in the proof of the Mordell-Lang conjecture given in [23]; more precisely, the Mordell-Lang conjecture follows quickly from it (see par. 3.2 in [23] for the argument), once the existence of jet schemes (see for instance par.…”

“…This last theorem is the main tool in the proof of the Mordell-Lang conjecture given in [23]; more precisely, the Mordell-Lang conjecture follows quickly from it (see par. 3.2 in [23] for the argument), once the existence of jet schemes (see for instance par. 2.1 in [23] for the latter) is established.…”

Mordell–Lang in positive characteristic

“…An algebro-geometric proof of Theorem 1.1 has previously been given by Rössler [24]. We give an overview of the structure of this article.…”