THE DYNAMICAL MORDELL–LANG CONJECTURE FOR ENDOMORPHISMS OF SEMIABELIAN VARIETIES DEFINED OVER FIELDS OF POSITIVE CHARACTERISTIC. (24th March 2021)

• Record Type: Journal Article
• Authors: Corvaja, Pietro; Ghioca, Dragos; Scanlon, Thomas; Zannier, Umberto
• Is Part Of: Journal of the Institute of Mathematics of Jussieu. Volume 20:Number 2(2021)
• Page Start: 669
• Page End: 698
• Publication Date: 2021-03-24
• Subjects: 11G10, -- 37P55; dynamical Mordell–Lang problem, -- endomorphisms of semiabelian varieties defined over fields of characteristic p; Mathematics -- Periodicals; 510.5
• DOI: 10.1017/S1474748019000318
• Languages: English
• ISSNs: 1474-7480

Abstract

Abstract: Let $K$ be an algebraically closed field of prime characteristic $p$, let $X$ be a semiabelian variety defined over a finite subfield of $K$, let $\unicode[STIX]{x1D6F7}:X\longrightarrow X$ be a regular self-map defined over $K$, let $V\subset X$ be a subvariety defined over $K$, and let $\unicode[STIX]{x1D6FC}\in X(K)$ . The dynamical Mordell–Lang conjecture in characteristic $p$ predicts that the set $S=\{n\in \mathbb{N}:\unicode[STIX]{x1D6F7}^{n}(\unicode[STIX]{x1D6FC})\in V\}$ is a union of finitely many arithmetic progressions, along with finitely many $p$ -sets, which are sets of the form $\{\sum _{i=1}^{m}c_{i}p^{k_{i}n_{i}}:n_{i}\in \mathbb{N}\}$ for some $m\in \mathbb{N}$, some rational numbers $c_{i}$ and some non-negative integers $k_{i}$ . We prove that this conjecture is equivalent with some difficult diophantine problem in characteristic 0. In the case $X$ is an algebraic torus, we can prove the conjecture in two cases: either when $\dim (V)\leqslant 2$, or when no iterate of $\unicode[STIX]{x1D6F7}$ is a group endomorphism which induces the action of a power of the Frobenius on a positive dimensional algebraic subgroup of $X$ . We end by proving that Vojta's conjecture implies the dynamical Mordell–Lang conjecture for tori with no restriction.