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THE DYNAMICAL MORDELL–LANG CONJECTURE FOR ENDOMORPHISMS OF SEMIABELIAN VARIETIES DEFINED OVER FIELDS OF POSITIVE CHARACTERISTIC

  • Pietro Corvaja (a1), Dragos Ghioca (a2), Thomas Scanlon (a3) and Umberto Zannier (a4)

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.

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The second author has been partially supported by a discovery grant from the National Science and Engineering Board of Canada. The third author has been partially supported by grant DMS-1363372 of the United States National Science Foundation and a Simons Foundation Fellowship.

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THE DYNAMICAL MORDELL–LANG CONJECTURE FOR ENDOMORPHISMS OF SEMIABELIAN VARIETIES DEFINED OVER FIELDS OF POSITIVE CHARACTERISTIC

  • Pietro Corvaja (a1), Dragos Ghioca (a2), Thomas Scanlon (a3) and Umberto Zannier (a4)

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