THE DYNAMICAL MORDELL–LANG CONJECTURE FOR ENDOMORPHISMS OF SEMIABELIAN VARIETIES DEFINED OVER FIELDS OF POSITIVE CHARACTERISTIC. (24th March 2021) - Corvaja, Pietro; Ghioca, Dragos; Scanlon, Thomas; Zannier, Umberto
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 [STIX]x1D6F7:XX be a regular self-map defined over K, let VX be a subvariety defined over K, and let [STIX]x1D6FCX(K) . The dynamical Mordell–Lang conjecture in characteristic p predicts that the set S={nN:[STIX]x1D6F7^n([STIX]x1D6FC)V} is a union of finitely many arithmetic progressions, along with finitely many p -sets, which are sets of the form {_i=1^mc_ip^k_in_i:n_iN} for some mN, 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 (V)2, or when no iterate of [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.