A fusion variant of the classical and dynamical Mordell-Lang conjectures in positive characteristic

Publication date: 5 May 2022

Abstract: We study an open question at the interplay between the classical and the dynamical Mordell-Lang conjectures in positive characteristic. Let

K

be an algebraically closed field of positive characteristic, let

G

be a finitely generated subgroup of the multiplicative group of

K

, and let

X

be a (irreducible) quasiprojective variety defined over

K

. We consider

K

-valued sequences of the form

a_{n} : = f (v a r p h i^{n} (x_{0}))

, where

v a r p h i c o l o n X i g h t a r r o w X

and

f c o l o n X i g h t a r r o w m a t h b b P^{1}

are rational maps defined over

K

and

x_{0} i n X

is a point whose forward orbit avoids the indeterminacy loci of

v a r p h i

and

f

. We show that the set of

n

for which

a_{n} i n G

is a finite union of arithmetic progressions along with a set of upper Banach density zero. In addition, we show that if

a_{n} i n G

for every

n

and the

v a r p h i

orbit of

x

is Zariski dense in

X

then {there is} a multiplicative torus

m a t h b b G_{m}^{d}

and maps

P s i : m a t h b b G_{m}^{d} o m a t h b b G_{m}^{d}

and

g : m a t h b b G_{m}^{d} o m a t h b b G_{m}

such that

a_{n} = g c i r c P s i^{n} (y)

for some

y i n m a t h b b G_{m}^{d}

. We then describe various applications of our results.

Mathematics Subject Classification ID

Positive characteristic ground fields in algebraic geometry (14G17) Arithmetic dynamics on general algebraic varieties (37P55)

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