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Preprint Number 1859

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1859. Jason Bell, Dragos Ghioca, and Rahim Moosa
Effective isotrivial Mordell-Lang in positive characteristic
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Submission date: 16 October 2020

Abstract:

The isotrivial Mordell-Lang theorem of Moosa-Scanlon describes the intersection of X and Gamma when X is a subvariety of a semiabelian variety G over a finite field and Gamma is a finitely generated subgroup of G that is invariant under the Frobenius endomorphism F. That description is here made effective, and extended to arbitrary commutative algebraic groups G and arbitrary finitely generated Z[F]-submodules Gamma. The approach is to use finite automata to give a concrete description of the intersection. These methods and results have new applications even when specialised to the case when G is an abelian variety over a finite field, X is a subvariety defined over a function field K, and Gamma = G(K). As an application of the automata-theoretic approach, a dichotomy theorem is established for the growth of the number of points in X(K) of bounded height. As an application of the effective description of their intersection, decision procedures are given for the following three diophantine problems: Is X(K) nonempty? Is it infinite? Does it contain an infinite coset?

Mathematics Subject Classification: 14Q20, 14G05, 14G17, 11B85, 11G10

Keywords and phrases: effectivity, commutative algebraic groups, Mordell-Lang, automata, F-sets

Full text arXiv 2010.08579: pdf, ps.


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