Effective isotrivial Mordell-Lang in positive characteristic
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866914987399708672 |
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| author | Bell, Jason Ghioca, Dragos Moosa, Rahim |
| author_facet | Bell, Jason Ghioca, Dragos Moosa, Rahim |
| contents | The isotrivial Mordell-Lang theorem of Moosa and Scanlon describes the set $X\capΓ$ when $X$ is a subvariety of a semiabelian variety $G$ over a finite field $\mathbb{F}_q$ and $Γ$ is a finitely generated subgroup of $G$ that is invariant under the $q$-power Frobenius endomorphism $F$. That description is here made effective, and extended to arbitrary commutative algebraic groups $G$ and arbitrary finitely generated $\mathbb{Z}[F]$-submodules $Γ$. The approach is to use finite automata to give a concrete description of $X\cap Γ$. These methods and results have new applications even when specialised to the case when $G$ is an abelian variety over a finite field, $X\subseteq G$ a subvariety defined over a function field $K$, and $Γ=G(K)$. 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 $X\capΓ$, decision procedures are given for the following three diophantine problems: Is $X(K)$ nonempty? Is it infinite? Does it contain an infinite coset? |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_08579 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Effective isotrivial Mordell-Lang in positive characteristic Bell, Jason Ghioca, Dragos Moosa, Rahim Number Theory Algebraic Geometry The isotrivial Mordell-Lang theorem of Moosa and Scanlon describes the set $X\capΓ$ when $X$ is a subvariety of a semiabelian variety $G$ over a finite field $\mathbb{F}_q$ and $Γ$ is a finitely generated subgroup of $G$ that is invariant under the $q$-power Frobenius endomorphism $F$. That description is here made effective, and extended to arbitrary commutative algebraic groups $G$ and arbitrary finitely generated $\mathbb{Z}[F]$-submodules $Γ$. The approach is to use finite automata to give a concrete description of $X\cap Γ$. These methods and results have new applications even when specialised to the case when $G$ is an abelian variety over a finite field, $X\subseteq G$ a subvariety defined over a function field $K$, and $Γ=G(K)$. 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 $X\capΓ$, decision procedures are given for the following three diophantine problems: Is $X(K)$ nonempty? Is it infinite? Does it contain an infinite coset? |
| title | Effective isotrivial Mordell-Lang in positive characteristic |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2010.08579 |