Arithmetic hyperbolicity and a stacky Chevalley–Weil theorem

Javanpeykar, Ariyan; Loughran, Daniel

doi:10.1112/jlms.12394

Cited by 17 publications

(34 citation statements)

References 53 publications

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“…Let k be an algebraically closed field of characteristic zero. Following [25,Sect. 4], we say that a finite type separated scheme X over k is arithmetically hyperbolic over k if, for all Z-finitely generated subrings A ⊂ k and all finite type separated schemes X over A with X k ∼ = X , the set X (A) is finite.…”

Section: The Green-griffiths-lang Conjecturementioning

confidence: 99%

Arithmetic hyperbolicity: automorphisms and persistence

Javanpeykar

2021

Math. Ann.

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We show that if the automorphism group of a projective variety is torsion, then it is finite. Motivated by Lang’s conjecture on rational points of hyperbolic varieties, we use this to prove that a projective variety with only finitely many rational points has only finitely many automorphisms. Moreover, we investigate to what extent finiteness of S-integral points on a variety over a number field persists over finitely generated fields. To this end, we introduce the class of mildly bounded varieties and prove a general criterion for proving this persistence.

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Section: The Green-griffiths-lang Conjecturementioning

confidence: 99%

Arithmetic hyperbolicity: automorphisms and persistence

Javanpeykar

2021

Math. Ann.

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“…intermediate Jacobian): complete intersections of Hodge niveau ≤ 1, prime Fano threefolds of index 2, sextic surfaces etc. These results can often be reinterpreted as the finiteness of O F,S -points in certain moduli spaces; see works of Javanpeykar and his coauthors [29], [27], [30], [31] for related studies from this point of view of arithmetic hyperbolicity. More recently, Lawrence and Sawin [37] proved some analogous finiteness result for hypersurfaces in abelian varieties based on the techniques in [38].…”

Section: Introductionmentioning

confidence: 99%

Unpolarized Shafarevich conjectures for hyper-Kähler varieties

Fu¹,

Li²,

Zou³

2022

Preprint

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Shafarevich conjecture/problem is about the finiteness of isomorphism classes of a family of varieties defined over a number field with good reduction outside a finite collection of places. For K3 surfaces, such a finiteness result was proved by Y. She. For hyper-Kähler varieties, which are higher dimensional analogues of K3 surfaces, Y. André has verified the Shafarevich conjecture for hyper-Kähler varieties of a given dimension and admitting a very ample polarization of bounded degree. In this paper, we provide a unification of both results by proving the (unpolarized) Shafarevich conjecture for hyper-Kähler varieties in a given deformation type. In a similar fashion, generalizing a result of Orr and Skorobogatov on K3 surfaces, we prove the finiteness of geometric isomorphism classes of hyper-Kähler varieties of CM type in a given deformation type defined over a number field with bounded degree. A key to our approach is a uniform Kuga-Satake map, inspired by She's work, and we study its arithmetic properties, which are of independent interest.

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“…These results can often be reinterpreted as the finiteness of O K,S -points in certain moduli spaces; see works of Javanpeykar and his coauthors [36], [32], [37], [38] for related studies from this point of view of arithmetic hyperbolicity. More recently, Lawrence and Sawin [46] proved some analogous finiteness result for hypersurfaces in abelian varieties based on the techniques in [47].…”

Section: Introductionmentioning

confidence: 99%

Unpolarized Shafarevich conjectures for hyper-Kähler varieties

Fu¹,

Li²,

Takamatsu³

et al. 2022

Preprint

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The Shafarevich conjecture/problem is about the finiteness of isomorphism classes of a family of varieties defined over a number field with good reduction outside a finite collection of places. For K3 surfaces, such a finiteness result was proved by Y. She. For hyper-Kähler varieties, which are higher-dimensional analogues of K3 surfaces, Y. André proved the Shafarevich conjecture for hyper-Kähler varieties of a given dimension and admitting a very ample polarization of bounded degree. In this paper, we provide a unification of both results by proving the (unpolarized) Shafarevich conjecture for hyper-Kähler varieties in a given deformation type. We also discuss the cohomological generalization of the Shafarevich conjecture by replacing the good reduction condition by the unramifiedness of the cohomology, where our results are subject to certain necessary assumption on the faithfulness of the action of the automorphism group on cohomology. In a similar fashion, generalizing a result of Orr and Skorobogatov on K3 surfaces, we prove the finiteness of geometric isomorphism classes of hyper-Kähler varieties of CM type in a given deformation type defined over a number field with bounded degree. A key to our approach to these results is a uniform Kuga-Satake map, inspired by She's work, and we study its arithmetic properties, which are of independent interest.

show abstract

Arithmetic hyperbolicity and a stacky Chevalley–Weil theorem

Cited by 17 publications

References 53 publications

Arithmetic hyperbolicity: automorphisms and persistence

Arithmetic hyperbolicity: automorphisms and persistence

Unpolarized Shafarevich conjectures for hyper-Kähler varieties

Unpolarized Shafarevich conjectures for hyper-Kähler varieties

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