Arithmetic hyperbolicity: automorphisms and persistence

Javanpeykar, Ariyan

doi:10.1007/s00208-021-02155-0

Cited by 13 publications

(36 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If 𝑋 𝐿 is arithmetically hyperbolic over 𝐿 for all algebraically closed field extensions 𝐿 ⊃ 𝑘, then we say that 𝑋 is absolutely arithmetically hyperbolic. This extends to quasi-projective varieties Lang's notions [36] of the Mordell/Siegel property for projective/affine varieties; see also [3,55,59] and [6,20,21,[24][25][26]30]. We refer to [3, 8, 14-16, 41, 45, 57, 58] for examples of arithmetically hyperbolic varieties.…”

Section: Introductionmentioning

confidence: 98%

Urata's theorem in the logarithmic case and applications to integral points

Javanpeykar

Levin

2022

Bulletin of London Math Soc

Self Cite

View full text Add to dashboard Cite

Urata showed that a pointed compact hyperbolic variety admits only finitely many maps from a pointed curve. We extend Urata's theorem to the setting of (not necessarily compact) hyperbolically embeddable varieties. As an application, we show that a hyperbolically embeddable variety over a number field 𝐾 with only finitely many  𝐿,𝑇 -points for any number field 𝐿∕𝐾 and any finite set of finite places 𝑇 of 𝐿 has, in fact, only finitely many points in any given ℤ-finitely generated integral domain of characteristic zero. We use this latter result in combination with Green's criterion for hyperbolic embeddability to obtain novel finiteness results for integral points on symmetric self-products of smooth affine curves and on complements of large divisors in projective varieties. Finally, we use a partial converse to Green's criterion to further study hyperbolic embeddability (or its failure) in the case of symmetric self-products of curves. As a by-product of our results, we obtain the first example of a smooth affine Brody-hyperbolic threefold over ℂ which is not hyperbolically embeddable.

show abstract

Section: Introductionmentioning

confidence: 98%

Urata's theorem in the logarithmic case and applications to integral points

Javanpeykar

Levin

2022

Bulletin of London Math Soc

Self Cite

View full text Add to dashboard Cite

show abstract

“…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

View full text Add to dashboard Cite

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.

show abstract

“…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

View full text Add to dashboard Cite

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: automorphisms and persistence

Cited by 13 publications

References 31 publications

Urata's theorem in the logarithmic case and applications to integral points

Urata's theorem in the logarithmic case and applications to integral points

Unpolarized Shafarevich conjectures for hyper-Kähler varieties

Unpolarized Shafarevich conjectures for hyper-Kähler varieties

Contact Info

Product

Resources

About