Recent developments of the Uniform Mordell-Lang Conjecture

Gao, Ziyang

doi:10.48550/arxiv.2104.03431

Cited by 2 publications

(13 citation statements)

References 59 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is explained [Gao21,Lem.10.4] that Theorem 1.1 self-improves to the following stronger statement conjectured by Conj.1.8].…”

Section: Introductionmentioning

confidence: 67%

“…When X ⊆ A is a geometrically connected smooth projective curve C of genus g ≥ 2 embedded into its Jacobian J, i.e. Mazur's question [Maz86, top of pp.234], Theorem 1.1 has recently been proved by Dimitrov, Habegger, the first-and third-named authors [DGH20b,Küh21]; see the survey [Gao21] for a summary and more detailed comments.…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 1.1 ′ ) compared to [Gao21, Conj.10.3] (resp. [Gao21,Conj.10.3'] which is [DP07, Conj.1.8]) is that deg L A is no longer involved in the constant. This can be done because Theorem 1.1 and Theorem 1.1 ′ can easily be reduced to the case where X generates A, and then deg L A is bounded from above solely in terms of g and deg L X; see Lemma 2.5.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Uniform Mordell-Lang Conjecture

Gao,

Ge,

Kühne

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…It is explained [Gao21,Lem.10.4] that Theorem 1.1 self-improves to the following stronger statement conjectured by Conj.1.8].…”

Section: Introductionmentioning

confidence: 67%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Uniform Mordell-Lang Conjecture

Gao,

Ge,

Kühne

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…In the above proof, we have used the following basic result, which improves [DGH1,Lem. 6.3] and [Gao2,Lem. 7.3].…”

Section: Consequence On Small Pointsmentioning

confidence: 99%

“…

Recently, a uniform version of this theorem was proved by Dimitrov-Gao-Habegger [DGH1] and Kühne [Kuh]. In fact, the new gap principle in [Gao2, Thm. 4.1], as a combination of [DGH1, Prop.

…”

mentioning

confidence: 99%

Arithmetic bigness and a uniform Bogomolov-type result

Yuan¹

2021

Preprint

View full text Add to dashboard Cite

Recently, a uniform version of this theorem was proved by Dimitrov-Gao-Habegger [DGH1] and Kühne [Kuh]. In fact, the new gap principle in [Gao2, Thm. 4.1], as a combination of [DGH1, Prop. 7.1] and [Kuh, Thm. 3], asserts that there are constants c 1 , c 2 > 0 depending only on g > 1 such that for any projective and smooth curve C over Q of genus g, and for anyHere h Fal (C) = h Fal (J) denotes the stable Faltings height of the Jacobian variety J.The new gap principle has a significant consequence to the uniform Mordell-Lang problem proposed by Mazur [Maz, pp.234]. Recall that the Mordell conjecture was proved by Faltings [Fal1], and a different proof was given by Vojta [Voj]. Vojta's proof was simplified and extended by Faltings [Fal2] to prove the Mordell-Lang conjecture for subvarieties of abelian varieties, and was further simplified by Bombieri [Bom] in the original case of curves. The proofs of [Voj,Fal2,Bom] actually gave an upper bound of the number of points of large heights, which was further refined by de Diego [dDi] and Rémond [Rem]. Combining the upper bound with the new gap principle, we obtain the uniform bound on the number of rational points in [Kuh, Thm. 4], which asserts that there is a constant c > 0 depending only on g > 1 such that for any projective and smooth curve C of genus g over an algebraically closed field F of characteristic 0, for any y ∈ C(F ), and for any subgroup Γ ⊂ J(F ) of finite rank,Our first result is the following uniform version of the Bogomolov conjecture, which strengthens and generalizes the new gap principle of [DGH1,Kuh].Theorem 1.1 (Theorem 4.6). Let K be either the field Q or the field k(t) for a field k. Let g > 1 be a positive integer. There are constants c 1 , c 2 > 0 depending only on g such that for any projective and smooth curve C over K of genus g, and for any line bundle α ∈ Pic(C) of degree 1, with the extra assumption that (C, α) is non-isotrivial over k in the case K = k(t), one has

show abstract

Recent developments of the Uniform Mordell-Lang Conjecture

Cited by 2 publications

References 59 publications

The Uniform Mordell-Lang Conjecture

The Uniform Mordell-Lang Conjecture

Arithmetic bigness and a uniform Bogomolov-type result

Contact Info

Product

Resources

About