Degeneration of Riemann theta functions and of the Zhang-Kawazumi invariant with applications to a uniform Bogomolov conjecture

Wilms, Robert

doi:10.48550/arxiv.2101.04024

Cited by 3 publications

(4 citation statements)

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“…A related uniform bound on the essential height minima of smooth algebraic curves of a fixed genus g has been published recently by Wilms [71,Corollary 1.5]. However, it seems not yet to imply our proposition because a uniform treatment of points of height below the essential minimum is needed.…”

Section: Anmentioning

confidence: 84%

“…It should also be said that we opted for a simpler proof at the cost of introducing some additional assumptions in the proposition (e.g., the existence of a principal polarization). Most recently, Wilms [72] has communicated to the author a proof of the function field case of Theorem 2 using the same approach as in [71], with an explicit constant c 2 (g).…”

Section: Anmentioning

confidence: 99%

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Equidistribution in Families of Abelian Varieties and Uniformity

Kühne¹

2021

Preprint

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Using equidistribution techniques from Arakelov theory as well as recent results obtained by Dimitrov, Gao, and Habegger, we deduce uniform results on the Manin-Mumford and the Bogomolov conjecture. For each given integer g ≥ 2, we prove that the number of torsion points lying on a smooth complex algebraic curve of genus g embedded into its Jacobian is uniformly bounded. Complementing other recent work of Dimitrov, Gao, and Habegger, we obtain a rather uniform version of the Mordell-Lang conjecture as well. In particular, the number of rational points on a smooth algebraic curve defined over a number field can be bounded solely in terms of its genus and the Mordell-Weil rank of its Jacobian.

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Section: Anmentioning

confidence: 84%

Section: Anmentioning

confidence: 99%

Equidistribution in Families of Abelian Varieties and Uniformity

Kühne¹

2021

Preprint

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“…Finally, we note that Theorem B may be thought of as an analogue, in the nonarchimedean setting, of a remarkable identity established by Wilms [28,Theorem 1.1] between analytic invariants of Riemann surfaces. In fact, in [15,29], Theorem B is used together with [28,Theorem 1.1] to derive a formula for the asymptotic behavior of the so-called Zhang-Kawazumi invariant [20,21,31] in arbitrary one-parameter semistable degenerations of Riemann surfaces.…”

Section: Applications and Context For Our Formulamentioning

confidence: 99%

Tropical moments of tropical Jacobians

Jong

Shokrieh²

2022

Can. J. Math.-J. Can. Math.

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Each metric graph has canonically associated to it a polarized real torus called its tropical Jacobian. A fundamental real-valued invariant associated to each polarized real torus is its tropical moment. We give an explicit and efficiently computable formula for the tropical moment of a tropical Jacobian in terms of potential theory on the underlying metric graph. We show that there exists a universal linear relation between the tropical moment, a certain capacity called the tau invariant, and the total length of a metric graph. To put our formula in a broader context, we relate our work to the computation of heights attached to principally polarized abelian varieties.

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“…For g ≥ 2 and degeneration to isolated singularities, the asymptotic formula was later proved by [JS2,Thm. 7.1] and [Wil3,Cor. 1.2].…”

Section: Bigness In the Arithmetic Casementioning

confidence: 99%

Arithmetic bigness and a uniform Bogomolov-type result

Yuan¹

2021

Preprint

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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

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Degeneration of Riemann theta functions and of the Zhang-Kawazumi invariant with applications to a uniform Bogomolov conjecture

Cited by 3 publications

References 22 publications

Equidistribution in Families of Abelian Varieties and Uniformity

Equidistribution in Families of Abelian Varieties and Uniformity

Tropical moments of tropical Jacobians

Arithmetic bigness and a uniform Bogomolov-type result

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