# Zhang's Conjecture and the Effective Bogomolov Conjecture over function fields

Zirve University

^{1}TL;DR: In this article, the Effective Bogomolov Conjecture was proved over a function field of characteristic 0 by proving Zhang's Conjectures about certain invariants of metrized graphs, which were previously known to be true only for curves of good reduction, for curve of genus at most 4 and a few other special cases.

Abstract: We prove the Effective Bogomolov Conjecture, and so the Bogomolov Conjecture, over a function field of characteristic 0 by proving Zhang’s Conjecture about certain invariants of metrized graphs. In the function field case, these conjectures were previously known to be true only for curves of good reduction, for curves of genus at most 4 and a few other special cases. We also either verify or improve the previous results. We relate the invariants involved in Zhang’s Conjecture to the tau constant of metrized graphs. Then we use and extend our previous results on the tau constant. By proving another Conjecture of Zhang, we obtain a new proof of the slope inequality for Faltings heights on moduli space of curves.

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TL;DR: In this article, the authors present measures on non-archimedean analytic varieties associated to metrized line bundles and some of its applications, as well as their applications.

Abstract: This paper has two goals. The first is to present the construction, due to the author, of measures on non-archimedean analytic varieties associated to metrized line bundles and some of its applications. We take this opportunity to add remarks, examples and mention related results.

48 citations

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TL;DR: In this paper, the second variation of the λ-invariant, recently introduced by S. Zhang, on the complex moduli space Mg of curves of genus g ≥ 2, using work of N. Kawazumi.

Abstract: We compute the second variation of the λ-invariant, recently introduced by S. Zhang, on the complex moduli space Mg of curves of genus g ≥ 2, using work of N. Kawazumi. As a result we prove that (8g +4)λ is equal, up to a constant, to the β-invariant introduced some time ago by R. Hain and D. Reed. We deduce some consequences; for example we calculate the λ-invariant for each hyperelliptic curve, expressing it in terms of the Petersson norm of the discriminant modular form. 1. Introduction. Recently, independently S. Zhang (27) and N. Kawazumi (16) introduced a new interesting real-valued function ϕ on the moduli space Mg of complex curves of genus g ≥2. Its value at a curve (X) ∈M g is given as follows. Let H 0 (X, ωX ) be the space of holomorphic differentials on X, equipped with the

26 citations

### Cites background or methods from "Zhang's Conjecture and the Effectiv..."

...jα : Cg[2]−→J (H)[2]...

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...It will be useful to pass from Mg , Cg and J (H) to the level-2 moduli orbifolds Mg[2], Cg[2] and J (H)[2]; see for example [10], Section 7....

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...The orbifold Mg[2] can be endowed with a universal theta characteristic α, i....

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...Actually, Cinkir in [2], Theorem 2....

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...On the other hand we have [2]∗wH = 4wH , which follows from the fact that wH restricts to a translation-invariant (1,1)-form in each fiber, and κ= 2jα which together give κ∗wH = 4j∗ α(wH)....

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Kyoto University

^{1}TL;DR: In this article, the minimal dimension of the components of a canonical measure on the tropicalization of the closed subvariety of an abelian variety is shown to be the same as that of the abelians.

Abstract: In this paper, we formulate the geometric Bogomolov conjecture for abelian varieties, and give some partial answers to it. In fact, we insist in a main theorem that under some degeneracy condition, a closed subvariety of an abelian variety does not have a dense subset of small points if it is a non-special subvariety. The key of the proof is the study of the minimal dimension of the components of a canonical measure on the tropicalization of the closed subvariety. Then we can apply the tropical version of equidistribution theory due to Gubler. This article includes an appendix by Walter Gubler. He shows that the minimal dimension of the components of a canonical measure is equal to the dimension of the abelian part of the subvariety. We can apply this result to make a further contribution to the geometric Bogomolov conjecture.

20 citations

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TL;DR: In this paper, the geometric Bogomolov conjecture over a function field of characteristic 0 was shown to hold for any function field with characteristic 0, and it was shown that the conjecture holds also for functions with characteristic 1.

Abstract: In the following, we prove the geometric Bogomolov conjecture over a function field of characteristic 0 .

18 citations

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TL;DR: The notion of monocritical toric metrized divisors was introduced in this article, where it was shown that for any generic D-small sequence of algebraic points of X and every place v of K, the sequence of their Galois orbits on the analytic space X an converges to a measure.

Abstract: We study the distribution of Galois orbits of points of small height on proper toric varieties, and its application to the Bogomolov problem. We introduce the notion of monocritical toric metrized divisor. We prove that a toric metrized divisor D on a proper toric variety X over a global eld K is monocritical if and only if for every generic D-small sequence of algebraic points of X and every place v of K, the sequence of their Galois orbits on the analytic space X an converges to a measure. When this is the case, the limit measure is a translate of the natural measure on the compact torus sitting in the principal orbit of X. The key ingredient is the study of the v-adic modulus distribution of Ga- lois orbits of generic D-small sequences of algebraic points. In particular, we characterize all their cluster measures. We generalize the Bogomolov problem by asking when a subvariety of the principal orbit of a proper toric variety that has the same essential minimum than the ambient variety, must be a translate of a subtorus. We prove that the generalized Bogomolov problem has a positive answer for monocritical toric metrized divisors, and we give several examples of toric metrized divisors for which the Bogomolov problem has a negative answer.

17 citations

##### References

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18 Jul 2002

TL;DR: In this paper, the authors present some topics in commutative algebra, including general properties of schemes, morphisms and base change, and local properties of sheaves of differentials.

Abstract: Introduction 1. Some topics in commutative algebra 2. General Properties of schemes 3. Morphisms and base change 4. Some local properties 5. Coherent sheaves and Cech cohmology 6. Sheaves of differentials 7. Divisors and applications to curves 8. Birational geometry of surfaces 9. Regular surfaces 10. Reduction of algebraic curves Bibilography Index

1,130 citations

### Additional excerpts

...We refer to [24 ]a nd [ 16 ] for the proofs of the mentioned facts....

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29 Aug 1983

TL;DR: In this article, the authors present a review of Lang's Diophantine Geometry, by L. J. Mordell and S. Lang, as well as a discussion of the relation between absolute values and proper sets of absolute values.

Abstract: 1 Absolute Values.- 2 Proper Sets of Absolute Values. Divisors and Units.- 3 Heights.- 4 Geometric Properties of Heights.- 5 Heights on Abelian Varieties.- 6 The Mordell-Weil Theorem.- 7 The Thue-Siegel-Roth Theorem.- 8 Siegel's Theorem and Integral Points.- 9 Hilbert's Irreducibility Theorem.- 10 Weil Functions and Neron Divisors.- 11 Neron Functions on Abelian Varieties.- 12 Algebraic Families of Neron Functions.- 13 Neron Functions Over the Complex Numbers.- Review of S. Lang's Diophantine Geometry, by L. J. Mordell.- Review of L. J. Mordell's Diophantine Equations, by S. Lang.

1,005 citations

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01 Jan 2000

TL;DR: In this paper, an introduction to diophantine geometry at the advanced graduate level is given, which contains a proof of the Mordell conjecture and several exercises for advanced graduate students and professional mathematicians.

Abstract: This is an introduction to diophantine geometry at the advanced graduate level. The book contains a proof of the Mordell conjecture which will make it quite attractive to graduate students and professional mathematicians. In each part of the book, the reader will find numerous exercises.

487 citations

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TL;DR: Gauthier-Villars as discussed by the authors implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.php).

Abstract: © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1988, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

267 citations