Geometric Bogomolov conjecture for abelian varieties and some results for those with some degeneration (with an appendix by Walter Gubler: the minimal dimension of a canonical measure)

2016

J. Amer. Math. Soc.

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Abstract. The Bogomolov conjecture for a curve claims finiteness of algebraic points on the curve which are small with respect to the canonical height. Ullmo has established this conjecture over number fields, and Moriwaki generalized it to the assertion over finitely generated fields over Q with respect to arithmetic heights. As for the case of function fields with respect to the geometric heights, Cinkir has proved the conjecture over function fields of characteristic 0 and of transcendence degree 1. However, the conjecture has been open over other function fields.In this paper, we prove that the Bogomolov conjecture for curves holds over any function field. In fact, we show that any non-special closed subvariety of dimension 1 in an abelian variety over function fields has only a finite number of small points. This result is a consequence of the investigation of non-density of small points of closed subvarieties of abelian varieties of codimension 1. As a by-product, we remark that the geometric Bogomolov conjecture, which is a generalization of the Bogomolov conjecture for curves over function fields, holds for any abelian variety of dimension at most 3. Combining this result with our previous works, we see that the geometric Bogomolov conjecture holds for all abelian varieties for which the difference between its nowhere degeneracy rank and the dimension of its trace is not greater than 3.

“…Conjecture 1.5 (Conjecture 2.9 of [29]). Let A be an abelian variety over K. Let X be a closed subvariety of A.…”

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confidence: 99%

Non-density of small points on divisors on Abelian varieties and the Bogomolov conjecture

2016

J. Amer. Math. Soc.

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“…Unlike the arithmetic cases, some abelian varieties over function fields actually have closed subvarieties which are not torsion subvarieties but have dense small points. Taking into account that fact, we have proposed in [6] [6]). Let A be an abelian variety over K. Let X be a closed subvariety of A.…”

Section: Definition 22])mentioning

confidence: 99%

“…Here, we recall the definition of special subvarieties defined in [6]. To define them, we use the K/k-trace of A.…”

Section: Definition 22])mentioning

confidence: 99%

See 1 more Smart Citation

Geometric Bogomolov conjecture for nowhere degenerate abelian varieties of dimension $5$ with trivial trace

Mathematical Research Letters

2017

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“…It follows from [21, Lemma 2.1] that whether or not X(ǫ; L) is dense in X for any ǫ > 0 does not depend on the choice of even ample L. Therefore, it makes sense to say that X has dense small points if X(ǫ; L) is dense in X for any ǫ > 0 and for some (and hence any) even ample line bundle L on A (cf. [21,Definition 2.2]).…”

Section: Introductionmentioning

confidence: 99%

Trace of abelian varieties over function fields and the geometric Bogomolov conjecture

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

2016

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Abstract. We prove that the geometric Bogomolov conjecture for any abelian varieties is reduced to that for nowhere degenerate abelian varieties with trivial trace. In particular, the geometric Bogomolov conjecture holds for abelian varieties whose maximal nowhere degenerate abelian subvariety is isogenous to a constant abelian variety. To prove the results, we investigate closed subvarieties of abelian schemes over constant varieties, where constant varieties are varieties over a function field which can be defined over the constant field of the function field.