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

2016 ◽  
Vol 30 (4) ◽  
pp. 1133-1163 ◽  
Author(s):  
Kazuhiko Yamaki
Keyword(s):  
Abelian Varieties ◽  
Bogomolov Conjecture

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

2018 ◽  
Vol 2018 (741) ◽  
pp. 133-159
Author(s):  
Kazuhiko Yamaki
Keyword(s):  
Abelian Variety ◽  
Function Field ◽  
Function Fields ◽  
Abelian Varieties ◽  
Constant Field ◽  
Abelian Schemes ◽  
Bogomolov Conjecture

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.

scholarly journals Strict supports of canonical measures and applications to the geometric Bogomolov conjecture

2015 ◽  
Vol 152 (5) ◽  
pp. 997-1040
Author(s):  
Kazuhiko Yamaki
Keyword(s):  
Recent Work ◽  
Dense Subset ◽  
Abelian Varieties ◽  
Special Kind ◽  
Number Fields ◽  
Good Reduction ◽  
Full Generality ◽  
Geometric Setting ◽  
Closed Subvariety ◽  
Bogomolov Conjecture

The Bogomolov conjecture claims that a closed subvariety containing a dense subset of small points is a special kind of subvariety. In the arithmetic setting over number fields, the Bogomolov conjecture for abelian varieties has already been established as a theorem of Ullmo and Zhang, but in the geometric setting over function fields, it has not yet been solved completely. There are only some partial results known such as the totally degenerate case due to Gubler and our recent work generalizing Gubler’s result. The key in establishing the previous results on the Bogomolov conjecture is the equidistribution method due to Szpiro, Ullmo and Zhang with respect to the canonical measures. In this paper we exhibit the limits of this method, making an important contribution to the geometric version of the conjecture. In fact, by the crucial investigation of the support of the canonical measure on a subvariety, we show that the conjecture in full generality holds if the conjecture holds for abelian varieties which have anywhere good reduction. As a consequence, we establish a partial answer that generalizes our previous result.

scholarly journals Classification of nonrigid families of abelian varieties

1993 ◽  
Vol 45 (2) ◽  
pp. 159-189
Author(s):  
Masa-Hiko Saitō
Keyword(s):  
Abelian Varieties
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