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The Bogomolov conjecture for a curve claims finiteness of algebraic points onthe curve which are small with respect to the canonical height. Ullmo hasestablished this conjecture over number fields, and Moriwaki generalized it tothe assertion over finitely generated fields over $\mathbb{Q}$ with respect toarithmetic heights. As for the case of function fields with respect to thegeometric heights, Cinkir has proved the conjecture over function fields ofcharacteristic $0$ and of transcendence degree $1$. However, the conjecture hasbeen open over other function fields.  In this paper, we prove that the Bogomolov conjecture for curves holds overany function field. In fact, we show that any non-special closed subvariety ofdimension $1$ in an abelian variety over function fields has only a finitenumber of small points. This result is a consequence of the investigation ofnon-density of small points of closed subvarieties of abelian varieties ofcodimension $1$. As a by-product, we remark that the geometric Bogomolovconjecture, which is a generalization of the Bogomolov conjecture for curvesover function fields, holds for any abelian variety of dimension at most $3$.Combining this result with our previous works, we see that the geometricBogomolov conjecture holds for all abelian varieties for which the differencebetween its nowhere degeneracy rank and the dimension of its trace is notgreater than $3$.

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