Equidistribution in Families of Abelian Varieties and Uniformity

Abstract

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.

Arithmetic bigness and a uniform Bogomolov-type result

“…Our first result is the following uniform version of the Bogomolov conjecture, which strengthens and generalizes the new gap principle of [DGH1,Kuh].…”

“…Our proof is very different from that of [DGH1,Kuh], and the key ingredient is a bigness result of adelic line bundles over the universal curve. We will come back to that in the next subsection.…”

“…Our theorem is stronger and more general than the new gap principle of [DGH1,Kuh] by the following aspects:…”

“…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.…”

The tropical Manin-Mumford conjecture

“…Katz, Rabinoff, and Zureick-Brown [11] used tropical methods to prove a uniform bound on the number of torsion points on an algebraic curve of fixed genus, which satisfy an additional technical constraint on the reduction type. Kühne [13] (in characteristic zero) and Looper, Silverman, and Wilms [14] (in positive characteristic) recently proved uniform bounds on the number of torsion points on an algebraic curve.…”