Arithmetic bigness and a uniform Bogomolov-type result

Abstract

Recently, a uniform version of this theorem was proved by Dimitrov-Gao-Habegger [DGH1] and Kühne [Kuh]. In fact, the new gap principle in [Gao2, Thm. 4.1], as a combination of [DGH1, Prop. 7.1] and [Kuh, Thm. 3], asserts that there are constants c 1 , c 2 > 0 depending only on g > 1 such that for any projective and smooth curve C over Q of genus g, and for anyHere h Fal (C) = h Fal (J) denotes the stable Faltings height of the Jacobian variety J.The new gap principle has a significant consequence to the uniform Mordell-Lang problem proposed by Mazur [Maz, pp.234]. Recall that the Mordell conjecture was proved by Faltings [Fal1], and a different proof was given by Vojta [Voj]. Vojta's proof was simplified and extended by Faltings [Fal2] to prove the Mordell-Lang conjecture for subvarieties of abelian varieties, and was further simplified by Bombieri [Bom] in the original case of curves. The proofs of [Voj,Fal2,Bom] actually gave an upper bound of the number of points of large heights, which was further refined by de Diego [dDi] and Rémond [Rem]. 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. There are constants c 1 , c 2 > 0 depending only on g such that for any projective and smooth curve C over K of genus g, and for any line bundle α ∈ Pic(C) of degree 1, with the extra assumption that (C, α) is non-isotrivial over k in the case K = k(t), one has

Elliptic Curves with Long Arithmetic Progressions Have Large Rank

“…A first breakthrough was obtained in 2020 by DeMarco, Krieger, and Ye [4] when they proved the conjectured uniform bound in the case E 1 and E 2 are given in Legendre form y 2 = x(x − 1)(x − λ) and π j are the corresponding projections onto the x-coordinate. As noted in [6], the Bogomolov-Fu-Tschinkel conjecture is now completely solved thanks to the recent proof of the Uniform Manin-Mumford conjecture [5,9,7,11]. In this work we prove the following generalization: Theorem 1.3 (Main Theorem for torsion).…”

“…In this way we first obtain a purely geometric result in Section 3 (Theorem 3.3) from which Theorem 1.3 is deduced in Section 5. Finally, Theorem 1.4 is also proved in Section 5 combining our geometric result with the Uniform Mordell-Lang conjecture [5,9,7,11] (see Section 4) instead of the Uniform Manin-Mumford conjecture.…”

“…After the recent works [5,9,7,11], the Uniform Mordell-Lang conjecture is proved. Here we recall the version obtained in [7] in the case of (possibly singular) curves contained in abelian varieties.…”

Dynamics on ℙ1: preperiodic points and pairwise stability

“…Dimitrov,Gao,and Habegger [DGH21b] and Kühne [Küh21] established uniformity in the Mordell-Lang conjecture following the blueprint they laid out in [DGH21a,GH19,Hab13]. Kühne [Küh21] established an arithmetic equidistribution theorem in families of abelian varieties and combined it with Gao's results [Gao20a,Gao20b] to prove uniformity in the Manin-Mumford and Bogomolov conjectures for curves in their Jacobians, a result that Yuan also obtained later with a different approach [Yua21]. Kühne's approach has been extended to higher dimensional subvarieties of abelian varieties by Gao,Ge,and Kühne [GGK21]; see Gao's survey article and the references therein for more about these developments [Gao21].…”

The Relative Manin-Mumford Conjecture

“…For each integer g ≥ 2, there exists a constant c = c(g) > 0 with the following property. Let C be an irreducible, smooth, projective curve of genus g defined over C. Let x 0 ∈ C(C), and let C − x 0 be the image of the Abel-Jacobi embedding based at x 0 in the Jacobian Jac(C) of C. Then A second proof of the Uniform Manin-Mumford Conjecture for curves was given by Yuan in [Yua21], based on the theory of adelic line bundles over quasi-projective varieties of Yuan-Zhang [YZ21]. Priori to Kühne's proof of the full conjecture, DeMarco-Krieger-Ye [DKY20] proved the case where g = 2 and C is bi-elliptic, using method of arithmetic dynamical systems.…”