Let \(A\) be an abelian variety over a number field \(K\). Then the uniform boundedness conjecture for abelian varieties allows us to predict that the order of the group of \(K\)-rational torsion points of \(A\) can be bounded uniformly in terms of the dimension of \(A\) and the number field \(K.\) From now on, we take \(K\) to be \(\mathbb{Q}.\) The conjecture for the case when \(A\) is an elliptic curve over \(\mathbb{Q}\) was proven by B. Mazur, but the cases for higher dimensional abelian varieties \(A\) over \(\mathbb{Q}\) are currently wide open. An easy consequence of the conjecture is that if we are given a family of abelian varieties over \(\mathbb{Q}\) and a section of infinite order, then the set of \(\mathbb{Q}\)-rational points of the base where the section becomes torsion is not Zariski dense. Other than the case of a family of elliptic curves over \(\mathbb{Q},\) it is also known unconditionally for a family of abelian varieties of arbitrary dimension over \(\mathbb{Q}\) if the dimension of the base of the family is \(1,\) which follows from a theorem of J. H. Silverman [J. Reine Angew. Math. 342, 197–211 (1983; Zbl 0505.14035)] and J. Tate [Am. J. Math. 105, 287–294 (1983; Zbl 0618.14019)]. In this regard, the first main result of the paper in review is that the uniform boundedness conjecture for abelian varieties is equivalent to the statement that the set of points where a section of infinite order becomes torsion is not Zariski dense (in the sequel, this latter statement is denoted by (F)), where the proof of the desired equivalence follows mainly from the induction on the dimension of the base and the fact that the stack of principally polarized abelian varieties of fixed dimension is a noetherian Deligne-Mumford stack. It is also noted that in view of a result of Cadoret and Tamagawa, it is sufficient to consider (F) for Jacobian varieties of curves to verify the uniform boundedness conjecture for abelian varieties.
For other interesting results of this paper, we introduce the following notion: for a given variety \(S\) over \(\mathbb{Q}\), an abelian scheme \(A/S\), and a section \(\sigma \in A(S)\), we say that the pair \((A/S, \sigma)\) has sparse small points if for all integers \(d \geq 1,\) there exists an \(\epsilon>0\) such that the set \[ \mathbf{T}_{\epsilon}(d):=\{s \in S(\overline{\mathbb{Q}})~|~[\kappa(s):\mathbb{Q}] \leq d~\text{and}~\widehat{h}(\sigma(s)) \leq \epsilon \} \] is not Zariski dense in \(S.\) (Here \(\kappa(s)\) (resp.\(\widehat{h}\)) denotes the residue field of the point \(s\) (resp. Néron-Tate height).) Then the author proposed a conjecture saying that every pair \((A/S, \sigma)\) with \(\sigma\) of infinite order has sparse small points, one of whose evidences is the aforementioned theorem of Silverman and Tate (for the case when \(S\) is one dimensional). This conjecture is also closely related to the uniform boundedness conjecture for abelian varieties in light of the fact that the torsion points have height zero, and the author proved the conjecture for families of elliptic curves under the assumption that a conjecture of Lang on lower bounds for heights in families is true. Also, the above conjecture is verified in certain special cases, namely, of families of nodal curves \(C/S\) with suitable conditions, especially, on the base \(S\) (e.g.\(\dim_{\mathbb{Q}} \text{Pic}(S) \otimes_{\mathbb{Z}} \mathbb{Q} = 1\)), whose proof is essentially obtained by introducing the notions like height jump and Green’s functions on resistive networks to show that the height jump is bounded for curves the Jacobian varieties of whose generic fibers admit Néron models, together with an inequality that is obtained by comparing ways of associating heights to line bundles which are not ample. (For more precise statements, see Theorem 3.17 of the paper.)