The optimality of the Boundedness Height Conjecture

关于主导有理自映射在射影簇上周期点和前周期点高度有界的性质

On the height boundedness of periodic and preperiodic points of dominant rational self-maps on projective varieties

本文推翻了关于周期点高度有界的猜想，并证明了在射影簇上某些主导有理自映射的周期点集高度有界，同时指出前周期点可能不满足这一性质。

Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves

We prove a theorem giving the asymptotic number of binary quartic forms having bounded invariants; this extends, to the quartic case, the classical results of Gauss and Davenport in the quadratic and cubic cases, respectively. Our techniques are quite general, and may be applied to counting integral orbits in other representations of algebraic groups. We use these counting results to prove that the average rank of elliptic curves over $\mathbb{Q}$, when ordered by their heights, is bounded. In particular, we show that when elliptic curves are ordered by height, the mean size of the 2-Selmer gr…

多项式上的统一有界性猜想与动力学Lang猜想

The Uniform Boundedness and Dynamical Lang Conjectures for polynomials

本文在假设算术几何标准猜想的前提下，证明了多项式数域上的统一有界性猜想，并提出了动态Lang猜想的类比结果，尤其在单临界多项式情况下结果无条件成立。

On the number of rational points on special families of curves over function fields

We construct families of curves which provide counterexamples for a uniform boundedness question. These families generalize those studied previously by several authors. We show, in detail, what fails in the argument of Caporaso, Harris, Mazur that uniform boundedness follows from the Lang conjecture. We also give a direct proof that these curves have finitely many rational points and give explicit bounds for the heights and number of such points.

Torsion points and height jumping in higher-dimensional families of abelian varieties

In 1983 Silverman and Tate showed that the set of points in a 1-dimensional family of abelian varieties where a section of infinite order has `small height' is finite. We conjecture a generalisation to higher-dimensional families, where we replace `finite' by `not Zariski dense'. We show that this conjecture would imply the Uniform Boundedness Conjecture for torsion points on abelian varieties. We then prove a few special cases of this new conjecture.

$abcd$ 猜想、一致有界性与动力系统

The $abcd$ conjecture, uniform boundedness, and dynamical systems

本文综述了Vojta对$abc$猜想和Szpiro猜想的高维推广及其在算术动力学中的应用，特别指出$abcd$猜想蕴含了挠点一致有界性及典范高度下界的动力学类比猜想。