The talk will discuss some properties concerning algebraic numbers of small Weil height, more specifically properties (N) and (B) introduced by Bombieri and Zannier. A field of algebraic numbers has property (N) if it contains finitely many elements of bounded height, while it has property (B) if the height of its elements, when nonzero, is lower bounded by an absolute constant. While it is easy to see that number fields enjoy both properties, a generally difficult problem is to decide their validity for infinite extensions of the rationals. After surveying what is known in this area, I will present some results obtained in collaboration with Arno Fehm and some perpectives.

The nonabelian Cohen—Lenstra program studies the distribution of the Galois group of maximal unramified extensions of a family of global fields. In this talk, we will discuss some new developments for this type of question. We will introduce a cohomological invariant of a Galois extension of $\mathbb{F}_q$. We show that by keeping track of this invariant we can generalize the nonabelian Cohen—Lenstra Heuristics given by Liu, Wood and Zureick-Brown to cover the case when the base field contains extra roots of unity; and moreover, we show that the new conjecture is a nonabelian generalization of the work by Lipnowski, Tsimerman and Sawin. We will prove the conjecture with large $q$ limit, and discuss how to make a similar conjecture for number fields.

Let $E/\mathbb{Q}$ be an elliptic curve. By Mordell's 1922 theorem, the points on $E$ with coordinates in $\mathbb{Q}$ form a finitely generated abelian group. In particular, the torsion subgroup of $E(\mathbb{Q})$ is a finite abelian group, and the groups that occur as $E(\mathbb{Q})_{\text{tors}}$ are known due to work of Mazur in 1977. The past decade has seen a renewed interest in studying torsion points on rational elliptic curves: from identifying the torsion subgroups that can arise on $E/\mathbb{Q}$ under base extension to a field of higher degree to a near complete classification of the image of $\ell$-adic Galois representations associated to elliptic curves over $\mathbb{Q}$. In this talk, we will discuss recent results which leverage this knowledge to begin to understand a new class of elliptic curves, namely, those geometrically isogenous to an elliptic curve defined over $\mathbb{Q}$. Motivated by the problem of producing low degree points on modular curves, we seek to characterize the elliptic curves within a fixed geometric isogeny class producing a point of prime-power order in least possible degree.

The number of integral points on any given elliptic curve is finite. Taking a family of elliptic curves and imposing some ordering, we expect that very few curves have non-trivial integral points. In certain quadratic and cubic twist families, we prove that almost all curves contain no nontrivial integral points. The proof uses a correspondence by Mordell between integral points on elliptic curves and integral binary quartic forms.