I am looking for an article or book where the theory of heights over function fields (in any characteristic) is treated. I am especially interested in Northcott-type statements. Fo …

I have three questions regarding height pairings: In [Serre, Lectures on the Mordell-Weil theorem], p. 85 f., it is stated that the following function is a local height function: …

Hello, Suppose $A$ is an Abelian variety of dimension $g$ over a number field $k$. Then using height functions one can show that there are non-torsion points in $A(\bar k)$. This …

The base field is a number field. It is known that the Shafarevich conjecture implies the Mordell conjecture (Kodaira-Parshin). Is the converse also true? Note that both conjec …

I know (edit: three) families of smooth projective connected curves over $\bar{\mathbf{Q}}$ for which the Belyi degree is not hard to bound from above. The modular curves $X(n)$ …

The divisor bound asserts that for a large (rational) integer $n \in {\bf Z}$, the number of divisors of $n$ is at most $n^{o(1)}$ as $n \to \infty$. It is not difficult to prove …

Let $Y$ be a smooth projective connected curve of genus $g>0$ over $\overline{\mathbf{Q}}$. Let $h_{\textrm{Fal}}(Y)$ be the Faltings height of $Y$. Question 1. Can one classify …

Let $\bar{\mathbf{Q}}$ be an algebraic closure of the rationals, and $\alpha$ denote an algebraic number in $\bar{\mathbf{Q}}$. We define the height of $\alpha$, denoted by $H(\alp …

Fix a number field $K$ and a polynomial $F(x)\in K[x]$ of degree at least $4$. For a squarefree integer $d$, define the curve $X_d$ over $K$ by the equation $dy^2 = F(x)$. Note th …

Is there a modern reference for Néron's "Quasi-fonctions et Hauteurs sur les Varietes Abeliennes" http://www.jstor.org/pss/1970644 i.e. using Grothendieck's language of schemes and …

Elkies maintains a list of nontorsion points of low height on elliptic curves over Q; does anyone know of anything similar for curves over number fields? Everest and Ward give exa …

Let $h$ be a function on the moduli space of abelian varieties of dimension $g$ over $\overline{\mathbf{Q}}$. Let $K$ be a number field and let $g\geq 2$ be an integer. Fix a rea …

Jon Hanke and I were just chatting and realized we didn't know the answer to the following question. If E is an elliptic curve over a number field, is there in any sense a "canoni …

In the literature there are several different definitions of what is often referred to as Beilinson's height pairing (see for example section 4.3.8 of Gillet and Soulé's paper Arit …

Let $K$ be a real number field, together with a fixed immersion in $\mathbb{R}$, and for each positive real number $M$ consider the set $S_M(K)$ of elements in $K \cap [0,1]$ havin …