# Questions tagged [heights]

The heights tag has no usage guidance.

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### Number fields ordered by discriminant

Since the discriminant of a number field $K \neq \mathbb{Q}$ is bounded from below by an exponential of the degree $[K:\mathbb{Q}]$, for instance by Minkowski's Geometry of Numbers bound, there are ...

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### Comparison between Faltings height and Modular Height

Motivation/Context: In Faltings’ proof of the Mordell conjecture, there is a theorem that establishes a finiteness of abelian varieties with respect to the Faltings height under certain conditions. ...

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### Is this a possible strengthening of the Lehmer conjecture?

Here is another possible refinement of the Lehmer conjecture.
For $\alpha \in \overline{\mathbb{Q}}^{\times}$, let $C_{\alpha} \subseteq \mathbb{Q}(\alpha)$ be the maximal cyclotomic field contained ...

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### Faltings height variation "at place of bad reduction''

Is there any example in the literature where someone has considered the problem of bounding the variation of Faltings height at a place of bad reduction? Specifically, if $A_i$ for $i\in \{1,2\}$ are ...

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### Transformation of height on projective varieties

I am trying to find a reference for the following statement: let $X$ and $Y$ be projective varieties defined over $ \mathbb{Q}$ and $\phi: X \to Y$ be a rational map defined over $ \mathbb{Q}$. Denote ...

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157 views

### Heights of multiples of rational points on elliptic curves

Let $E/\mathbb{Q}$ be an elliptic curve, given by some minimal Weierstrass equation (say $Y^2 = X^3 + aX + b$ for some integer $a$ and $b$), and let $P$ be a rational point on $E$ which is not the ...

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282 views

### Mahler measure of a totally positive, expanding algebraic integer

Consider a degree-$d$ algebraic integer $\alpha$ all of whose conjugates (including itself) are real numbers greater than 1. Its Mahler measure $M(\alpha)$ is simply equal to the norm $N(\alpha)$. ...

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255 views

### Can we define a height function for a variety over a finite field?

That is, is there a way to measure the complexity of a point over a finite field the same way we do it over number fields?

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### algorithm to find a new point of small height in a number field extension

By the height of an algebraic number $\alpha$, I mean the absolute, logarithmic (additive) Weil height $h(\alpha)$; e.g. $h(2^{1/n}) = (\log 2)/n.$
If $K$ is a number field, let $\delta(K)$ denote ...

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303 views

### Weil height of an Abelian Variety with everywhere (potentially) good reduction

Background: Suppose that $E$ is an elliptic curve over $\mathbb{Q}$ with everywhere (potentially) good reduction. there are many ways to define the height of $E$, and I will be concerned with the ...

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### How did height in algeb. number theory/elliptic curves started?

Maybe this is obvious but it isn't to me yet. What is the history of heights used in say points of the project plane over a number field or of elliptic curve over a number field? I would guess people ...

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### Asymptotics for algebraic numbers of height less than one

The question. Is an asymptotic equivalent known or conjectured for the number $N(d)$ of $\alpha \in \bar{\mathbb{Q}}$ with $h(\alpha) < 1$ and $[\mathbb{Q}(\alpha):\mathbb{Q}] \leq d$?
The rather ...

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122 views

### Points with minimal height

Let $K$ be an algebraically number field and $$\phi : \mathbb P^n (K) \to \mathbb P^m (K)$$ a polynomial map, such that $\forall \alpha \in \mathbb P^n$, where $\alpha = [\alpha_0, \dots , \alpha_n]$, ...

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134 views

### Lang's height conjecture over $\mathbb{F}_q(T)$?

Is the canonical height of a non-torsion $\mathbb{F}_q(T)$-rational point of an elliptic curve over $\mathbb{F}_q(T)$ known or supposed to be bounded from below by an absolute positive number (or ...

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146 views

### Must the coordinates of a polynomial iteration have about the same size?

Original post. The following statements seem plausible (not to say intuitively obvious), but I do not see how to prove them.
Let us say that a polynomial mapping of $\mathbb{C}^2$ is reducible if ...

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### Examples of “nice” properties of algebraic extensions of $\mathbb{Q}$

I am writing a short survey of some "nice'' properties of algebraic extensions of $\mathbb{Q}$. Let's say a property (P) is nice if
every finite extension of $\mathbb{Q}$ satisfies (P), and
if $K \...

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291 views

### algebraicity of Néron-Tate canonical height for Abelian varieties over global function fields

(transcendence of canonical heights)
Is the Néron-Tate canonical height for an Abelian variety $A$ over a global function field $K$, $\hat{h}: A(K) \times A^\vee(K) \to \mathbf{R}$ known to always ...

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331 views

### relation between Faltings height and periods

Let $E$ be an elliptic curve defined by an equation $y^2=4x^3+ax+b$ where $a$ and $b$ are algebraic numbers. What is the relation between the Faltings height $h_F(E)$ and the periods
$$
\int_{\gamma} ...

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531 views

### Reference request for the theory of heights over function fields

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. For instance, over a ...

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### questions on Néron-Tate canonical height

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:
"Let $V/R$ be a ...

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661 views

### Does the Mordell conjecture imply the Shafarevich conjecture

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 conjectures are now ...

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299 views

### modern reference for Néron's “Quasi-fonctions et Hauteurs sur les Varietes Abeliennes”

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 in English?

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### Almost Northcott properties for heights of abelian varieties

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 real number $C$. Does ...

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### Is there an easier argument to prove that almost all of these curves have no semi-stable reduction

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 that the curves $X_d$ ...

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### Records for low-height points on elliptic curves over number fields

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 examples of points of ...

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870 views

### Which curves have stable Faltings height greater or equal to 1

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 or describe the ...

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587 views

### Comparing the height of an algebraic number with the height of its conjugates

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 $H(\alpha)$, to be $$H(\...

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### The divisor bound in number fields

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 this bound using the ...

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### Families of curves for which the Belyi degree can be easily bounded

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 $X(n)$. They are ...

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### Beilinson's height pairing vs Neron-Tate

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 Arithmetic intersection ...

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284 views

### Equidistribution in the unit interval of numbers in a real field with bounded Mahler measure

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 $K \cap [0,1]$ having Mahler measure ...

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### Torsion points in Abelian varieties over number fields

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 looks like an ...

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### Is there a canonical height on the Weil-Chatelet group?

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 "canonical height" on the ...