Questions tagged [heights]

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14
votes
3answers
512 views

Height functions on $\mathcal{M}_g(\overline{\Bbb{Q}})$ defined via dessins d'enfants?

Belyi's theorem establishes a correspondence between smooth projective curves defined over number fields and the so called dessins d'enfants which are bipartite graphs embedded on an oriented surface ...
6
votes
0answers
154 views

Rational points on varieties whose anticanonical bundle is nef but not ample

Is the following plausible? "If $X$ is a variety over $\mathbf{Q}$ whose anticanonical bundle $L$ is nef but not ample, there is a number field $K$ such that $X(K)$ contains an infinite set of ...
11
votes
3answers
1k views

Integration of a function over 7-sphere

Suppose we have $x_1^2 + y_1^2 + x_2^2 + y_2^2 + x_3^2 + y_3^2 + x_4^2 + y_4^2 = 1$ and we define $z_j = x_j + iy_j$, where $j = 1,\,2,\,3,\,4$. The problem is finding or approximating the ...
3
votes
1answer
217 views

What is the "geometric height" mentioned by Moriwaki?

Let $K$ be a finitely generated field over $\mathbb{Q}$ of transcendence degree 1, and take a curve $C$ over a number field $k$ such that $k(C)=K$. In "Arithmetic height functions over finitely ...
1
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0answers
88 views

how often can a fixed prime be anomalous?

Let $p$ be a fixed prime. Say for simplicity $p>5$. As we vary over all elliptic curves $E/\mathbb{Q}$ of height $< X$, can one (expect to) say anything about what proportion of elliptic curves ...
2
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0answers
109 views

Faltings' height theorem for isogenies over finite fields

For an Abelian scheme over a ring of integers in a number field, Faltings has a theorem that describes how the Faltings' height changes through an isogeny. There are multiple references for this ...
1
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0answers
88 views

Clarification regarding the definition of absolute height of an algebraic number

According to page 1-2 of this paper (https://arxiv.org/abs/0906.4286), Mahler has established the inequality $$|\alpha_1 - \alpha_2| \geq H(\alpha_1)^{-(d-1)} \tag 1 $$ to be valid for all pairs of ...
2
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0answers
147 views

References for heights of algebraic or projective variety

In my research I am using the notion of a $\textbf{rational function f on a domain}\: U\subset \mathbb C^{n}$. By this I mean that the graph of $f$ is an analytic component of an affine variety $X$ ...
5
votes
1answer
312 views

Deligne's example of $\deg \pi_{*}\Omega_{X/Y}<0$

While reviewing Lang's book on Arakelov theory, I saw the following comment by Paul Vojta: "...Deligne has found an example when $\deg \pi_{*}\Omega_{X/Y}$ can be negative, because Green's functions ...
3
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0answers
57 views

Heights of semiabelian varieties

Fix a prime number $l$. Let $K$ be a finite extension of $\mathbb{Q}$ and $R$ be the ring of integers in $K$. In Chapter 2 of the Storrs volume (Cornell-Silverman) it is claimed that it is not ...
8
votes
0answers
248 views

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 ...
7
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0answers
252 views

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. ...
9
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0answers
288 views

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 ...
6
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0answers
211 views

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 ...
1
vote
1answer
125 views

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 ...
1
vote
1answer
306 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 ...
5
votes
2answers
334 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 $N(\alpha)$. ...
2
votes
1answer
321 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?
2
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0answers
154 views

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 ...
7
votes
1answer
358 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 ...
9
votes
2answers
729 views

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 ...
14
votes
3answers
771 views

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 ...
1
vote
0answers
124 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 $\alpha = [\alpha_0, \dots , \alpha_n]$, ...
3
votes
0answers
163 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 ...
2
votes
0answers
151 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 ...
5
votes
1answer
625 views

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 \...
2
votes
0answers
336 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 ...
4
votes
1answer
419 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} ...
5
votes
2answers
690 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 ...
7
votes
2answers
1k views

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 ...
5
votes
1answer
735 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 ...
2
votes
0answers
318 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?
3
votes
2answers
589 views

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 ...
2
votes
1answer
229 views

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 $X_d$ ...
12
votes
2answers
402 views

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 ...
11
votes
1answer
1k 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 ...
3
votes
2answers
664 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(\...
23
votes
2answers
2k views

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 ...
13
votes
2answers
1k views

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 ...
6
votes
1answer
930 views

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 ...
1
vote
2answers
301 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 ...
14
votes
4answers
1k views

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 ...
12
votes
1answer
655 views

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 ...