Questions tagged [heights]
The heights tag has no usage guidance.
46
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Does the $p$-adic regulator depend on Weierstrass model?
I am a little confused on the $p$-adic regulator on elliptic curves and what happens when you switch to different Weierstrass models. Restrict to ell. curves over $\mathbb Q$ for simplicity.
From my ...
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3
votes
1
answer
227
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Are there any quadratic functions on an abelian variety not from the height machine?
Let $X$ be an abelian variety defined over a number field $K$. We know that the Neron--Tate height machine associates to a class in the Picard group of $X$ a unique quadratic function which is zero at ...
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7
votes
1
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463
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Weil height vs Moriwaki height
Let $X$ be a projective veriety over a number field. After fixing an embedding into $\mathbb P^n$ (i.e. a very ample line bundle $L$), one can define the Weil height $\hat h_{L}$ by restriction of the ...
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3
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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 ...
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0
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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 ...
- 18.8k
14
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3
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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 ...
- 143
3
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1
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279
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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 ...
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1
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0
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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 ...
- 1,079
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0
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126
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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 ...
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0
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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 ...
- 709
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0
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168
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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 $X$ ...
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5
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1
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344
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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 ...
- 6,061
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57
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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 ...
user140765
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0
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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 ...
- 13.5k
6
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0
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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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1
answer
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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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1
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369
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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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votes
2
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352
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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 $N(\alpha)$. ...
- 765
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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?
user76098
2
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0
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160
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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 ...
- 1,489
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votes
1
answer
377
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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 ...
- 2,696
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2
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776
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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 ...
- 2,060
14
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3
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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 ...
- 13.5k
1
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0
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130
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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 $\alpha = [\alpha_0, \dots , \alpha_n]$, ...
user24815
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172
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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 ...
- 13.5k
2
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0
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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 ...
- 13.5k
5
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1
answer
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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 \...
2
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0
answers
367
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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 ...
user19475
4
votes
1
answer
443
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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
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2
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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 ...
user19475
5
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1
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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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0
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329
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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?
user19475
3
votes
2
answers
666
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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 ...
- 31
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1
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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$ ...
- 9,087
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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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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 ...
- 9,087
3
votes
2
answers
733
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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 ...
- 95.1k
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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 ...
- 9,087
6
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1
answer
976
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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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2
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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 ...
- 1,269
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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 ...
- 1,332
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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 ...
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