33
votes
Integration of a function over 7-sphere
The result is
$$
I(k)=\frac{2\pi^4\ \Gamma\left(\frac{k}{2}+1\right)\Gamma\left(\frac{k}{2}+2\right)}{\Gamma(k+4)}\ ,
$$
which can be simplified a bit more using the Legendre Duplication Formula.
More ...
11
votes
Accepted
Height functions on $\mathcal{M}_g(\overline{\Bbb{Q}})$ defined via dessins d'enfants?
If $X$ is a (smooth projective) curve over $\overline{\mathbb{Q}}$, we define
The Belyi degree $\deg_B(X)$ of $X$ to be the minimum degree of a Belyi map $X\to \mathbb{P}^1_{\overline{\mathbb{Q}}}$.
...
- 9,497
9
votes
Accepted
Constant term arising in general exact expression for the canonical height of a Mordell-Weil generator point on an elliptic curve $E$
You've decomposed the canonical height into a sum of local heights. If the curve has (split) multiplicative reduction at $p$, then $c_p=\operatorname{ord}_p(\Delta_E)$, the valuation of the minimal ...
- 47.4k
9
votes
Accepted
A Simplification of the computation of local heights in Gross-Zagier
The reference is his 1995 Math Annalen paper "On the $p$-adic height of Heegner cycles". See e.g. the discussion on page 6.
The fact that the $q$-expansion of height pairings is modular is ...
- 2,606
8
votes
Integration of a function over 7-sphere
$\newcommand{\R}{\mathbb{R}}\newcommand{\C}{\mathbb{C}}$Here's an approach I like that takes advantage of the fact that the integrands are homogeneous. I'll just show how to compute
$$ \int_{S^7} |...
- 27.5k
8
votes
Weil height vs Moriwaki height
One can define a height with respect to a special metric so that some particular useful equation involving the height holds exactly, instead of approximately as for the Weil height.
For example, one ...
- 148k
6
votes
Height functions on $\mathcal{M}_g(\overline{\Bbb{Q}})$ defined via dessins d'enfants?
If I am not mistaken, this precise issue was attacked by Dirk Smit during 1990s in a series of papers on Communications of Mathematical Physics. I am not sure why his papers are not quoted more widely ...
- 6,249
6
votes
Height functions on $\mathcal{M}_g(\overline{\Bbb{Q}})$ defined via dessins d'enfants?
I think there is basically no hope of proving such a comparison (and it is probably not true, but my argument for this is less convincing).
Such a comparison theorem would have two parts:
(1) Proof ...
- 148k
6
votes
Accepted
Integration of a function over 7-sphere
This is not a complete answer but shows how to simplify your integral:
As pointed out by mlk in the comments the Hopf fibration plays a crucial role: Consider $\mathbb S^7\subset\mathbb C^4.$ On $\...
- 6,825
5
votes
Accepted
What is the "geometric height" mentioned by Moriwaki?
The geometric height is easiest to define for points on $\mathbb P^n(K)$. These define maps $C \to \mathbb P^n$ and we take the line bundle $\mathcal O(1)$ on $\mathbb P^n$, pull back to $C$, and take ...
- 148k
5
votes
Asymptotics for algebraic numbers of height less than one
Dubickas, Algebraic numbers with bounded degree and Weil height, Bull Aust Math Soc 98 (2018) 212-220, writes,
For a positive integer $d$ and a nonnegative number $\xi$, let $N(d,\xi)$ be the number ...
- 39.9k
5
votes
How should multiplicative height on projective space interact with automorphisms?
Daniel Loughran is correct that the naive height may be regarded as merely one among many possible heights, and for naive counting problems, which one you choose will not affect the order of growth, ...
- 47.4k
4
votes
Accepted
Definition of intersection pairing on an arithmetic surface
You should have a look at Chapter 4 of Moriwaki's Book "Arakelov Geometry". In particular, your question seems to be answered in Proposition 4.19.
- 408
4
votes
Accepted
Transformation of height on projective varieties
First, your statement can't be true as you've stated it, because you say it holds for any (by which I assume you mean all) rational points on $X$. But your map is a rational map, so there will be some ...
- 47.4k
3
votes
Accepted
Inequality between archimedean and non-archimedean height function on number fields
Such a bound cannot exist if $K$ has infinitely many units, because for a unit $h_0=0$ but $h_1$ can be large. Hence assume $K=\mathbb Q(\sqrt{-d})\subset\mathbb C$ is imaginary quadratic. Then $h_1^K(...
- 651
3
votes
Accepted
Deligne's example of $\deg \pi_{*}\Omega_{X/Y}<0$
This example should be the elliptic curve with j-invariant 0 and can be found in
Deligne's paper Preuve des conjectures de Tate et de Shafarevitch (p. 29). If you are more interested in small values ...
- 408
3
votes
Accepted
Effective version for Silverman’s specialization theorem
All of the height estimates used to prove Theorem B can in principal be made effective, in terms of the equations defining your family of abelian varieties and the equations defining the section. The ...
- 47.4k
2
votes
Accepted
How should multiplicative height on projective space interact with automorphisms?
The naive height is not at all intrinsic. It is just a convenient choice to work with for notational simplicity. If one is doing things properly one should be counting rational points of bounded ...
- 21.3k
2
votes
Accepted
Does the $p$-adic regulator depend on Weierstrass model?
Here a long comment to settle this question.
This is really a bug in the implementation of $p$-adic heights in sagemath. I have announced it as a bug here on the sage trac list. I hope to add the code ...
- 8,945
1
vote
Constant term arising in general exact expression for the canonical height of a Mordell-Weil generator point on an elliptic curve $E$
To add to Silverman's great answer, I managed to locate this text. Proposition 3.4.1 allows us to give $(1)$ fully.
Let $$E:\quad y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$ be the minimal Weierstrass model ...
- 533
1
vote
Does the $p$-adic regulator depend on Weierstrass model?
After some trial and error, here's what I think is the origin of the bug from my understanding.
It's not a complete answer, but there's not enough space to write it as a comment.
The algorithm for the ...
- 207
1
vote
Accepted
Are there any quadratic functions on an abelian variety not from the height machine?
The source has countable dimension over $\mathbb R$, since $A$ has countably many divisors defined over a finite extension of $K$, while the target, being the space of quadratic functions on a ...
- 148k
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