# Questions tagged [arakelov-theory]

The arakelov-theory tag has no usage guidance.

50
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### Definition of intersection pairing on an arithmetic surface

$\def\div{\operatorname{div}} \def\Spec{\operatorname{Spec}}$Let $K$ be a number field, $O_K$ be the ring of integers, and $X \to \Spec(O_K)$ be a regular arithmetic surface. I want to understand how ...

2
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0
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### Is there any relation between Berkovich spaces over $\Bbb Z$ and Arakelov theory?

As I understand it, both Arakelov geometry and Berkovich geometry over $\Bbb Z$ (or $\mathcal O_K$) consider geometric objects that contain in some sense information about both archimdean and ...

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### What is the status of N. Durov's PhD thesis?

N. Durov Phd thesis "New Approach to Arakelov Geometry" is ofted mentioned as a beautiful approach to Arakelov geometry and it includes also a treatment of $\mathbb F_1$. It is a very long ...

6
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### Why do Chern forms show up in Arakelov geometry?

Let $X$ be a regular, projective flat scheme over $\Bbb{Z}$, let $\bar{L}$ be a hermitian line bundle on $X$. In order to define the height of an integral closed subset $Y$ we define it on closed ...

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### Why does the Manin-Mumford conjecture over number fields imply the conjecture over arbitrary fields of characteristic 0?

The Manin-Mumford conjecture states that for an abelian variety A over a field F of characteristic 0 the torsion points are dense in an integral closed subvariety Z if and only if it is an abelian ...

4
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### Green currents in Arakelov theory

Let $K$ be number field and $\mathcal{O}_K$ its ring of integers. In Arakelov theory the idea is to enrich an arithmetic scheme $X$ over $\mathcal{O}_K$ "at infinity", that is to add data at ...

1
vote

1
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145
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### Arithmetic ampleness and scalings of the metric

Let $\overline L= (L, h)$ be a hermitian $C^
\infty$ line bundle on an arithmetic variety $X\to\operatorname{Spec }\mathbb Z$ (I am reasoning in terms of higher Arakelov geometry, like in Gillet & ...

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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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### Symmetric spaces as the moduli spaces of Arakelov vector bundles

Over a function field of a curve $K = k(C)$, there is the Weil uniformization
$$\mathrm{Bun}_{GL_n}(C) = GL_n(K) \backslash GL_n(\mathbb{A}_K) / GL_n(\mathcal{O}_K).$$
This equality is (for example)...

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### Why isn’t there an arithemetic Riemann Roch for closed immersions?

I read Faltings’s works and Soule’s works on ARR and found that both of them proved this for proper maps which are smooth over Q. But GRR holds for arbitrary proper maps between smooth varieties, so I ...

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### Durov approach to Arakelov geometry and $\mathbb{F}_1$

Durov's thesis on algebraic geometry over generalized rings looks extremely intriguing: it promises to unify scheme based and Arakelov geometry, even in singular cases, as well as including geometry ...

5
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357
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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 ...

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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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### What happened to Suren Arakelov? [closed]

I heard that Professor Suren Arakelov got mental disorder and ceased research. However, a brief search on the Russian wikipedia page showed he was placed in a psychiatric hospital because of political ...

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### Analogies of Quillen metric for non-archimedean places

It is well known that the Quillen metric is defined on the determinant line bundle of a Riemann surface. I am wondering if there is any non-archimedean analog of this.
I remember this is one of the ...

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### Discrete Gaussian free field for a closed manifold

I want to ask if a construction of discrete Gaussian free field has been done for a closed Riemannian manifold. Most of the literature I surveyed either need extra boundary condition and consider ...

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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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### Dualizing sheaf and determinant of cohomology

Let $\pi:X\to S=\operatorname{Spec } O_K$ be an arithmetic surface in the sense of Arakelov geometry. Here $K$ is a number field $\pi$ is a flat map and $X$ is a projective surface. For any coherent ...

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### hyperbolic "Green function" on a product of upper half-planes

Let $\Delta_{hyp}=\Delta_{hyp,1}=-y^2(\partial_x^2+\partial_y^2)$ be the hyperbolic Laplacian acting on functions of $\mathfrak{h}$ (the Poincare upper half-plane) and consider its resolvent
$$
R(s)=(...

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### Why Green functions and not Neron functions?

Arakelov constructed a nice intersection theory on arithmetic surfaces. A key point is the notion of Green function for a Riemann surface, which will be involved in the ''part at infinity'' of the ...

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### Is there any definition of $H^1$ in one dimensional Arakelov geometry

Consider a number field $K$ with ring of integers $O_K$. On the affine scheme $\overline X=\operatorname{Spec}(O_K)$ we have the well known one dimensional Arakelov geometry.
Let $\overline D=\sum_{\...

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### Meaning of the determinant of cohomology

The Arakelov intersection number on arithmetic surfaces is defined as an "extension" of the classical intersection number on algebraic surfaces. It was introduced to get a nice intersection theory ...

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### Why it is difficult to define cohomology groups in Arakelov theory?

I have been puzzled by the following Faltings' remark in his paper Calculus on arithemetic surfaces (page 394) for a few months. He says:
If $D$ is a divisor on $X$, we would like to define a ...

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### Arakelov Motivic Cohomology and Hodge Theory

Lately I have been studying these two papers (first and second) that introduce a new cohomology theory called Arakelov motivic cohomology. While most of the applications presented in the papers are ...

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### Is there a notion of hyperbolicity for number rings?

For algebraic curves over a nice enough field $k$, we have a notion of what it means to be hyperbolic: If $\overline{C}$ is a smooth projective curve of genus $g$ and $P_1,\dots,P_n$ are closed points,...

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### Why are Green functions involved in intersection theory?

I've been learning Arakelov geometry on surfaces for a while. Formally I've understood how things work, but I'm still missing a big picture.
Summary:
Let $X$ be an arithmetic surface over $\...

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### Does exist a "product formula" for arithmetic surfaces?

Let $K$ be a number field, then the Arakelov geometry of $\operatorname{Spec }O_K$ can be interpreted by means of the adelic ring $\mathbb A_K$. In particular, a key ingredient is the product formula ...

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### Analogies between classical geometry on complex surfaces and Arakelov geometry

This is my first question on this wonderful site. The following question about Arakelov geometry is gonna be quite long and wide; to be clear one of that kind of questions that are usually ignored. ...

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### The notions of $H^0(\widehat{ D})$ and $h^0(\widehat{D})$ are not satisfactory

Let $K$ be a number field with ring of integers $O_K$. Moreover consider an Arakelov divisor $\widehat{D}\in\overline{\operatorname{Div }(\operatorname {Spec }O_K)}$, namely
$$D=\sum_{\mathfrak p\;\...

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### Arakelov divisor on $\operatorname{Spec } O_F$: places or embeddings?

Let $F$ be a number field such that $[F:\mathbb{Q}]=n$ and with ring of integers $O_F$. Let's put $B=\operatorname{Spec } O_F$, then an Arakelov divisor is an element of:
$$Div(X)\times \bigoplus_\...

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### Günter Tamme's course "Arakelov theory and Grothendieck-Riemann-Roch"

On chapter III.4 ("Metrized $\mathcal{o}$-modules") of this book on algebraic number theory, Neukirch credits his treatment of the theory of finitely generated $\mathcal{o}$-modules to the course "...

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### Mochizuki's Gaussian Integral Analogy

In his latest 115-page overview, Mochizuki spends some time explaining "alien copies" by the analogue of evaluating the Gaussian integral by squaring it and introducing a second variable/dimension. In ...

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### Arakelov divisors and the meaning of real coefficients

I'm learning Arakelov theory on arithmetic surfaces and I have the following general question.
Let $K$ be a number field and consider its ring of integers $O_K$. Moreover let $S:=\operatorname{Spec} ...

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### Fiber at infinity of an arithmetic surface $X$ as an element of $\widehat{\operatorname{Div}(X)}$

Introduction:
Let $M$ be a Riemann surface, then a Green function on $M$ is an element $g\in C^\infty(V)$ where $V=M\setminus\{x_1,\ldots,x_r\}$ and around each point $p\in M$ we have:
$$g=a\log\...

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### Smoothness of the "Archimedean special fiber" in Arakelov geometry

If $X$ is a scheme over, let's say, $\mathbb{Z}_p$, one can consider its special fiber obtained by reduction modulo $p$ ans it certainly makes sense to ask if this special fiber is smooth or not.
...

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### Computing intersection number of two arithmetic line bundles

I have some questions in Arithmetic Arakelov geometry
Let $\mathcal X\to Spec(\mathcal O_K)=C$ be an arithmetric projective variety over $C$ , where $\mathcal O_K$, ring of number filed $K$ and $\...

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### Meaningful review of Moriwaki's "Arakelov Geometry"

I have been asked to write a mathscinet review for Atsushi Moriwaki's Arakelov Geometry
book:
http://www.ams.org/bookstore-getitem/item=mmono-244
I could do the review the standard way in a day or ...

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### What is an infinite prime in algebraic topology?

The links between algebraic topology (stable homotopy theory in particular) and number theory are nowadays abundant and fruitful. In one direction, there is chromatic homotopy theory, exploiting the ...

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### Mumford-Ramanujam examples in characteristic p [and in Arakelov geometry]

For a compact Riemann surface $B$ of genus $\geq 2$, it is a consequence of the Narasimhan-Seshadri theorem that there exist rank-$2$ vector bundles $E \to B$ of degree zero, all of whose symmetric ...

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### Inequality regarding sum of gaussian on lattices

When S is a subset of an inner product space, let d(S) denote ${\sum\limits_{s \in S} e^{- \langle s,s \rangle}}$
Suppose L is a discrete additive subgroup of $\mathbb{R^n}$, M is a subgroup of L, ...

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### What analysis should I know for studying Arakelov Theory?

Hi!
I have a fairly good background in Algebraic Geometry (say at the level of Hartshorne's book and some Intersection Theory from Fulton) and since I think working over $\text{Spec } \mathbb{Z}$ is ...

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### Points of minimum Arakelov height and harmonic arithmetical varieties

Added. (28/2) To put it less pompously (and more vaguely, less concretely), I wanted to relate the impression that it is the general rule that an Arakelov (i.e., geometric) height on an arithmetical ...

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### Overview of Arakelov intersection theory and the Arakelov Chow ring

I'm looking for a reference that gives an overview of the most important properties of Arakelov intersection theory (on arithmetic varieties of arbitrary dimension) and that describes basic properties ...

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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 ...

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### What should I read before reading about Arakelov theory?

I tried reading about Arakelov theory before, but I could never get very far. It seems that this theory draws its motivation from geometric ideas that I'm not very familiar with.
What should I read ...

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### What does the theta divisor of a number field know about its arithmetic?

This question is about a remark made by van der Geer and Schoof in their beautiful article "Effectivity of Arakelov divisors and the theta divisor of a number field" (from '98) (link).
Let ...

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### On the relation of special values of motivic L functions and partial zetas

Let $K$ be a number field, $L$ a finite abelian extension and $\chi \in \widehat{Gal(L/K)}$ a (non-trivial) character. If we multiply out the associated Artin L-function $L(\chi,s)$ we can write this ...

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### is there any way to bound the number of CM points by height functions?

It is known that if $X$ is a curve over a number field $F$ equipped with a flat regular model over $O_F$ the integer ring, one can define, using a suitable ample line bundle with an Hermitian metric, ...

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### Implications of the abc conjecture in Arakelov theory

It is apparent that the abc conjecture is deeply related to Arakelov theory. In one direction, it is shown in S. Lang, "Introduction to Arakelov Theory", that a certain height inequality in Arakelov ...

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### Is there a category-theoretic definition of the arithmetic Grothendieck group

Let $X$ be a regular scheme which is flat over $\mathbf{Z}$. The arithmetic Grothendieck group $\hat{K}(X)$ is defined to be the quotient of $\hat{G}(X)$ by $\hat{G}^\prime(X)$. This is actually quite ...