# Questions tagged [arakelov-theory]

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42
questions

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votes

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151 views

### 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 ...

**21**

votes

**1**answer

496 views

### 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**

votes

**1**answer

253 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 ...

**8**

votes

**0**answers

180 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. ...

**110**

votes

**1**answer

7k views

### 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 ...

**2**

votes

**0**answers

69 views

### 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 ...

**4**

votes

**0**answers

104 views

### Exact sequence of Quillen metrics

Let $M$ be a compact Riemann surface equipped with its Arakelov metric. Let $\xi$ be a holomorphic line bundle on $M$ equipped with an admissble metric. For any point $P\in M$ we get an admissible ...

**2**

votes

**1**answer

450 views

### 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 ...

**6**

votes

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194 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 ...

**4**

votes

**1**answer

348 views

### 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 ...

**2**

votes

**1**answer

220 views

### 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)=(...

**7**

votes

**1**answer

338 views

### 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 ...

**5**

votes

**1**answer

323 views

### 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_{\...

**13**

votes

**2**answers

1k views

### 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 ...

**23**

votes

**2**answers

997 views

### 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 ...

**7**

votes

**0**answers

437 views

### 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 ...

**10**

votes

**1**answer

348 views

### 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,...

**37**

votes

**4**answers

2k views

### 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 $\...

**6**

votes

**1**answer

194 views

### 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 ...

**27**

votes

**1**answer

851 views

### 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. ...

**5**

votes

**2**answers

439 views

### 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\;\...

**8**

votes

**2**answers

326 views

### 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_\...

**20**

votes

**1**answer

563 views

### 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 "...

**11**

votes

**1**answer

2k views

### 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 ...

**3**

votes

**1**answer

378 views

### 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} ...

**5**

votes

**1**answer

327 views

### 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\...

**9**

votes

**1**answer

400 views

### 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.
...

**5**

votes

**0**answers

429 views

### 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 $\...

**12**

votes

**0**answers

1k views

### 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 ...

**41**

votes

**2**answers

1k views

### 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 ...

**3**

votes

**1**answer

214 views

### 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 ...

**12**

votes

**1**answer

747 views

### 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, ...

**9**

votes

**1**answer

1k views

### 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 ...

**8**

votes

**0**answers

473 views

### 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 ...

**8**

votes

**1**answer

1k views

### 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 ...

**11**

votes

**1**answer

905 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 ...

**36**

votes

**2**answers

6k views

### 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 ...

**40**

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**0**answers

2k views

### 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 me first ...

**11**

votes

**0**answers

841 views

### 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 ...

**9**

votes

**1**answer

457 views

### 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, ...

**11**

votes

**1**answer

3k views

### 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 ...

**7**

votes

**1**answer

647 views

### 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 ...