Questions tagged [arakelov-theory]
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51
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The local global principle for differential equations
Are there any good reference to tackle the problem below?
Or, are there any know result?
Problem
Let $f_1...f_n\in \mathbb{Z}[x_1,..,x_n]$ and $V:\mathbb{R}^n\rightarrow \mathbb{R}^n$ be a vector ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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