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Questions tagged [arithmetic-geometry]

A subject that lies at the intersection of algebraic geometry and number theory dealing with varieties, the Mordell conjecture, Arakelov theory, and elliptic curves.

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Unipotent Groups and Torsors

I've been doing some reading in some arithmetic geometry, and there's a subtle point that is confusing me slightly: say $U$ is some unipotent affine group scheme over a field $L$, and $T$ is some ...
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Questions about Zeta Function of Singular Plane Curve

I am working on a project which involves learning about the zeta function (weil zeta function) for plane curves, but I do not know much algebraic geometry. (I do not know anything about schemes). I ...
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The set of the ideals of $\mathbb{C}$ corresponding to the points of complex geometry

There is parallel relation between the ideals over $\mathbb{C}$ corresponding to the points of complex geometry and the ideals of the field of the complex function, now how to construct the ideals ...
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Degree of the field extension $\bar{k}(Z_f) /\bar{k}(x)$

I came across an argument on the lecture notes. It does not have a proof of the fact and I want to understand the argument. Let $k$ be a field, $f(x,y) \in \bar{k}[x,y]$. Define $Z_f := \{(a,b) \in ...
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Does Faltings's theorem imply the set of Diophantine equations are decidable?

Actually, we know all Diophantine equations are not decidable. Does Faltings's theorem imply the sets of Diophantine equations are decidable? That is , there is an algorithm that decide whether those ...
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Rational points of a scheme X over a field k (Qing Liu Problem 3.7)

I'm working on problem 3.7 from Qing Liu's text on algebraic geometry, and I'm having a bit of difficulty formalizing my intuition. I searched around, and was unable to find the answer elsewhere on ...
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why the number of irreducible vertical divisors on X is finite?

I am reading Qing liu - gebraic geometry and arithmetic curves Page 356-358 8.3.3 contraction On the head he says the number of irreducible vertical divisors on X is finite ,but I can't know why ...
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Period matrices of isogenous abelian varieties

It is well-known that two principally polarized abelian varieties of dimension $g$ over $\mathbb{C}$ are isomorphic to each other if and only if their period matrices lie in the same $\text{Sp}_{2g}(\...
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Arithmetic intersection number

Let $\mathcal{X}$ be a smooth proper model of $X$ over $O_{K,S}$, where $K$ is a number field, $X$ is a projective curve over $K$, $O_{K,S}$ is the ring corresponding to the set $S$ of bad reductions ...
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Notation Confusion Reading Nick Katz' Proof of RH for (Projective, Smooth, Geometrically Connected) Curves

Trying to read this so-called "Note" by Nick Katz and cannot get over some of his notation. The fundamental group stuff is fine, but then he...seems to define some notation with the same notation in ...
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406 views

How can we use sin/cos for things like simple harmonic motion or any periodic type?

The sine and cosine functions are both well defined periodic function with fixed period and amplitude. My understanding is that for phenomena that are periodic, since sine and cosine are well defined ...
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Lifting points via étale morphism of adic spaces

This question was suggested to me during the reading of Huber's book about Etale Cohomology of Adic Spaces. I formulate this question here in the context of adic spaces, but I think, since a morphism ...
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1answer
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Recursive writing involving arithmetic progression

I've been trying to figure out this recursion problem but I'm getting stuck trying to find the nth-term sequence for the last recursion. I found one but the second i'm so clueless about. I don't know ...
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1answer
138 views

field of rational functions of a curve

Let $C$ be the algebraic curve defined by the modular polynomial $\phi_N$ of order $N>1$ over the rational numbers, i.e. \begin{equation}C:=\text{specm}(\mathbb{Q}[X,Y]/\phi_N(X,Y)). \end{equation} ...
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135 views

Are there Neron models for algebraic groups of multiplicative type?

Let $K$ be a number field with Galois group $G$ and $N$ be a finitely generated abelian group which is also a discrete $G$-module. Let $D(N)$ be the algebraic group defined as $D(N)(R)=Hom_{\mathbb{Z}...
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Trouble with finding geometric progression pattern

I have this system: $$ b_2-b_1 = 18 $$ $$ b_4-b_3 = 162 $$ I have to find $b_1$ (the first element) and $q$ (common ratio). Any ideas how to solve it?
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An isogeny from a split algebraic torus

Suppose that there is an isogeny (in the category of commutative algebraic groups) from a split algebraic torus to a semi-abelian variety. Does it follows that this semi-abelian variety is also an ...
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How do I sum 4 binary numbers?

I know how to do the simple binary addition, where there are only 2 binary numbers to be summed, but now I am dealing with 4 binary numbers: ...