# 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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### In search for methods for finding a closed form for the roots of a non-homogenous diophantine equation.

I'm trying to solve a special case of a number theoretical problem, and it relies on finding a closed form for the roots of a non-homogenous diophantine equation with four variables, but I could only ...
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### Singularities of arithmetic surfaces

I have a problem understanding the discussion of example 8.3.54 in Liu's Algebraic Geometry and Arithmetic Curves. The setting is the following: We have a DVR with uniformiser $t$, characteristic of ...
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### If E is non-singular then it has non-singular weierstrass equation

Let $E$ be a non-singular projective curve of genus one. There exist regular functions $x,y$ on $E$ satisfying a Weiestrass equation $$y^2 = x^3 + ax + b$$ Is this equation necessarily non-singular? ...
1 vote
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### Existence of a model for $(X,L)$

Recently I was reading Moriwaki's work about the heights on arithmetic varieties, and I came across the following doubt about the setting. Let $(X,L)$ be a couple where $X$ is a projective nonsingular ...
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### Torsionness of the kernel of the pullback map of Picard groups of a normalization map

Let $X$ be a projective variety over a number field $k$, $\pi: \tilde X \to X$ be its normalization, and $\pi^{*}: \mathrm{Pic}(X) \to \mathrm{Pic}(\tilde X)$ be the corresponding map of Picard groups....
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### What is the needed background to study Euler system?

I'm interested in Euler system.I heated this theory was used to prove the finiteness of Tate-Shafarevich group of certain elliptic curves by rubin. To read Karl Rubin's 'Euler system', what is the ...
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### Decomposition of the regular representation of $S_3$ into irreducible ones.

The regular representation of $S_3 = \langle e, a, a^2, b, ab, a^2b \rangle$ is realised into the $6 \times 6$ matrices acting on the six dimensional vector space spanned by each element of $S_3$. OK, ...
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### Injection from $WC(E/K)$ to $H^1(G_K,E)$ from Silverman's 'the arithmetic of elliptic curves'

This is question regarding Silverman's 'the arithmetic of elliptic curves'(AEC), p325, theorem 3.6. I want prove there is bijection between $WC(E/K)$ and $H^1(G_K,E)$. The bijection is given by {$C/K$}...
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### Morphism between algebraic (smooth) curves of degree 1 is an isomorphism. Does the converse holds?

Morphism between algebraic (smooth) curves of degree 1 is an isomorphism. Does the converse holds ? In other words, isomorphism between algebraic smooth curves has always degree 1 ? Degree of morphism ...
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### Let $C$ and $C'$ be algebraic (smooth) curves.If morphism $φ:C→C'$ satisfies the condition that $＃φ^{-1}(Q)＝1$ for all $Q∈C'$, why $φ$ is isomorphism?

Let $C$ and $C'$ be algebraic (smooth) curves. If morphism $φ:C→C'$ satisfies the condition that $＃φ^{-1}(Q)＝1$ for all $Q∈C'$, why $φ$ is isomorphism ? If I could prove $φ$ is a degree one morephism, ...
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### Elliptic surfaces: translation by a torsion section preserves intersection numbers?

Let $S$ be a rational elliptic surface with elliptic fibration $f:S\to\Bbb{P}^1$ (with a section) over an algebraically closed field. Let $E(K)$ be the Mordell-Weil group of $S$. In this text, Example ...
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### Why does transforming an Elliptic curve into Weierstrass form not affect the group of rational points?

There are various steps involved like realising the curve in projective space, multiple coordinate transformations, homogenising/dehomogenising the curve and converting a curve with rational ...
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### The affine scheme Spec $\mathcal{O}_K$ is a curve over F1 too?

I read that we can considerate the affine scheme $\mathrm{Spec}(\mathbb{Z})$ like a curve over $\mathrm{Spec}(\mathbb{F}_1)$ where $\mathbb{F}_1$ is the field with one element. So, can we also ...
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### $p$-adic valuation on algebraic numbers

Is there a way to define in a canonical way a $p$-adic valuation on complex number or at least on algebraic number, that extend the $p$-adic valuation on $\mathbb{Q}$? That is, is there a canonical ...
1 vote
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### What is an isomorphism between $L:aX+bY=Z$ and $\Bbb P^1_K$?

It is known that genus $0$ smooth curve over field $K$ with base point is isomorphic to $\Bbb P^1_K$ over $K$. I want to understand this with a lot of examples. For example, let $a,b\in K^\times$ and ...
1 vote
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### Translation of Loic Merel's Paper on Bounds for Elliptic Curves

I have been reading elliptic curves and had a discussion with my professor regarding the torsion subgroup of the Mordell-Weil group. He advised me to read Loïc Merel's "Bornes pour la torsion ...
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### How does the geometric Frobenius act on stalk?

How does the geometric Frobenius act on stalk? The first image is from http://virtualmath1.stanford.edu/~conrad/Weil2seminar/Notes/L19.pdf page 9. The second image is from Kiehl-Weissauer page 7.
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### The textbooks of number theory including p-adic cyclotomic character and Teichmuller lift

I would like to learn number theory, and it seems like necessity in number thoery that ingredients which is p-adic cyclotomic character and Teichmuller lift. Therefore, I wish to learn these, but in ...
1 vote
$\newcommand\spec{\operatorname{spec}}\newcommand\P{\mathbb P}\newcommand\msO{\mathscr O}\newcommand\push{{#1}_*}$ Say I have some relative Weierstrass curve $f:W\to S$ (so $f$ is flat and proper ...