# 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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### What condition on an abelian variety ensures that the associated formal group has integral coefficients?

I am new in the study of abelian variety or in general algebraic variety. I am looking for a sufficient condition that ensures the formal group associated with an abelian variety has integral ...
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### Twist of elliptic curve by degree $n$ extension

Let $E/K : y^2=x^3+ax+b$ be an elliptic curve over number field $K$. For quadratic extension $L=K(\sqrt{D})/K$, $E_D/K : Dy^2=x^3+aX+b$ is called a twist of $E/K$ by $D$. This curve $E_D$ has ...
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### Accessible result motivating Arithmetic Geometry

My understanding of arithmetic geometry is that it applies concepts originating from algebraic geometry to schemes that are quite different from these initially studied in this field (those associated ...
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### $\mathbb{F}_1$-schemes under $\text{Spec}_\mathbb{Z}$

$\text{Spec}_\mathbb{Z}$ is the terminal object in the category of schemes. In the context of Borger's absolute geometry, we can treat the category of schemes as a subcategory of "the category of ...
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### Silverman AEC Exercise 7.4

The following problem in chapter 7 of Silverman's book has been bothering me: Here's my progress: I first tried to solve it in the $\text{char}(k) \ne 2,3$ case, wherein we can assume the equation is ...
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### Is $H^1(G_{\Bbb{Q}_p},E[2])=0$ for good prime of $E/\Bbb{Q}$?

Let $E/\Bbb{Q}$ be an elliptic curve over number field $\Bbb{Q}$. Let $p$ be a prime number. $H^1(G_{\Bbb{Q}_p},E)$ is $0$ for good prime of $E/\Bbb{Q}$ because every genus $1$ curve over $\Bbb{F}_p$ ...
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### Some questions about almost étale extension of local field

I'm in trouble understandig the example and proof of Example 4.2.2 in Denis Benois's lecture note about $p$-Adic Hodge Theory, he gives the following definition: A finite extension $E/F$ of non-...
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### Using étale fundamental group to show unramifiedness of Tate module

Let $E$ be an elliptic curve over $\mathbb{Q}_p$ with good reduction at $p$, i.e., there exists an elliptic curve $\mathcal{E}$ over $\mathbb{Z}_p$ whose generic fiber is isomorphic to $E$. I have ...
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### Pair of torsors and divisors and Galois cohomology

Let $E/\Bbb{Q}$ be an elliptic curve. $H^1(G_{\Bbb{Q}},E[2])$ is bijection with the set of pair of torsors and divisor of degree $2$. Let call the latter set $WCD(E/\Bbb{Q})$. I tried to construct a ...
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### Example of an elliptic curve which is not defined over $\Bbb{Q}$

An elliptic curve $E$ is defined over $\Bbb{Q}$ if only if it is isomorphic over $\overline{\Bbb{Q}}$ to an elliptic curve whose coefficients are in $\Bbb{Q}$. My question is, what are examples of an ...
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### Why is the Function Field of the Modular Curve $X(N)$ defined over $\mathbb{Q}(\mu_n)$?

Following Section 7.6 in Diamond & Shurman, the algebraic model of $X(N)$ is constructed (a priori) over $\mathbb{Q}$ by first defining its function field as \begin{equation*} \mathbb{Q}(j,f_{(0,1)...
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