# 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.

468 questions
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### Algebraic 1-cocycles and Galois gerbs

We have the following set up: $K/F$ is Galois, $D$ is an algebraic group of mult. type and $E$ is an extension of groups: $$1\to D(K)\to E\to Gal(K/F)\to 1$$ Now take a linear algebraic group G over F....
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### Galois representation on Tate module of a twist of an elliptic curve

Let $E/K$ be an elliptic curve with a twist $E'/K$. Let $f: E_{\overline K} \to E'_{\overline K}$ be an isomorphism. Let $m(\sigma) = f^{-1}\circ f^\sigma$ be the 1-cocycle corresponding to this twist....
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### Conjugations in the comparison isomorphisms between Betti cohomology and algebraic de Rham cohomology

For a smooth projective variety $X$ defined over $k$ which admits a real embedding $\sigma:k \rightarrow \mathbb{C}$, its Betti cohomology is defined by H^*_{B,\sigma}(X):=H^*(X \...
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### Problems on showing that a reduction map is defined, and that a certain scheme is finite.

I am currently on the last chapters in Liu's book and I am trying to solve the following problem, which is the first step in showing that a certian reduction map is well-defined: Let $X \rightarrow T$ ...
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### Involution On elliptic curve

In an elliptic curve E given by $y^2=x^3 + ax^2 +bx+ c$ and origin at the point of infinity, why does the map $i$ sending $(x, y)$ to $(x, -y)$ send $i(P)=-P$, where $-P$ denotes the inverse of $P$ ...
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### Quasi-coherent $\mathcal{O}_X$-modules

Suppose $X$ is a separated scheme, $F$ an abelian sheaf on $X$ in the Zariski topology. In order to assign an $\mathcal{O}_X$-module structure on $F$, is it enough to do it on a single open affine ...
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### $\pi^{tame}(\mathbb{A}^1_k)$ is trivial

Fixed an algebraically closed field of characteristic $p>0$, it is well known the result of the title: $\pi^{tame}(\mathbb{A}^1_k)\simeq 1$. Where the tame fundamental group, in this situation, ...
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### Fraction field of $p$-adic power series ring

Let $L$ be a finite extension of $\mathbf{Q}_p$. Write \mathcal{O}_{\mathcal{E}} = \left\{ f = \sum_{k \in \mathbf{Z}} a_kT^k \in \mathcal{O}_{L}[[T,T^{-1}]] \mid \lim_{k \to -\infty} a_k = 0\right\...
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### elliptic k3 surface and Shioda Inose structure

We know that suppose given two elliptic curves $E$ and $E'$, there is a Kummer surface $km(E,E')$. And I'm curious suppose we know a $K3$ surface is kummer, how do we recover the pair $(E,E')$? For ...
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### Krull's intersection theorem in the q-expansion principle

I'm currently reading the proof of the q-expansion principle in Katz'73 paper "p-adic properties of modular schemes and modular forms" . The principle itself is a Corollary (1.6.2) of Theorem 1.6.1, ...
Suppose I have a (smooth, projective) variety $X/\mathbb{Q}$. Is the notion of a primes of bad/good reduction well-defined from this data? Motivation and attempt at an answer: The question should be ...