# 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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### Finding formula that solves $w^4+x^4=y^4+z^4$ over the integers.

Several formulae that solve the diophantine equation $$w^4 + x^4 = y^4+z^4 \tag{1}$$ are presented in this collection. The simplest one bases on $$f_1 = a^7 + a^5 - 2 a^3 + 3 a^2 + a \tag{2}$$ and ...
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### Why are the Weil Conjectures stated via zeta functions?

Let $C$ be a smooth curve over $\mathbf{F}_p$, and let $\zeta$ be the zeta function of $C$. The Weil Conjectures for $C$ are usually stated something like this: The zeta function $\zeta(s)$ is a ...
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### Reference request- Galois Coverings over genus 1 curves

Would anyone have any good recommendations on learning about Galois coverings of genus 1 curves/torsors over genus 1 curves? More specifically learning about $H^{1}_{ét}(E,\mathbb{Q}/\mathbb{Z})$ ...
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### Can $P$ in $Q=[(E,P)] \in X_1(N)(K)$ always be defined over $K$?

Let's say we have an algebraic number field $K$ and a point $Q\in X_1(N)$ that is not a cusp. Now $Q$ can be represented as $Q=[(E,P)]$, where $E$ is an elliptic curve and $P$ is a pont on $E$ of ...
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### On well-definedness of Shimura curve.

Suppose that we have a quaternion algebra $D$ over a totally real number field $K$ such that $[K \colon {\Bbb Q}] = {\mathrm{odd}}$. We assume that $D$ splits everywhere at finite places of $K$ and at ...
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### How should I think of the nearby cycle $\mathrm R\Psi\overline{\mathbb Q_{\ell}}$ with its Galois action on a smooth projective scheme?

Let $F$ be a $p$-adic field with ring of integers $\mathfrak o$ and residue field $k$. Let $X$ be a smooth equidimensional projective scheme over $\mathrm{Spec}(\mathfrak o)$ of dimension $d$. Denote ...
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### To prove $Φ$：$\Bbb C\to E(\Bbb C)$ : $t\mapsto (\wp(t),\wp'(t))$ is a group hom

This is a question from Silverman's 'the arithmetic of elliptic curves', p171. If $\Lambda$ is a lattice in $\Bbb C$ the map $$z\mapsto (\wp(z),\wp'(z))$$ is a parametrisation of the complex points of ...
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### Definitions of CM abelian varieties

I was reading through a presentation by Oort (https://www2.math.upenn.edu/~chai/UPenn2013-beamer.pdf) and noticed something which disturbed me: he defines (slide 38) a simple CM abelian variety to be ...
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### Group action on ringed space

I am trying to understand the action of a finite group on a ringed space. let $G$ be a finite group acting on $(X, O_X)$. I know that $g \in G$ induces a morphism $g: X\longrightarrow X$ and for the ...
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### Computing the endomorphism ring of an elliptic curve over a finite field (using SAGE)

$\newcommand{\End}{\mathrm{End}} \newcommand{\Gal}{\mathrm{Gal}} \newcommand{\kb}{\overline{k}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\F}{\mathbb{F}} \newcommand{\Q}{\mathbb{Q}}$ I would like to ...
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### The Segre embedding and addition on elliptic curves

Let $$E:Y^2Z=X^3+AXZ^2+BZ^3$$ be an elliptic curve with $A,B\in \mathbb{Z}$. I would like to understand the map \begin{align} G:E\times E &\to E\times E\\ (P,Q)&\mapsto (P+Q,P-Q). \end{align} ...
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I have a question in the following setting: Let $k$ be a number field with ring of integers $O_k$. For a finite set of places $S$ of $k$ we can form the ring of $S$-integers $O_{k,S}$. Let $\mathbb{A}... 0answers 63 views ### How to calculate which primes have good reduction? I'm trying to find for which primes the elliptic curve over$\mathbb{Q}$defined by$y^2z+yz^2=x^3-xz^2$, for example, has good reduction. I think the way to go would be to find the minimal ... 0answers 25 views ### Transcendence degree of intersection of Field extensions So my question goes as follows: Suppose we have the function field$\mathbb{C}(x)$(no particular reason for choosing$\mathbb{C}$- in principle, choosing$\mathbb{Q}$might be a better idea), where$...
Let $K$ be an algebraically closed field. There is for any non-singular hypersurface $Y \subseteq \mathbb{P}^n_K$ a "Chow-Kunneth decomposition" (see page 38 in the link L1 below). Given any ...