# 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.

461 questions
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### has the ABC conjecture actually been solved?

I have heard that the proof announced by Shinichi Mochizuki has considered a flawed proof by several expert mathematicians who are specialised in arithmetic geometry including Peter Scholze (a field ...
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### Quotients and exact sequences of algebraic group schemes

Now I study group schemes to understand fundamental properties about semi abelian varieties and generalized jacobian varieties. Let $G$ be an algebraic group scheme over a field $k$ ($=$ $k$ group ...
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### Using Velu's formulas in MAGMA

Most isogeny-based cryptographic schemes rely on constructing an isogeny having a given kernel. That is, given an elliptic curve $E$ and a subgroup $G$ of points of $E$, there is interest in ...
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### Finding Smallest Radius of a Sphere that can Inscribe a Circular Cylinder

Hello I am trying to solve for the smallest possible radius $r$ for a sphere that can inscribe a circular cylinder of volume 8 cubic units(so the volume is given and is a constant 8 cubic units). I ...
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### Smooth proper variety over $\mathbb Q$ with everywhere bad reduction

$\newcommand{\Spec}{\operatorname{Spec}}$ It is a well-known fact that a smooth projective variety over $\mathbb Q$ has good reduction almost everywhere, i.e. everywhere apart from finitely many ...
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### Strong Approximation theorems in Function Fields versus Groups

In the context of global function fields, the strong approximation theorem can be stated as follows: Let $F$ be a global function field, and $P_1,P_2,\dots,P_r$ be a finite set of places of $F$ (with ...
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### Brauer group of global fields

Is the Brauer group $\text{Br}(K)$ of a global field $K$ an $\ell$-divisible group for some prime $\ell$? If so, what $\ell$? Is $\text{Br}(K)[n]$ finite, for $n$ integer? I know from class field ...
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### What are arithmetic curves?

Suppose we are given a curve $C$ such that the natural morphism $C\to \mathrm{Spec}\:\mathbb Z$ is integral (resp. flat or surjective). Is it true that in all three cases $C$ is affine? Are there ...
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### Are problems in “Arithmetica” of Diophantus all solved now?

It's well-known that Diophantus had written ”Diophantus“ which contains many problems about solving arithmetic equations. I wonder whether all of them has been solved using modern techniques, as some ...
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### Proposition 1.6.6, Etale Cohomology theory, Lei Fu

I have difficulties in understanding a gap of the proof of proposition 1.6.6 in Etale Cohomology Theory written by Lei Fu. $\mathrm{Proposition} 1.6.6$ Let $g:S'\rightarrow S$ be a quasi-compact ...
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### How to compute explicitly the covering map in the modularity theorem?

The modularity theorem (original Shimura-Taniyama-Weil conjecture) asserts the existence of a covering (uniformization) map $\pi:X_0(N) \to E$ for every $E$, an elliptic curve defined over $\mathbb{Q}$...
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### Torsion of elliptic curves and abelian extensions

Let $L/K$ be an abelian $p$-extension of number fields and $E$ be an elliptic curve over $\Bbb Q$. If $E[p](K)=0$, does it follow that $E[p](L)=0$ ? The converse is obviously true, but I don't have ...
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### Is $p$ an anomalous prime?

Let $E$ be an elliptic curve defined over the number field $K$ and $p$ be a rational prime such that for all $v\mid p$, $E$ has good ordinary reduction. If $E[p]\subseteq E(K)$, can we conclude $p$ is ...
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### Bad reduction at prime numbers for the elliptic curve $y^2+y=x^3-x^2+2x-2$

Consider the elliptic curve $$E:y^2+y=x^3-x^2+2x-2.$$ My goal is to compute the conductor of the elliptic curve, the example is from https://planetmath.org/conductorofanellipticcurve. My problem isn'...
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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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### Base change to make singular points on singular fibers split

Let $C$ be a semi-stable curve over $Spec(O_K)$ where $K$ is an algebraic number field. Assume that $X_s$ is singular for $s\in Spec(O_K)$. How do we know that there exists a finite separable ...
Let $X\rightarrow Spec(O_K)$ be a semi-stable curve, where $K$ is an algebraic number field. Let $D$ be a horizontal divisor and finite extension $K'\supset K$ contains the splitting field of the ...
This is a question that I just came up with, so it may be completely stupid and I sincerely apologize if that's the case. The question is; Take $A$ to be an integral domain. Let $B$ be an integral ...