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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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slightly different from AGM

$a_{n+1}=\frac{a_n+b_n}{2}$ $b_{n+1}=\sqrt{a_{n+1}b_{n}}$ also, $a_1=a$ and $b_1=b$ evaluate $$\lim_{n\to\infty}a_{n}$$ This question is from guillaume musso's novel La Jeune Fille et la Nuit . ...
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What is a supersingular elliptic curve over arbitrary rings

I read in Katz Mazur's book on moduli spaces of elliptic curves that an elliptic curve over an $\mathbb{F}_p$-algebra $R$ is called ordinary if its geometric points are all ordinary. Now the question ...
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Galois invariants of the function field of a projective variety defined over K

I apologize if this question is too simple but I cannot see why the following is true: Let $ K $ be a perfect field and $ \overline{K} $ a fixed algebraic closure. Let $ X \subset \mathbb{P}^n(\...
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52 views

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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1answer
51 views

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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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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57 views

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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69 views

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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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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49 views

$p$-th coefficient of weight $2$ new form with $p | N$ must be $1$ or $0$ or $-1$?

Let $f= \sum_{n=1}^{\infty}a_nq^n \in S_2^{new}(N)$ be a normalized new form of weight $2$ with respect to $\Gamma_0(N)$ and assume $p|N$ is a prime. Then must $a_p=0$ if $p^2|N$ and belongs to $\{-1,...
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Moduli interpretation of the integral anticanonical tower

This question is related to my reading of On torsion in the cohomology of locally symmetric varieties by Scholze. In chapter $3$, using the theory of canonical subgroup, he produces Frobenius maps ...
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Are there counterexamples of isogeny elliptic curves with non-isomorphic integral Tate modules?

Let $K$ be a field and $G_K$ be its absolute Galois group. Let $E_1,E_2$ be two elliptic curves over $K$. Assume that there exists an isogeny $f:E_1\rightarrow E_2$. Let $p$ be a prime number. Then $f$...
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How to compute the $p$-adic period of $\mathbb G_m$?

I am reading one paper by Colmez about periods of abelian varieties with complex multiplication. As a motivation, he says we can compute the periods of $\mathbb G_m$: I know the example for the ...
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Roadmap to rigid cohomology

I am interested in the study of p-adic geometry. Unfortunately what I know is basic Algebraic geometry and basic number theory. To get an idea of the amount of material to study what could be a "...
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Finding elliptic curves achieving the upper and lower bounds of Hasse's Interval

I always thought that Hasse's bound is sharp (at least for elliptic curves). In other words I always thought that given a prime number $p$, I can find two elliptic curves $E_1,E_2$ over $\mathbb F_p$ ...
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46 views

Is Hom scheme between projective curves of large genus finite etale?

Let $K$ be a number field, $T$ be a finite set containing some finite places of $K$, and $S=\operatorname{Spec} O_{K,T}$. If $X,Y$ are two projective smooth curves over $S$ with genus large than $1$ ...
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An abelian variety not isogenous to a Jacobian

In the L-functions and Modular Forms Database is an isogeny class of an abelian variety of dimension $2$ over $\mathbb F_5$. They claim that it is principally polarizable but not the Jacobian of a ...
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Modular parametrization from equality of $L$-functions

In the literature, an elliptic curve $E/\mathbb{Q}$ is defined to be modular in two different ways 1) if there exists a nonconstant morphism $X_0(N) \to E$, 2) if there exists a modular form $f$ ...
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88 views

decomposition group and inertia group, the minimal polynomial,surjectivity of the map $D_{M/P}\rightarrow Gal$

Can anyone explain the underlined sentence? For notation, A:Dedekind domain, K=Frac(A), L/K:Galois extension, B:The integral closure of A in L, M:A maximal ideal of B, P:The intersection of M and A (...
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Finding the integral closure of $k[x]$ in $k(x)(\sqrt{f})$, where $f(x)=x^6+tx^5+t^2x^3+t\in k[x]$.

This is an exercise (1.15) of Dino Lorenzini' s An Invitation to Arithmetic Geometry. Let $F$ be a field of characteristic $2$. Let $k:=F(t)$, the field of rational functions in the variable $t$. Let ...
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1answer
141 views

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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1answer
48 views

$\mathbb{C}$-valued points and flatness

Let $X$ be an integral scheme over $\mathrm{Spec}(\mathbb{Z})$, we denote $X(\mathbb{C})$ as the set of $\mathbb{C}$-valued points in $X$. Then $Y\rightarrow \mathrm{Spec}(\mathbb{Z})$ is flat if $Y(\...
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1answer
63 views

Torsion points of an elliptic curve (example in Silverman)

Let $E$ be the elliptic curve $$y^2=x(x-2)(x-10)$$ Silverman claims (Example. X. 1.5 p.315 Arithmetic of Elliptic Curves) that $E(\mathbb{Q})_{tors}$ injects into the reduction $\widetilde{E}(\mathbb{...
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Prerequisites for reading the paper “Birational Calabi–Yau n-folds have equal Betti numbers" by Victor V. Batyrev?

As the title suggests, I'm trying to read this paper. But it demands a lot of prerequisite knowledge that I'm not just unaware of, but I'm having trouble narrowing down what books and notes I can take ...
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Why é​t​a​l​e​?

Background: The notion of an étale morphism has proved itself to be ubiquitous within the realm of algebraic geometry. Apart from carrying a rich intuitive idea, it is the first ingredient in notions ...
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Frobenius for modular curves.

Actually, I am a bit confused about the notation and what is called Frobenius for modular curves. Let $N\geq 4$ be an integer, let $R$ be an $\mathbb{F}_p$-algebra, where $p$ is a prime not dividing $...
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Pullback of a scheme along the Frobenius morphism

Let $S$ be an $\mathbb{F}_p$-scheme for a given prime $p$. Let $X$ be a scheme over $S$. Then, we can consider the Frobenius morphism of $X$ relative to $S$, defined by taking the unique morphism ...
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Base change of horizontal divisor in semi-stable curves

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 ...
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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 ...
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Is an integer extension of a ring the integral closure of this ring in some extension?

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 ...
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What would a arithmetic surface look like

By arithmetic surface we mean a projective, regular and flat scheme of dimension 1 over $Spec(O_K)$ for some algebraic number field $K$. So we can view such a scheme $X$ as a closed subscheme of $...
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Global sections of a Vojta divisor: a lemma for Faltings' theorem and Vojta's inequality

This is E.5 of Hindry, Silverman's Deophantine Geometry. I want a global section of a Vojta divisor from some polynomials. Let $K$ be a number field, $C$ a smooth projective $K$-curve of genus $g \ge ...
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44 views

Degree of variety over $\mathbb{Q}$ versus over $\mathbb{F}_p$

Let $V$ be a projective variety (possibly reducible) in $\mathbb{P}^n$ defined over $\mathbb{Z}$. What is the relation between the degree of $V$ seen as a variety over $\overline{\mathbb{Q}}$ and ...
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Is rational connectedness a constructible property?

Let $f:X\rightarrow S$ be a morphism of varieties (say, over an algebraically closed field $k$). Is the locus of points above which the fibers of $f$ are rationally connected $\{s\in S:X_s \text{ is ...
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Finiteness of the Brauer group and torsion-freeness of Picard group of a rationally connected variety over an algebraically closed field

Let $X$ be a rationally connected variety over an algebraically closed field $k$. Then (1) is the Brauer group $\text{Br}(X)$ finite? (2) is the Picard group $\text{Pic}(X)$ torsion-free?
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1answer
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Rational points on elliptic curve over a local field correspond to integral points on minimal Weierstrass model

Let $R$ be a complete discrete valuation ring, with fraction field $L$. Let $E$ be an elliptic curve over $L$, and $W$ a minimal Weierstrass model over $R$. Why is $W(R) \simeq E(L)$? We have a map $...
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Formal completion of modular curves

Let $N\geq 4$ be an integer coprime with $p$, where $p$ is a fixed prime number. Then we know that there exists a scheme $Y_N$ over $\text{Spec}(\mathbb{Z}_p)$ whose $\text{Spec}(R)$-points are ...
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108 views

topology on the ring of Witt vectors in the theory of period rings of Fontaine

For a $p$-adic field $K$ with perfect residue field $k$, we know the standard construction of the ring $R$. I will recall it briefly. It is $\varprojlim_{x \rightsquigarrow x^p} O_{C_K}/pO_{C_K}$, ...
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Points of Scholze's Anticanonical Tower

In his paper "On torsion of locally symmetric spaces", Schole introduces a perfectoid space, which is the anticanonical tower. He claims that points coming from $\text{Spa}(C,C^+)$, where $C$ is a ...
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55 views

Endomorphism of the elliptic curve $y^2 = x^3 - ax$

For the elliptic curve $$E:y^2 = x^3 - ax$$I know that $E\left(\mathbb{C}\right)\cong\mathbb{C}/\left(\mathbb{Z}i + \mathbb{Z}\right)$ and hence $\text{End}_{\mathbb{C}}\left(E\right)\cong\mathbb{Z}\...
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Geometrical proof of the impossibility of angle trisection by straightedge and compass

There's a fascinating (almost) geometrical proof of the impossibility of angle trisection by straightedge and compass by Terrence Tao which is somehow more comprehensible (and more directly to the ...
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Infinite subgroups of elliptic curves and quotients

Let $E_1$ be an elliptic curve over $\Bbb Q$ and $S \subset E_1(\Bbb Q)$ be a subgroup. Is there an elliptic curve $E_2$ with an algebraic group morphism $E_1 \to E_2$, and such that $E_2(\Bbb Q) \...
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Smooth rational points of affine curves under birational maps

Suppose $K$ is an infinite field of char. $0$. Let $C$ be an $K$-irreducible affine algebraic $K$-curve and suppose that $P_1,\ldots, P_n$ are non-singular $K$-points of $C$. Can one always find a ...
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Tate module of an Abelian Scheme

It is well known that if $A/k$ is an Abelian Variety over (the spectrum of) a field, a very important object to consider is its Tate module $T(A):=\underset{\underset{n}{\longleftarrow}}{\lim}A[p^n](\...
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Construction of arbitrary regular polygons with ruler and compass

A lot of arithmetic geometry has to do and took off with constructions or proofs of non-constructibility of specific figures only with ruler and compass. See e.g. the non-constructibility of the ...
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Analogies between Hodge conjecture and Tate conjecture

I hear sometimes that there is many analogies between the Hodge conjecture and the Tate conjecture. If we take a look at the statements of this two conjectures, we have the followings : The Tate ...