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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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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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1answer
40 views

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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1answer
148 views

Tate module of an elliptic curve.

In the case of an elliptic curve $E$ defined over a field $K$, I know that there is a good definition of the so-called Tate module, for every prime $p$, which is the $\mathbb{Z}_p$-module $T_p(E)=\...
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30 views

Why is the kernel of reduction contained in the image of this isogeny? (From a paper of Cassels)

$\newcommand{\fp}{\mathfrak{p}} \renewcommand{\phi}{\varphi}$ I'm currently trying to understand a paper of Cassels [see page 189 for the relevant content, esp. (4.9)] and I've hit a little snag. The ...
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2answers
744 views

genus of normalization of stable curve

Let $X$ be a stable curve of genus $g$ over the field $k$, i.e., a $k$-rational point of the Deligne-Mumford stack $M_g$. What is the genus of the normalization of $X$? Does it depend on the number ...
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15 views

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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27 views

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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1answer
7k views

Translation for EGA/SGA

People often recommend Grothendieck's EGA (Elements de Geometrie Algebrique) and SGA (seminaire de geometrie algebrique) as a good reference for learning arithmetic geometry. However, as the title ...
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1answer
2k views

Learning path to the proof of the Weil Conjectures and étale topology

Assume you are an algebraic geometry advanced student who has mastered Hartshorne's book supplemented on the arithmetic side by the introduction of Lorenzini - "An Invitation to Arithmetic Geometry" ...
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5answers
2k views

Why is it “easier” to work with function fields than with algebraic number fields?

I just bought a copy of Jürgen Neukirch's book Algebraic Number Theory. While browsing through it I found a section titled § 14. Function Fields in chapter I. In it the author describes ...
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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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1answer
62 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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1answer
64 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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90 views

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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2answers
141 views

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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1answer
130 views

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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80 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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0answers
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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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0answers
35 views

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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0answers
47 views

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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1answer
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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0answers
64 views

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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1answer
500 views

How do I sum 4 binary numbers?

I know how to do the simple binary addition, where there are only 2 binary numbers to be summed, but now I am dealing with 4 binary numbers: ...
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0answers
82 views

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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0answers
35 views

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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0answers
52 views

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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2answers
79 views

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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1answer
89 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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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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0answers
88 views

CAS for counting points of varieties over finite fields

I am looking for a computer algebra system that is able to do some of the following (equivalent in theory) things for a smooth projective variety defined over a finite field: Count the number of ...
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1answer
92 views

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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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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1answer
21 views

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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1answer
751 views

What is Spec of the Adeles?

Let $K$ be a global field and $A_K$ the ring of adeles. What are the prime ideals of $A_K$? I have been told that a full proof of this is quite subtle, but have been unable to find a reference ...
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1answer
49 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
150 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
77 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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2answers
730 views

Intuition for Class Numbers

So I've been thinking about the analytic class number formula lately, and class numbers in general and I'm trying to develop a good intuition for them. My basic question, which may be too general/...
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36 views

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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1answer
251 views

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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1answer
52 views

Prime $\ell \in \mathbb{Z}$ where $\ell\mathcal{R}_i$ prime ideal of $\mathcal{R}_i$ for all $i = 1, 2, \ldots, n$? (Exercise 5.5, Silverman)

Let $\mathcal{K}/\mathbb{Q}$ be an imaginary quadratic field, and let $\mathcal{R}_1, \ldots, \mathcal{R}_n$ be orders in $\mathcal{K}$. Is there a prime $\ell \in \mathbb{Z}$ such that $\ell\mathcal{...
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0answers
43 views

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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0answers
57 views

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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1answer
98 views

Arakelov Theory and Arakelov curves

There exists a definition of Arakelov Curve in Arakelov theory? My question is because Neukirch (Algebraic Number Theory, Chapter III) defined Arakelov divisors in the set $X=Spec(\mathcal O_K)\...
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1answer
116 views

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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15 views

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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1answer
54 views

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 ...