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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151 views

Is any irrational algebraic number normal number?

Suppose number$R$ expands in decimal,and $\lim_{n\rightarrow\infty}\frac{C_{n}(d)}{n}=\frac{1}{10}$ where $d$ is one of the ten digit,and ${C_{n}(d)}$ the counting numbers of $d$ from first digit to $...
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68 views

Value of transcendental function $f(x)$ at rational point $a$

$f(x)$ is a transcendental function over $\mathbb{Q}(x)$,and analytic in disk with natural boundary. If $a\gt 0$ and $a\in \mathbb{Q}$, then $f(a)$ is a transcendental number. Has this assertion been ...
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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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51 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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116 views

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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62 views

(Reference request) How to show elliptic curve has positive Mordell-Weil rank

I know there must be a lot of ways to show an elliptic curve has positive Mordell-Weil rank if it really does. And I guess that I am supposed to collect them by myself. But since I am not working in ...
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1answer
287 views

Leray spectral sequence

Let $f : X\to Y$ be a continuous map of topological spaces, $A$ an abelian sheaf on $X$. We have the Leray spectral sequence $$E_2^{p,q} := H^p(Y, R^qf_*A)\Rightarrow H^{p+q}(X, A).$$ Could someone ...
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149 views

Endomorphisms of products of elliptic curves

Suppose that we have a product $E_1 \times \cdots \times E_n$ of elliptic curves $E_i$ and that we know the endomorphisms $\text{End} E_i$ for each elliptic curve. Is it possible to use this to ...
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92 views

Will someone kindly explain Kato's dual exponential map?

I am reading this article by Rubin. Will somebody how to derive the formula given in equation 2 of section 5? It states thus: $z$ corresponds to the map $$x\mapsto Tr_{\mathbb{Q}_{n,p}/{\mathbb{Q}...
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Points on an elliptic curve over $\mathbb F_p$

Let $E$ be an elliptic curve over $\mathbb F_p$ (the finite field of $p$ elements) defined by $y^2=x(x-n)(x-m)$ where $p\nmid nm(n-m)$. Let $N$ be the number of $\mathbb F_p$-valued points of $E$. ...
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186 views

multiplicity of schemes in positive characteristic

let $K$ a field and $X$ a scheme of finite type over $K$. let $X_1,\dots,X_r$ the irreducible components of $X$ with generic points $\eta_i$. Let $\bar{X}=X\times Spec(\bar{K})$ where $\bar{K}$ is an ...
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28 views

$\mu_p$ as a $Z/p$ torsor

Let us fix the base field $k=\mathbb Q$. Then, say on the etale site, $\mu_p$ is a $\mathbb Z/p$ torsor since locally (after base change to $Q(\mu_p)$), we can pick a root of unity, aka a section ...
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1answer
107 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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47 views

The inclusion $E(K)/2E(K)\hookrightarrow K^*/(K^*)^2$ for Weak Mordell-Weil theorem

I'm reading a master's dissertation where the author proves the weak Mordell-Weil theorem for $\mathbb{Q}$ (i.e.: for any elliptic curve $E:y^2=f(x)=x^3+Ax+B$ over $\mathbb {Q}$, the group $E(\mathbb{...
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134 views

Explanation for the coefficients of L-function of an elliptic curve

I am reading about L-functions of Elliptic curves from Milne's notes. The definition of the L-function of an Elliptic curve defined over $\mathbb Q$ is $$L(E,s)=\prod_{p\text{ good}} (1-a_pp^{-s}+p^{1-...
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185 views

Discriminant of an Elliptic Curves and Good/Bad Reduction

Here is a remark from Silverman's Arithmetic of Elliptic Curves: [in his notation, $K$ is a number field and $M_K^0$ is the set of non-archimedian places in $K$] How does he know that $v(a_1),...,v(...
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24 views

Element is Integer in a Local Field

my question refers to a argumentation step of the proof of VII, Prop. 5.5 in Silverman's "The Arithmetic of Elliptic Curves" (page 199): Setting: $E/K$ is a elliptic over local field $K$. In the ...
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217 views

Properties of horizontal divisors on a fibered surface.

Let $X$ be an integral, regular, projective scheme of dimension $2$, flat over $\operatorname{Spec }\mathbb Z$. I have a couple of questions about horizontal and vertical divisors on $X$: Is it ...
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183 views

Representing a curve as a plane curve in different ways

Let $X$ be a (smooth projective geometrically connected) curve over $k$, where $k$ is a field of characteristic zero. We assume $X$ has genus at least $2$. I know that $X$ has a plane model. More ...
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94 views

theory for contraction maps on curves

do u know any reference for studying the maps $\bar{M}_{i,n}\rightarrow \bar{M}_{j,n-2}$ (moduli space of stable curves with marked points)where the map is given by identifying two marked points? ...
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32 views

Points on an Elliptic Curve, how to interpret $(x(2P):z(2P))$?

This is from J.S. Milne's book 'Elliptic Curves'- With $E(\mathbb{Q}) : Y^2Z = X^3 +aX Z^2+bZ^3$ and given a point on $P =(x,y)$ on it's Weierstrass equation dehomogenized $E: y^2= x^3+ax+b$ where $a,...
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26 views

Riemann Roch theorem for surfaces

Hi am a student of Maths at university; I am studying the theorem of Riemann-Roch for curves. I am interested in understanding what happens in the case of surfaces. I do not want to look for the whole ...
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54 views

Prerequisites/Background for Arithmetic of Shimura Curves

A proposed graduate course for next fall on the "Arithmetic of Shimura Curves" caught my eye for various reasons. I would like to know what background one would need to have to start learning about ...
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40 views

Ring of Definition of Morphism

Let $\phi \colon \mathbb{P}^n_\mathbb{Q} \to \mathbb{P}^m_\mathbb{Q}$ be a morphism over $\mathbb{Q}$. Is there a way in general to find the smallest positive integer $N$ such that $\phi$ is defined ...
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44 views

Pointless, non-singular, absolutely irreducible affine plane curves over finite fields

Crossposted from MO because of contradicting comments there. We think the following is true and have partial results: For all sufficiently large primes $p$ and all natural $g \ge 1$, there exists ...
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51 views

Existence of a $\mathbb{Q}$-rational point from the existence of an $\mathbb{F}_p$-rational point for all $p$?

Suppose I have some $f\in \mathbb{Z}[x]$ and some $a\in \mathbb{Z}$ such that $f(a)\equiv 0 \mod p$ for all primes $p$. Then of course $f(a)=0$. Suppose now I have a scheme $X$ over $\mathbb{Z}$ and ...
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35 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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32 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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66 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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109 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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41 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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18 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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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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30 views

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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62 views

Why do vertical divisor not contribute to the “intersection pairing”?

Let $X\to S=\operatorname{Spec}(O_K)$ be an arithmetic surface. We denote with $X_s$ the fiber over $s\in S$ and let $\operatorname{Div}_s(X)$ be the set of divisors on $X$ with support contained in $...
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91 views

motives in current research

Are motives still studied in current research? Searching various papers I can see that a lot of these works about motives are quite old. I'm interested in arithmetic geometry and i'm also really ...
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Relation between different definitions of modular forms.

I was trying to understand two different approaches to $p$-adic modular forms. First, there is Katz's definition of modular forms. Given a prime $p>5$ (the condition is not necessary for the ...
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42 views

Divisiblity of sum of powers

Let $a_i \in \Bbb C, i=0,\dots,k$ be several complex numbers such that $s_n=\sum_{i=0}^ka_i^n$ are all integers. By checking some examples, I have a question: does there exist a finite set of primes ...
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26 views

If a variety has a $Q$ model, does its blowing up also have $Q$ model?

To be precise, suppose $X$ is a smooth variety defined over the rational number field $Q$, fix a $Q$ model of it, and let $P$ be a $Q$ point of $X$ under this model. Then take $\pi:\widetilde{X}\to X$ ...
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Constant group schemes.

Let $G$ be any group. $R$ is a $Z$-module. Let $G_{SpecR}=\amalg_{g\in G}SpecR$. I see that $G_{SpecR}\times_{SpecZ}G_{Spec R} $ is isomorphic with $\amalg_{g,g^{'}\in G}SpecR$ from the note "finite ...
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65 views

Unipotent Groups and Torsors

I've been doing some reading in some arithmetic geometry, and there's a subtle point that is confusing me slightly: say $U$ is some unipotent affine group scheme over a field $L$, and $T$ is some ...
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80 views

Questions about Zeta Function of Singular Plane Curve

I am working on a project which involves learning about the zeta function (weil zeta function) for plane curves, but I do not know much algebraic geometry. (I do not know anything about schemes). I ...
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31 views

The set of the ideals of $\mathbb{C}$ corresponding to the points of complex geometry

There is parallel relation between the ideals over $\mathbb{C}$ corresponding to the points of complex geometry and the ideals of the field of the complex function, now how to construct the ideals ...
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22 views

Degree of the field extension $\bar{k}(Z_f) /\bar{k}(x)$

I came across an argument on the lecture notes. It does not have a proof of the fact and I want to understand the argument. Let $k$ be a field, $f(x,y) \in \bar{k}[x,y]$. Define $Z_f := \{(a,b) \in ...
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65 views

Does Faltings's theorem imply the set of Diophantine equations are decidable?

Actually, we know all Diophantine equations are not decidable. Does Faltings's theorem imply the sets of Diophantine equations are decidable? That is , there is an algorithm that decide whether those ...
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158 views

Rational points of a scheme X over a field k (Qing Liu Problem 3.7)

I'm working on problem 3.7 from Qing Liu's text on algebraic geometry, and I'm having a bit of difficulty formalizing my intuition. I searched around, and was unable to find the answer elsewhere on ...
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60 views

why the number of irreducible vertical divisors on X is finite?

I am reading Qing liu - gebraic geometry and arithmetic curves Page 356-358 8.3.3 contraction On the head he says the number of irreducible vertical divisors on X is finite ,but I can't know why ...
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63 views

Period matrices of isogenous abelian varieties

It is well-known that two principally polarized abelian varieties of dimension $g$ over $\mathbb{C}$ are isomorphic to each other if and only if their period matrices lie in the same $\text{Sp}_{2g}(\...
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119 views

Arithmetic intersection number

Let $\mathcal{X}$ be a smooth proper model of $X$ over $O_{K,S}$, where $K$ is a number field, $X$ is a projective curve over $K$, $O_{K,S}$ is the ring corresponding to the set $S$ of bad reductions ...
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57 views

Notation Confusion Reading Nick Katz' Proof of RH for (Projective, Smooth, Geometrically Connected) Curves

Trying to read this so-called "Note" by Nick Katz and cannot get over some of his notation. The fundamental group stuff is fine, but then he...seems to define some notation with the same notation in ...