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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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1k views

Tate conjecture for Fermat varieties

I've been looking at Tate's Algebraic Cycles and Poles of Zeta Functions (hard to find online... Google books outline here) and have a question about his work on (conjecturing!) the Tate conjecture ...
16
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0answers
349 views

Prop. 3.2.15 in Liu: Geometrically reduced algebraic variety

I have a problem with proposition 3.2.15 of Algebraic geometry and arithmetic curves of Qing Liu: Let $X$ be an integral algebraic variety over $k$. If $X$ is geometrically reduced then $K(X)$ is a ...
11
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0answers
651 views

Self-Intersection Number $-2$

I am new here, but hopefully you can help me with a concrete problem I have. I try to compute a Self-Intersection Number of a constructed curve in an analytic surface. I know the answer by some ...
8
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0answers
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 ...
8
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92 views

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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74 views

Is torsion of a topological module closed?

I was asking to myself the following question. Consider a $p$-adically complete and separated topological algebra $R$ over $\mathbb{Z}_{p}$. As $\mathbb{Z}_{p}$ is a domain, we know that the $\mathbb{...
7
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220 views

What are applications of etale cohomology and Abelian varieties? (And what is arithmetic geometry?)

First, I apologize for my poor English. I like number theory such as "when can prime $p$ be written as $x^2 + y^2$?" and "find the integer solutions of this equation." Because I've heard that these ...
7
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0answers
396 views

Do Neron models of hyperbolic curves exist

Let $X$ be a smooth projective geometrically connected curve over a number field $K$. Assume that $g\geq 2$. Does there exist a Neron model $\mathcal X$ for $X$ over $O_K$? By a Neron model, I mean ...
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137 views

A question on Grothendieck's paper On the de Rham cohomology of algebraic varieties

In the paper, http://www.numdam.org/article/PMIHES_1966__29__95_0.pdf Grothendieck proves the isomorphism between algebraic de Rham cohomology and Betti cohomology, i.e. for a smooth quasi-...
6
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136 views

Why must the universal cover of a variety with a rational point also have a rational point?

After reading: Why is the fundamental group a sheaf in the etale topology? I followed read the linked survey paper by Minhyong Kim: http://people.maths.ox.ac.uk/kimm/papers/leeds.pdf where at the ...
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92 views

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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141 views

Structure Sheaves of Rigid Analytic Spaces: What is the right Value Category?

Let $K$ be a complete non-archimedean field and let $X$ be an affinoid $K$-space. Then the structure sheaf $\mathcal O_X$ of $X$ is defined first on the "weak Grothendieck topology", ie on affinoid ...
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70 views

When does a variety have a point over a finite field for sufficiently large primes p?

Let $X$ be an algebraic variety over the rational numbers. Suppose that $X$ has positive dimension. I would like to say that $X(\mathbb{F}_p)$ is non-empty for sufficiently large primes $p$. One idea ...
5
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0answers
113 views

Involution On elliptic curve

In an elliptic curve E given by $y^2=x^3 + ax^2 +bx+ c$ and origin at the point of infinity, why does the map $i$ sending $(x, y)$ to $(x, -y)$ send $i(P)=-P$, where $-P$ denotes the inverse of $P$ ...
5
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0answers
77 views

Quasi-coherent $\mathcal{O}_X$-modules

Suppose $X$ is a separated scheme, $F$ an abelian sheaf on $X$ in the Zariski topology. In order to assign an $\mathcal{O}_X$-module structure on $F$, is it enough to do it on a single open affine ...
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233 views

Some questions about reduction of elliptic curves

Let $E \rightarrow S$ be an elliptic curve (i.e, a smooth proper curve of genus 1). If $S = \text{Spec (K)}$ where $K$ is a local field, the usual way of doing a reduction at a prime $\mathfrak{p} = ...
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172 views

Computing a contraction of an exceptional divisor.

For a few days, I have been working on the following problem, from Qing Liu's book: Let $\mathcal{O_K}$ be a discrete valuation ring with uniformizing parameter t and residue characteristic $\neq 2,3$...
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214 views

Why is the trace map on an abelian variety continuous

Let $X$ be a (EDIT) variety with a group structure. For $a\in A$, let $t_a$ be the translation on $X$: $t_a(x) = a+x$. Why is the function $f:X\to \mathbf{C}$ given by $$f(a) = \sum_{i} (-1)^i \...
4
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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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0answers
66 views

Galois representations viewed as the fundamental group of $2,3,5,… \infty$

In a paper of T. Saito http://www.ms.u-tokyo.ac.jp/~t-saito/pp/GR2.pdf he said that the Galois group $\text{Gal}(\mathbb{Q})$ could be seen as the fundamental group of the set $2,3,5,... \infty$. ...
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196 views

Construction of Elliptic Curve with given $j$-invariant

Recall the following well-known result. Let $K$ be a number field, and let $j_0 \in \overline{K} \setminus \{1728\}$. Then there is an elliptic curve defined over the field $K(j_0)$ with $j$-invariant ...
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188 views

Why is this cubic polynomial generic for cyclic field extensions?

[EDIT: There doesn't seem to be any interest in answering this question, so could anyone just provide me a reference for understanding (2), and if possible (1)? Hopefully that would be enough to help ...
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122 views

Which modular forms correspond to abelian varieties?

I have seen in a few places in the literature that given a weight two modular form, we can find an abelian variety that have the same $L$-function. But I haven't been able to find a precise statement ...
4
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0answers
125 views

What constitutes a good reading course in $p$-adic number theory?

I have had a course in number theory where I studied Marcus and also a course in differential geometry. I have read Koblitz's introductory book on $p$-adic numbers. I am roughly interested in both $p$-...
4
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146 views

What are some topics of advanced number theory every young geometers should know? (soft question)

By "advanced number theory", I mean topics like arithmetic/Diophantine geometry, modular/automorphic forms and Shimura varieties. I'm interested in derived/non-commutative algebraic geometry, some ...
4
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0answers
268 views

Tate curve and cusps

I know this is a naive question, but what is the relation between the Tate curve and cusps on a modular curve? Naive googling seems to suggest that level structures on the Tate curve (up to ...
4
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0answers
163 views

Contraction of curves on a surface and stalks

Let $$f:X \rightarrow S$$ be a fibered surface over a Dedekind scheme of dimension $1.$ Let $$s_1, \ldots, s_n$$ be closed points of S and $\{E_{ij}\}$ irreducible vertical divisors of $X$ with $E_{ij}...
4
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0answers
100 views

Trivialising cover for étale morphisms

Let $f:Y \to X$ be a finite étale morphism of smooth and proper schemes over a field $k$ (not necessarily separable closed). Is there a geometrically connected étale cover $\{U_i\}$ of $X$ which ...
4
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0answers
166 views

If $X\times C$ is isomorphic to $Y\times C$, does it follow that $X$ is isomorphic to $Y$

Let $k$ be a field. Let $X$, $Y$ and $C$ be varieties over $k$ such that $X\times_k C $ is isomorphic to $Y\times_k C$. Assume that $C$ is a curve. (The varieties $X$, $Y$ and $C$ are assumed to be ...
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123 views

Curve - non singular curve and its genus

Help me please with this problem:$ X \subset \mathbb{P}^{2}$ defined as $x^{3}y+y^{3}z+z^{3}x=0$ 1.Prove X - non singular curve and find its genus. 2.Prove X - maximal curve over $F_{8}$ field, and ...
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146 views

On local rings of a normalization

Let $X$ an irreducible singular curve with a singular point $x$. Consider $A$, the normalization of $\mathcal{O}_{X,x}$, and $x_1,\dots,x_r$ the points over $x$ in the normalization of $X$. Why $A\...
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93 views

find valuations

consider $f(x)=x^4+5x^3+1$. Let $\alpha$ be a root of this polynomial and consider $K=\mathbb{Q}(\alpha)$. Which are the absolute values on $K$ extending the usual absolute values on $\mathbb{Q}$? Are ...
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48 views

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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93 views

Qing Liu exercise 4.1.3: Non-affine Dedekind scheme

Exercise 4.1.3 of Qing Liu's Algebraic Geometry and Arithmetic Curves asks of us to prove, given a normal Noetherian local scheme $X$ of dimension $2$ with closed point $s$, that $X \backslash \{ s \}$...
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182 views

Different ideal and Kähler differentials

Let $A$ be a Dedekind domain with fraction field $K$, $L/K$ be a finite separable extension and $B$ be the integral closure of $A$ in $L$, which is also a Dedekind domain. Assume all residue ...
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61 views

What is the significance of algebraic functions in algebraic geometry/number theory?

I am especially interested in reading Chevalley's 1951 book on the topic. I would like to know how people were led to the study of algebraic functions historically and also the geometrical way to ...
3
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0answers
66 views

Where does the derivation on the graded algebra of modular forms come from?

Let $M$ be the graded algebra of modular forms for $\operatorname{SL}_2(\Bbb Z)$. It is generated by $Q = E_4, R= E_6$, the Eisenstein series of weight $4$ and $6$ respectively. If we define: $$P = ...
3
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0answers
252 views

Frobenius morphisms of a scheme

I'm trying to undersand the various frobenuis morphism we can consider on a scheme. Those are concepts I just touched, so I'm not sure about some stuff I'm writing and things became clearer while I ...
3
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0answers
136 views

Is a quotient of group schemes well defined?

I will first define the notions of exact,surjective for group schemes and then ask my question. Let $B$ and $C$ be fppf group schemes over S. The book I am reading defines a homomorphism $f:B\to C$ ...
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0answers
49 views

Do there exist any elliptic curves $E\mathbb{F}_8$ satisfying either $\# E(\mathbb{F}_8) = 7$ or $\#E(\mathbb{F}_8) = 11$?

As the question title suggests, do there exist any elliptic curves $E(\mathbb{F}_8)$ satisfying either $\# E(\mathbb{F}_8) = 7$ or $\#E(\mathbb{F}_8) = 11$?
3
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0answers
146 views

Infinite primes (places) of a number field geometrically

Given a (global) number field $K$, thinking of the affine scheme $\mathrm{Spec}\mathcal{O}_K$ can gige an insight into (at least) some kf the number-theoretic terminology, e.g. ramification or local ...
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366 views

Kummer map and cohomology group for an elliptic curve

Let $E=E_q$ be the Tate ellipitc curve over a finite extension $K$ of $\mathbb{Q}_p$ for a $q$. Let $T$ be its p-adic Tate module. Let $\mathfrak m$ be the maximal ideal in $K$. I saw in this paper (...
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0answers
71 views

$E[n]$ is etale locally $(\mathbb{Z}/n\mathbb{Z})^2$

I don't think we need the entire setup below (from Katz and Mazur's elliptic curve book, pages 74 and 75), but, as a beginner, I am unable to identify the assumptions I need. Let $S$ be the open ...
3
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0answers
83 views

Asking for some exercises to help me understanding abelian varieties better?

I want to study Mumford's Abelian Varieties in the coming winter break. I tried to study it before, but I didn't find my self really understanding(or memorizing) too much. I guess a better and more ...
3
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0answers
95 views

Application of GRR in number theory

In Neukirch Book Algebraic Number Theory page 254, states the Grothendieck-Riemann Roch-Theorem, but missing of applications. Do you know references for applications for this theorem, or may be ...
3
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0answers
199 views

Where does the terminology “fake/false elliptic curve” come from?

In describing, say, the moduli of Shimura curves, people often refer to "fake" or "false elliptic curves" ("les fausses courbes elliptiques"), which are abelian surfaces whose endomorphism ring is an ...
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0answers
58 views

Why do we assume the ring to be torsion free when dealing with formal logarithms in the context of formal group laws?

Let $F$ be a formal group over a ring $R$. Why do we require that $R$ has no additive torsion before we discuss formal logarithms?
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88 views

degrees of L-functions and dimensions of Shimura Varieties

I try to grab a (very) little understanding of what a Shimura variety is, and although I still don't understand the formal definition of that notion, a few vague ideas have come to my mind. Hence ...
3
votes
0answers
128 views

Witt vector question

I've started reading various papers and notes on Schemes over the Witt Vectors. In example 8.8 of these: https://www.uni-due.de/~mat903/books/esvibuch.pdf W2 has addition defined as $k \oplus k\cdot ...
3
votes
0answers
91 views

Elliptic curve which attains potential good reduction over an Artin-Schreier extension.

I am looking for an elliptic curve $E$ over the field $\overline{\mathbb{F}_{p}}((t))$, which attains good reduction over an Artin-Schreier extension of $\overline{\mathbb{F}_{p}}((t))$, i.e.: an ...