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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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255 views

What is the significance of Coleman maps arising in Iwasawa theory?

I have come across two instances of "Coleman map" Let $E$ be an elliptic curve defined over $\mathbb{Q}_p$. Let $k_\infty$ be the unique $\mathbb{Z}_p$ extension of $\mathbb{Q}_p$ contained in $\...
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2answers
219 views

How does $\text{Gal}(L/K)$ act on the automorphism group of an elliptic curve?

Let $L/K$ be a finite Galois extension of number fields; I'm interested mainly in the case $K = \Bbb{Q}$ and $L= \Bbb{Q}(\sqrt{d})$. Let $X$ be an elliptic curve over $K$ and $\text{Aut}(X_L)$ the ...
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989 views

The canonical divisor of the projective line

Let $A$ be a ring. Assume it has some nice properties if necessary, e.g., $A$ is a Dedekind domain. Let $\mathbf{P}^1_A$ be the projective line over $A$. I want to show that $-2 [\infty]$ is a ...
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152 views

Very special rational points on curves over number fields

For some reason, I'm convinced the answer to the following question should be (obviously) negative, but I can't come up with a good reason. Does there exist a number field $K$, a smooth projective ...
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1answer
352 views

Characterization of Horizontal Irreducible Divisors

I´m asking for a proof of a fact used by Arakelov in his paper: Intersection Theory of Divisors on an Arithmetic Surface (page 1169 row 16). He gives no references or explanations for this fact. The ...
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381 views

What is the difference between algebraic number theory, arithmetic geometry and diophantine geometry?

So, arithmetic geometry studies objects like varities over any field not necessarly algebraiclly closed and it seems that diophantine geometry is closely related and I'm concerned about the relation ...
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301 views

The normalization of the peculiar curve $x^p + y^p - (x+y)^p = 0$

Fix a prime number $p$. Consider the affine curve $C$ in $\mathbf{A}^2$ over a number field $K$ given by the equation $x^p+y^p - (x+y)^p =0$. Its Jacobi matrix is $(px^{p-1} -p(x+y)^{p-1} \ py^{p-1} ...
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120 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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1answer
164 views

Galois representation on Tate module of a twist of an elliptic curve

Let $E/K$ be an elliptic curve with a twist $E'/K$. Let $f: E_{\overline K} \to E'_{\overline K}$ be an isomorphism. Let $m(\sigma) = f^{-1}\circ f^\sigma$ be the 1-cocycle corresponding to this twist....
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Conjugations in the comparison isomorphisms between Betti cohomology and algebraic de Rham cohomology

For a smooth projective variety $X$ defined over $k$ which admits a real embedding $\sigma:k \rightarrow \mathbb{C}$, its Betti cohomology is defined by \begin{equation} H^*_{B,\sigma}(X):=H^*(X \...
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Problems on showing that a reduction map is defined, and that a certain scheme is finite.

I am currently on the last chapters in Liu's book and I am trying to solve the following problem, which is the first step in showing that a certian reduction map is well-defined: Let $X \rightarrow T$ ...
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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 $P$ ...
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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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237 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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174 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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216 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 \...
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4answers
268 views

How do I write down a curve with exactly one rational point

Let $g\geq 1$. I would like to write down (for all $g$) a smooth projective geometrically connected curve $X$ over $\mathbf{Q}$ of genus $g$ with precisely one rational point. Is this possible? For ...
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2answers
294 views

isomorphisms of algebraic closures

let $K$ be an algebraically closed field. Consider the algebraic closure $\overline{K(X)}$ of $K(X)$, with $X$ trascendent over $K$. Are there cases in which $\overline{K(X)}\cong K$? where $\cong$ is ...
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2answers
91 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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2answers
145 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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1answer
99 views

Why is the rank of $f_\ast L$ the degree of $f$

Let $f:X\to Y$ be a finite morphism of curves. Let $L$ be a line bundle on $X$. Why is $f_\ast L$ a line bundle and is the degree of $f_\ast L$ equal to $\deg f$ or $\deg f+ \deg L$? Here is my ...
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1answer
132 views

The universal cover of the multiplicative group over the field of algebraic numbers

Let $X=\mathbf{A}^1_{\overline{\mathbf{Q}}}-\{0\} = \mathbf{G}_{m,\overline{\mathbf{Q}}}$ be the multiplicative over the field of algebraic numbers. Each finite etale cover $Y\to X$ (with $Y$ ...
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2answers
166 views

Bound for number of points on surface over $\mathbb{F}_p$

I know of the bound for the number of points on an elliptic curve over a finite field: $$|\# E(\mathbb{F}_q) - q - 1| < 2\sqrt{q}$$ where this includes the point at infinity. I have been told that ...
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1answer
103 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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1answer
136 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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2answers
120 views

${\mathbb{Q}_p^*}^2$ is open in $\mathbb{Q}_p^*$

Show that the set of squares in $\mathbb{Q}_p^*$ is open in $\mathbb{Q}_p^*$. Here $\mathbb{Q}_p$ is the $p$-adic numbers and $\mathbb{Q}_p^*$ is the set of units in $\mathbb{Q}_p$. I know that $\...
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1answer
181 views

Etale cohomology and algebraic closure

$\DeclareMathOperator{\h}{H}$Apologies in advance if this is overly stupid. Let $k$ be a field and $X$ a variety over $k$. Let $n$ be an integer which is invertible in $k$. One often looks at the ...
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Torsion in weird groups

Let $\mathbf{C}$ be the complex numbers, and $\mathbf{Q}_p$ the $p$-adic rationals, embedded into $\mathbf{C}$ by some field embedding $s$. Is the abelian group $\Gamma := \mathbf{C}/s(\mathbf{Q}_p)...
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1answer
281 views

Grothendieck's Vanishing Cycles

Suppose $S$ is the spectrum of a strict henselian ring $R$ which is also a discrete valuation ring (DVR), then $S$ consists of a closed point $s$ and a generic point $\eta$. We have a henselian trait, ...
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1answer
91 views

$E/\mathbb{F}_p$ supersingular elliptic curve, what's $\# E(\mathbb{F}_{p^n})$?

Let $E/\mathbb{F}_p$ be a supersingular elliptic curve with $p \ge 5$ prime, and let $n \ge 1$ be an integer. How do I see that$$\# E(\mathbb{F}_{p^n}) = \begin{cases} p^n + 1 & \text{if }n\text{ ...
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1answer
141 views

Zeta function of smooth conic?

Let $k$ be a finite field of characteristic different from $2$. Let $X = V(x^2 + y^2 + z^2) \subset \mathbb{P}_k^2$ be the smooth conic. What is the zeta function of $X$?
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363 views

Abelian subvarieties of a principally polarized abelian variety are principally polarized

Let $A$ be a principally polarized abelian variety. Let $X\subset A$ be an abelian subvariety. Is $X$ also principally polarized? Here's what I think should be a proof. Is it correct? We may and do ...
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1answer
177 views

Reduction map on torsion of elliptic curves

Let $E$ be an elliptic curve over $\mathbb{Q}$ with good reduction at a prime $p$. It is well-known that the map $$E[N]\to E_p[N]$$ is injective when $p\nmid N$. It is even a bijection since both ...
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1answer
258 views

Group law for an elliptic curve using schemes

I was trying to understand better the definition of the group law for an elliptic curve given in Katz and Mazur's book (http://books.google.com.br/books/about/Arithmetic_Moduli_of_Elliptic_Curves.html?...
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2answers
470 views

Genus of curves embedded into some projective space

The Plucker formula allows one to calculate the genus of a curve embedded into $\mathbf{P}^2$. Does there exist a "higher-dimensional" analogue of the Plucker formula? More precisely, let $f_1,\ldots,...
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1answer
232 views

$\pi^{tame}(\mathbb{A}^1_k)$ is trivial

Fixed an algebraically closed field of characteristic $p>0$, it is well known the result of the title: $\pi^{tame}(\mathbb{A}^1_k)\simeq 1$. Where the tame fundamental group, in this situation, ...
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1answer
199 views

Fraction field of $p$-adic power series ring

Let $L$ be a finite extension of $\mathbf{Q}_p$. Write $$\mathcal{O}_{\mathcal{E}} = \left\{ f = \sum_{k \in \mathbf{Z}} a_kT^k \in \mathcal{O}_{L}[[T,T^{-1}]] \mid \lim_{k \to -\infty} a_k = 0\right\...
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1answer
268 views

elliptic k3 surface and Shioda Inose structure

We know that suppose given two elliptic curves $E$ and $E'$, there is a Kummer surface $km(E,E')$. And I'm curious suppose we know a $K3$ surface is kummer, how do we recover the pair $(E,E')$? For ...
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1answer
153 views

Krull's intersection theorem in the q-expansion principle

I'm currently reading the proof of the q-expansion principle in Katz'73 paper "p-adic properties of modular schemes and modular forms" . The principle itself is a Corollary (1.6.2) of Theorem 1.6.1, ...
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1answer
398 views

Primes of good reduction for varieties

Suppose I have a (smooth, projective) variety $X/\mathbb{Q}$. Is the notion of a primes of bad/good reduction well-defined from this data? Motivation and attempt at an answer: The question should be ...
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1answer
302 views

Are these two notions of Galois morphism the same

Let $f:X\to Y$ be a finite morphism of integral schemes. Let $G$ be the automorphism group of $X$ over $Y$. Are the following two conditions equivalent? The function field extension $K(Y)\subset K(X)...
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1answer
289 views

discriminant of an étale cover of an elliptic curve

Let $\pi:X\to E$ be a finite étale morphism, where $E$ is an elliptic curve over a number field $K$. Assume $X$ to be connected, and to be of genus 1. Edit: Assume $X$ and $E$ have semi-stable ...
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1answer
100 views

coefficients of the zeta function of curve over a finite field $\mathbb{F}_q$

Let $C$ be a non-singular curve over $\mathbb{F}_q$. Denote by $d$ the degree map from the group of divisors to $\mathbb{Z}$ and denote by $P$ the set of prime divisors w.r.t. to the function field. ...
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67 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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205 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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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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214 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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195 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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124 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 ...
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130 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 $p$-...