# Questions tagged [global-fields]

The global-fields tag has no usage guidance.

34
questions

3
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### The second Tate-Shafarevich group of a permutation module is trivial

Suppose I have a global field $K$ and a finite Galois extension $L/K$ of Galois group $G$. It is often written without proofs (it seems that this is a very common statement) that for every $G$-module $...

3
votes

0
answers

175
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### Does every (ordinary) elliptic curve over a quite large prime field $\mathbb{F}_{p}$ have a lift to the function field with huge Mordell-Weil rank?

Ulmer (and then other authors) showed existence of elliptic curves $E$ over the function field $\mathbb{F}_{p}(t)$ (where $p$ is a prime) with huge Mordell-Weil ranks (i.e., depending on $p$). The ...

2
votes

0
answers

99
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### Can global fields be defined as certain topological fields like local fields?

It's known that local fields can be defined as a non-discrete, Hausdorff (equivalently non-indiscrete), locally compact, topological field, which is the same as non-trivial (i.e. neither discrete nor ...

4
votes

0
answers

125
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### Analog of a theorem on equidistribution in adeles

Is there a reference anywhere for the analog of Theorem 6 in chapter XV of Langs Algebraic Number Theory for global function fields?
In my research I have been using this theorem to prove density ...

4
votes

0
answers

97
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### Boundary divisor of an equivariant compactification of a non-trivial twist of $\mathbb{G}_a^n$

Let $K$ be a non-perfect field of positive characteristic. Then there exist non-trivial forms of $\mathbb{G}_a^n$ which split over finite purely inseparable extensions of $K$, i.e., algebraic groups $...

6
votes

0
answers

122
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### Finiteness of wildly ramified cohomology

$\newcommand\p[1]{\left(#1\right)}\newcommand\Char{\operatorname{char}}\newcommand\Gal{\operatorname{Gal}}\newcommand\b[1]{\left\{#1\right\}}$
Let $K$ be a global field. All cohomology below is fppf-...

2
votes

0
answers

196
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### Basic question on the Langlands conjectures for $GL_n$ over global field of positive characteristic

My field is far from the Langlands conjectures. I am just trying to understand some basic ideas.
At the moment I am interested in a global field $K$ of positive characteristic and the group $G=GL_n$. ...

7
votes

1
answer

234
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### Reference request. Finiteness of the Selmer group

Let $K$ be a global field (ie either a number field or the function field of a curve over a finite field). Let $A,B$ be abelian varieties over $K$ and let $\phi:A\to B$ be an isogeny. Associated with $...

2
votes

0
answers

73
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### What is the relationship between ramification in central simple algebras and in fields?

Suppose $K$ is the field of fractions of a Dedekind domain $R$, and let $L$ be a finite extension of $K$. There is a notion of ramification of primes of $K$ in $L$, which describes how $\mathfrak p \...

1
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0
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188
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### Computer algebra programs that can solve polynomial systems on algebraically closed fields besides MAGMA

I was wondering which computer algebra programs out there can solve polynomial systems on the algebraic closure of $\mathbb{Q}$ analytically and efficiently.
So far, I only found MAGMA with its ...

2
votes

0
answers

143
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### Artin map and profinite completion of the idèles

One way to formulate local class field theory is by saying that the local Artin map induces an isomorphism from the profinite completion of $K^\times$ to $\operatorname{Gal}(K^\text{ab}/K)$, which ...

1
vote

0
answers

223
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### Globalization of a local field

I am reading the paper ''Endoscopic classification of representations of quasi-split unitary groups'' by Chung Pang Mok, and cannot come up with the proof of theorem 7.2.1.
Here is the statement.
...

1
vote

0
answers

51
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### Do we have $\cap_{P\in S}\left(\mathcal{O}_P+(1-\tau)(K)\right)=\left(\cap_{P\in S}\mathcal{O}_P\right)+(1-\tau)(K)$?

Let $\mathbb{F}$ be a finite field and let $q$ be its number of elements. Let $(C,\mathcal{O}_C)$ be a geometrically irreducible smooth projective curve over $\mathbb{F}$ and let $K=\mathbb{F}(C)$ be ...

2
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0
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99
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### self dual character of local fields and global fields

There are two concepts of self dual character, one is for global and another is for local.
Let $K$ be an imaginary quadratic number field, and a Hecke character $\chi : \mathbb{A}_K^{\times}/K^{\...

4
votes

1
answer

769
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### Rings of $S$-integers are finitely generated as rings

Let $K$ be a global field (number field or algebraic function field over a finite field), $\mathcal{V}$ the set of $\mathbb{Z}$-valuations on $K$, $S \subseteq \mathcal{V}$ a finite set. The ring of $...

2
votes

1
answer

137
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### Dedekind criterion for function fields

Let $p$ be a prime, $f\in \overline{\mathbb F}_p[x]$ a polynomial of degree $>1$ and $t$ be transcendental over $\mathbb F_p$. Let $i\geq 0$ and let $M=\overline{\mathbb F}_p(t)(\alpha)$, where $\...

1
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0
answers

48
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### How explicitly write a projective transformation between the conics over the univariate function field?

Consider the quadratic forms
$$
Q_1 = x^2 + y^2 - (t^2+1)z^2,\qquad Q_2 = x^2 + y^2 - z^2
$$
over the rational function field $\mathbb{F}_p(t)$, where $p > 2$ is a prime such that $t^2 + 1$ is ...

4
votes

1
answer

219
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### Sums of squares in global fields (Reference Request)

There is a result due to Siegel that, for a number field $K$, any totally positive element of $K$ is the sum of four squares of $K$. This is discussed in another question (sum of squares in ring of ...

3
votes

1
answer

238
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### Quadratic equation over a global field of characteristic 2

Let $F=\mathbb F_{2^n}(t)$, and let $f=x^2+ax+b\in F[x]$. Is there any necessary and sufficient condition for $f$, depending on its coefficients, to have a root in $F$? I'm not interested in finding ...

5
votes

2
answers

242
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### Does a $K_{\upsilon}$-point of a variety $V$ give a point of $V$ in $K_{(\upsilon)}=K^{sep}\cap K_{\upsilon}$ for a global field $K$?

Let $K$ be a global field and $\upsilon$ a place of $K$. Let $K_{\upsilon}$ denote the completion of $K$ at $\upsilon$ and $K_{(\upsilon)}:=K^{sep}\cap K_{\upsilon}$ the henselization (which is the ...

6
votes

2
answers

752
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### Why is $K_{\upsilon}|K$ separable for a global field $K$?

I asked this question on math.stackexchange but maybe it fits here better. If not, I apologize in advance and will remove the question.
Let $K$ be a global field and $\upsilon$ a prime of $K$. Then ...

1
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0
answers

145
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### Are elliptic global function fields totally imaginary?

Let $K/\mathbb{F}_{q}(X)$ be a global function field extension of degree $2$, having minimal polynomial $Y^{2}=f(X)$, where $f(X)$ is a degree 3 polynomial in $X$ with coefficients in $\mathbb{F}_{q}$,...

4
votes

1
answer

642
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### Reference for: Every local field can be realized as the completion of a global field

It is well known that every local field (i.e. nondiscrete topological field locally compact with respect to the topology) is the completion of some global field. I know the argument, a nice ...

2
votes

0
answers

119
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### Property of a derivative in global field

Before posting I want to make it clear that I posted the same question in stack exchange awhile ago (https://math.stackexchange.com/questions/1533814/property-of-derivative-in-a-local-field) but didn'...

4
votes

0
answers

542
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### Explicit description/calculation of norm group of ideles of characteristic $p$ global field

I posted the same question earlier in stack exchange,
(https://math.stackexchange.com/questions/1130391/algebraic-proof-of-2nd-inequality-of-global-class-field)
thinking it is most definitely not a ...

5
votes

1
answer

297
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### Compactness of adelic quotients for unipotent groups over global fields

Let $K$ be a global field, $\mathbb{A}_K$ the ring of adeles, and $U$ a unipotent algebraic group over $K$. Why is $U(\mathbb{A}_K)/U(K)$, when endowed with the quotient topology, compact?

2
votes

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answers

415
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### Global Artin reciprocity law from Local class field theory

Let $K=\mathbb F_q((t)), p -$ prime ideal in $K$, $\psi_p$ be the local Artin map$K_p^* \to Gal(K_p^{ab}/K_p)=G_p \subset Gal(K^{ab}/K)$. Then I define global Artin map $\psi_K$as product of $\psi_p$, ...

9
votes

3
answers

3k
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### Elliptic Curves over Global Function Fields

I am currently thinking about a problem, and I feel that by knowing more about elliptic curves over extensions of $\mathbb{F}_q(T)$, for $q$ a power of $p$ say, might lead to insight. I am also ...

72
votes

3
answers

6k
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### Is there a "purely algebraic" proof of the finiteness of the class number?

The background is as follows: I have been whittling away at my commutative algebra notes (or, rather at commutative algebra itself, I suppose) recently for the occasion of a course I will be teaching ...

17
votes

4
answers

2k
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### Dimension of central simple algebra over a global field "built using class field theory".

If $F$ is a global field then a standard exact sequence relating the Brauer groups of $F$ and its completions is the following:
$$0\to Br(F)\to\oplus_v Br(F_v)\to\mathbf{Q}/\mathbf{Z}\to 0.$$
The ...

6
votes

1
answer

615
views

### Can algebraic number fields be generalized in a similar way to function fields in 1 variable over a finite field?

Global fields consist of finite extensions of $\mathbb{Q}$ (algebraic number fields) and finite extensions of $\mathbb{F}_q(x)$ (function fields in 1 variable over a finite field). The latter are ...

58
votes

9
answers

15k
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### Learning Class Field Theory: Local or Global First?

I've noticed that there seem to be two approaches to learning class field theory. The first is to first learn about local fields and local class field theory, and then prove the basic theorems about ...

13
votes

2
answers

643
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### Evidence for $Q^{\operatorname{solv}}$ being pseudo-algebraically-closed

This is a follow-up to the following answer:
Solvable class field theory
in which it is stated as a "folklore" conjecture that the maximal solvable extension of Q is pseudo algebraically closed (...

29
votes

5
answers

6k
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### Global fields: What exactly is the analogy between number fields and function fields?

Note: This comes up as a byproduct of Qiaochu's question "What are examples of good toy models in mathematics?"
There seems to be a general philosophy that problems over function fields are easier to ...