Questions tagged [global-fields]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
3 votes
0 answers
48 views

On the complexity of global fields isomorphism

Let $L$ and $K$ be isomorphic global fields (i.e. either function fields of curves over a finite field, or number fields). What is the complexity of finding an isomorphism between them? Is there a ...
Reyx_0's user avatar
  • 131
2 votes
0 answers
46 views

Constructing a cyclic extension $L$ with given local behavior of a global field $K$ such that $L$ is normal over a subfield $F$ of $K$

Let $F$ be a global field without real places (that is, a function field or a totally imaginary number field). Let $K/F$ be a cyclic extension of degree $n$. Let $S$ be a ${\rm Gal}(K/F)$-invariant ...
Mikhail Borovoi's user avatar
3 votes
1 answer
223 views

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 $...
Tuvasbien's user avatar
  • 156
3 votes
0 answers
175 views

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 ...
Dimitri Koshelev's user avatar
2 votes
0 answers
112 views

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 ...
Z Wu's user avatar
  • 340
4 votes
0 answers
128 views

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 ...
Boaz Moerman's user avatar
4 votes
0 answers
99 views

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 $...
Abdulmuhsin Alfaraj's user avatar
6 votes
0 answers
129 views

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-...
Niven's user avatar
  • 161
3 votes
0 answers
223 views

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 $G=GL_n$. ...
asv's user avatar
  • 21.1k
7 votes
1 answer
264 views

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 $...
Damian Rössler's user avatar
2 votes
0 answers
78 views

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 \...
user's user avatar
  • 121
1 vote
0 answers
205 views

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 ...
ArminJR's user avatar
  • 21
2 votes
0 answers
149 views

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 ...
Antoine Labelle's user avatar
1 vote
0 answers
228 views

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. ...
Aut's user avatar
  • 347
1 vote
0 answers
51 views

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 ...
Stabilo's user avatar
  • 1,469
2 votes
0 answers
102 views

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^{\...
Yi_Feng's user avatar
  • 47
4 votes
1 answer
818 views

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 $...
Bib-lost's user avatar
  • 267
2 votes
1 answer
137 views

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 $\...
Pirate1234's user avatar
1 vote
0 answers
48 views

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 ...
Dimitri Koshelev's user avatar
4 votes
1 answer
231 views

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 ...
user221330's user avatar
3 votes
1 answer
245 views

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 ...
Ferra's user avatar
  • 509
5 votes
2 answers
245 views

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 ...
Frida's user avatar
  • 111
6 votes
2 answers
768 views

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 ...
Frida's user avatar
  • 111
1 vote
0 answers
145 views

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 $\mathbb{F}_{q}$,...
Hair80's user avatar
  • 655
4 votes
1 answer
669 views

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 ...
jaycegetz's user avatar
  • 412
2 votes
0 answers
119 views

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'...
Jack Yoon's user avatar
  • 111
4 votes
0 answers
546 views

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 ...
Jack Yoon's user avatar
  • 111
5 votes
1 answer
310 views

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?
Question Mark's user avatar
2 votes
0 answers
420 views

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 $\psi_p$, ...
oznd's user avatar
  • 71
9 votes
3 answers
3k views

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 ...
Giuseppe's user avatar
  • 791
73 votes
3 answers
6k views

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 ...
Pete L. Clark's user avatar
17 votes
4 answers
2k views

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 ...
Kevin Buzzard's user avatar
6 votes
1 answer
622 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 ...
teil's user avatar
  • 4,261
58 votes
9 answers
15k views

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 ...
David Corwin's user avatar
  • 15.1k
13 votes
2 answers
649 views

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 (...
Brandon Levin's user avatar
30 votes
5 answers
6k views

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 ...
user avatar