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Questions tagged [global-fields]

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39 questions
Newest Active Bountied Unanswered
6 votes
2 answers
469 views

Let $k$ be a global function field of positive characteristic $p$ (e.g. $k = \mathbb{F}_p[t]$). Let $x \in k$ be non-zero and assume that $x$ is not a $p$th power. For each place $v$ of $k$, we can ...
Daniel Loughran's user avatar
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9 votes
3 answers
646 views

Let $L/K$ be a cyclic number field extension. The Hasse Norm theorem states that an element $x$ in $K$ is a norm from $L$ iff it's locally a norm, i.e. at each place $v$, $x$ is a norm from $L_v$ to $...
Milan Boutros's user avatar
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1 vote
0 answers
342 views

According to a remark from wikipedia the motivation of Lubin-Tate theory arose from the analogy to the way in which elliptic curves $E/K$ over a number field $K$ with extra endomorphisms (ie those ...
user267839's user avatar
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3 votes
0 answers
90 views

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
  • 201
2 votes
0 answers
121 views

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
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3 votes
1 answer
390 views

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
  • 186
3 votes
0 answers
199 views

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
  • 1,873
3 votes
0 answers
185 views

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
  • 632
4 votes
0 answers
187 views

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
  • 339
4 votes
0 answers
144 views

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
  • 81
6 votes
0 answers
176 views

$\newcommand\p[1]{\left(#1\right)}\newcommand\Char{\operatorname{char}}\newcommand\Gal{\operatorname{Gal}}\newcommand\b[1]{\left\{#1\right\}}$ Let $K$ be a global field. All cohomology below is fppf-...
Niven's user avatar
  • 161
3 votes
0 answers
287 views

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
  • 23.3k
7 votes
1 answer
430 views

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
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2 votes
0 answers
146 views

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
331 views

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
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