Questions tagged [local-fields]
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262 questions
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Image of norm map for local field
Let $F$ be a finite extension of $Q_2$ (2-adic field) or $F_2((x))$ (function field). Let $E/F$ be a separable extension of degree $2$.
What is the image of the norm map $N_{E/F}$?
In particular - ...
CommunityBot
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How to understand the representation theory of $SL(n)$ from $GL(n)$?
Let $F$ be a local field. Consider the group extension (split)
$$ PSL(n,F) \rightarrow PGL(n,F) \rightarrow F^\times / (F^\times)^n.$$
What knowledge about $PGL(n)$ is necessary in order to understand ...
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Automorphisms of local fields
It is an amusing coincidence (at least it appears to be a coincidence to me) that any completion of the field $\mathbb{Q}$ has trivial automorphism group as an abstract field, i.e. when ignoring the ...
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squares in dyadic local fields
Hello,
By the local square theorem I know that $1+4\alpha$ is square if $|\alpha|<1$ ($\alpha$ is not a unit). Now, Can I always get a unit $\alpha$ such that $1+4\alpha$ is not a square ?? For ...
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Brauer group of complete DVR
Let $A$ be a complete discrete valuation ring with fraction field $K$ and perfect residue field $\kappa$.
Let $K_{nr}$ be the maximal unramified extension of $K$ and let $A_{nr}$ be its ring of ...
- 5,670
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cuspidal types and Iwahori subgroup for $SL(2,F)$
Let $(J,\pi)$ be a cuspidal type in $SL(2,F)$, $F$ is a non-Arch. local field and let $I$ be the Iwahori subgroup of $SL(2,F)$.
Is there any possibility that $J\subset I$ or even a subgroup?
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integrating a character of a non-archimedean local field
By way of motivation, this computation comes from a proof in Bump's book Automorphic Forms and Representations where he shows that the Weil index of the reduced norm of a four-dimensional central ...
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Computing (on a computer) higher ramification groups and/or conductors of representations.
I am supervising an undergraduate for a project in which he's going to talk about the relationship between Galois representations and modular forms. We decided we'd figure out a few examples of weight ...
- 5,363
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What are the open normal subgroups of the inertia group of a local field?
Let $K$ be a non-Archimedean local field, i.e., complete with respect to a non-trivial, non-archimedean discrete absolute value, with finite residue field $k$ of characteristic $p\neq 0$. Also let $...
- 3,380
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Conductors of Weil-Deligne representations
Suppose $(V,N)$ is an $n$-dimensional semisimple $WD$ representation of $W_{\mathbb{Q}_p}$. This corresponds under local Langlands to an admissable representation $\pi$ of $GL_n(\mathbb{Q}_p)$. Is ...
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Is there any approximated version of Hilbert 90?
Suppose $K$ is a local field and $L$ a finite cyclic extension of $K$. By Hilbert 90, we know that if an element $a$ in $ L$ such that $N_{L / K}(a) =1 $ then $ a = b / \sigma(b) $ for some $b$ in $L$ ...
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Why isn't there a structure with two primes?
I don't know whether this question is a bit too vague for MO or not, so feel free to delete it if you see fit.
The p-adic integer is defined by taking the inverse limit $\ldots \mathbb{Z} / p^2 \...
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