Questions tagged [local-fields]
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262 questions
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Q_p*/(Q_p*)^2 and descent for elliptic curves
Is there a simple description of the group Q_p*/(Q_p*)^2 where Q_p denotes the p-adic integers?
I am doing descent calculations for elliptic curves, and so am most interested in the case p = 2. ...
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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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metaplectic group does not split
I'm trying to understand the Weil representation and hope there are some experts around who can set me straight. Let $F$ be a non-Archimedean local field (I don't mind assuming that the characteristic ...
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A remark of Mordell alluding to a local/global principle for cubic Diophantine equations
In Mordell Diophantine Equations he says:
In recent years it has been shown that there seems to be a close connection between the number of solutions of f(x,y) = 0 (mod $p^r$) and the existence of ...
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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 - ...
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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 $...
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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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Given $v,w$ primes of $k$, is there $K/k$ so $K_v\cap\Bbb Q^\text{cycl}=K_w\cap\Bbb Q^\text{cycl}=K\cap\Bbb Q^\text{cycl}$?
For any field $k$, let $\mu(k)$ denote the roots of unity in $k$. Now let $k$ be a number field and let $v, w$ be non-archimedean primes of $k$ with distinct residual characteristics. Does there ...
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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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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 ...
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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 ...
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