81 votes
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Motivating Lubin-Tate theory

Sorry I didn’t see this earlier. My memory is vague, and probably colored by subsequent events and results, but here’s how I recall things happening. Since I had read and enjoyed Lazard’s paper on ...
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  • 3,818
22 votes
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What is a tamely-ramified Weil-Deligne representation?

$\def\R{\mathbf{R}}$ $\def\Z{\mathbf{Z}}$ $\def\Q{\mathbf{Q}}$ $\def\Qbar{\overline{\Q}}$ $\def\F{\mathbf{F}}$ $\def\GL{\mathrm{GL}}$ $\def\Gal{\mathrm{Gal}}$ Here are some thoughts on your question ...
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18 votes

Does the discriminant of an irreducible polynomial of a fixed degree determine the discriminant of the number field it generates?

The discriminants of the irreducible polynomials $$(x^2-2)^2+60 = x^4 - 4 x^2 + 64, \qquad (x^2+2)^2+60 = x^4 + 4 x^2 + 64$$ are both equal to $58982400 = 2^{18} \cdot 3^2 \cdot 5^2$. However, the ...
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15 votes
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On unramified p-adic groups

Let us start with (b) => (a). We only need to show that if $\mathcal{G}$ is a reductive $O$-group scheme, then $\mathcal{G}_F$ is quasi-split and split after an unramified extension. Let us first ...
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15 votes
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Conductor as volume of the integers ring

Apply the Fourier Inversion Formula to the characteristic function $\Phi(x) = \chi_\mathcal{O}(x)$ of the ring $\mathcal{O}$ of integers in $F$. The Fourier transform is the integral $\widehat{\Phi }(...
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14 votes
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Is every connected reductive group over a local field already defined over a global field?

Pick a global field $E$ and finite place $w$ with $E_w=K$. The fraction field $k$ over $E$ of the henselization of the "algebraic" local ring at $w$ is the direct limit of finite separable sub ...
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  • 1,369
14 votes
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Totally ramified subextension in a finite extension of $\mathbf{Q}_p$

This is not a complete answer, but perhaps it's a roadmap to a counterexample. My strategy is to consider some non-Galois $K/\mathbf{Q}_p$ for which the result is true, and let's make some deductions ...
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13 votes
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Examples to keep in mind while reading the book 'The Admissible Dual...' by Bushnell and Kutzko and the importance of Interwining of representations

Your question (f) makes me suspect that you don't really know any of the representation theory of $p$-adic groups at all. You definitely should not try to read the Bushnell--Kutzko book before ...
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  • 574
13 votes
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Does $0\to I\to\mathrm{Gal}_K\to\mathrm{Gal}_k\to 0$ always split?

Good question! Let me try to guess what Gabber had in mind there. (Note that he only says "known" (to him), not "well-known"...) The claim is that the extension splits. Note that ...
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11 votes

Motivating Lubin-Tate theory

Abelian extensions of $\mathbb{Q}$ can be described using torsion points in the multiplicative group. If $K$ is a quadratic imaginary field, and $E$ is an elliptic curve where $\mathcal{O}_K$ acts by ...
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11 votes

Local inverse Galois problem

The short answer is (as far as I am aware) no, but there is a lot that is known. Jannsen and Wingberg have given an explicit presentation for $Gal(\overline{K}/K)$ in the case that the residue ...
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  • 17.8k
11 votes

Which groups are Galois over some p-adic field?

I'll upgrade my comment to an answer. Any finite Galois extension of $\mathbb{Q}_l$ of degree coprime to $l$ is tamely ramified. In particular, its Galois group is an extension of two cyclic groups. ...
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11 votes
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Rational points on varieties over local fields

Let us assume that $X$ is smooth and projective for simplicity, given by a number of polynomial equations with coefficients in the ring of integers $\mathcal O$ of $k$. Let $\kappa$ denote the residue ...
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11 votes
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Langlands correspondence for higher local fields?

The Langlands correspondence for higher local fields is still at an early stage of development. I haven't really kept up with it, but here's some key points. As the question stated, and Loren ...
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  • 12.5k
11 votes
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Why does the field norm on the field extension $ \mathbb C/\mathbb R $ induce a vector space norm?

The map $|N(\cdot)|^{1/n}$ is a continuous multiplicative extension of $|\cdot|$. By a multiplicative function I mean a function $\chi:L\to [0,\infty)$ such that $\chi(0)=0$, $\chi(1)=1$ and for every ...
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  • 10k
10 votes

Local inverse Galois problem

Three comments (which I don't have enough reputation to add as comments): The parenthetical claim in the statement of the question is false: Galois groups of local fields need not be supersolvable. ...
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10 votes
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Type of place versus type of unitary group

Things are perhaps a bit messier than you hope. In particular it is not true that the unitary group is non-quasi-split if and only if $v$ ramifies. Disclaimer: I did not know the answer to this ...
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10 votes

Totally ramified subextension in a finite extension of $\mathbf{Q}_p$

Note that the question is equivalent to the following: Given $K/\mathbb{Q}_p$, is there $L/\mathbb{Q}_p$ totally ramified so that $KL/K$ and $KL/L$ are unramified? You note that it is true for $K/\...
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  • 601
10 votes
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Is there any explicit description of the maximal totally ramified extension of $\mathbb{Q}_p$?

A composite of totally ramified extensions need not be totally ramified: Example 1. (As per LSpice's suggestion) Consider the extensions $\mathbb Q_p(\sqrt{p})$ and $\mathbb Q_p(\sqrt{\varepsilon p})$...
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9 votes

Are the abelian absolute Galois groups of these local fields isomorphic?

This is how to answer the question (but it's not an answer). (Edit: it and the comments below now form an answer). Let $K_1$ and $K_2$ be the fields in question. By local class field theory you're ...
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  • 355
8 votes
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Is $G \rightarrow G/P$ surjective on $K$-points over a local field?

The map G(K) to (G/P)(K) is surjective over any field K. Here is a link to an explanation by Brian Conrad. http://math.stanford.edu/~conrad/249CS13Page/handouts/parsurj.pdf
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8 votes

Which groups are Galois over some p-adic field?

When $p \neq \ell$, if $N/K$ has Galois group $G$ then $N/K$ is tamely ramified. It follows that $N = K(\sqrt[e]{\pi}, \zeta)$ where $e$ is the ramification degree of $N/K$, $\pi$ is some uniformizer ...
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  • 601
8 votes

Are the abelian absolute Galois groups of these local fields isomorphic?

Although @znt has gotten the answer through pari, I think it may be instructive to outline my argument. It all depends on the transition function of Higher Ramification Theory: for a finite extension ...
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  • 3,818
7 votes

Reference request for Kato's paper: A generalization of local class field theory by using K -groups

I found this old question while searching for Kato's paper myself. Just in case anyone else is also still looking for these, here's what I found. Kato's work was published in three installments in J. ...
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7 votes
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Is the intersection of ramification groups in upper numbering of a $p$-adic local field trivial?

Yes, at least if the upper ramification groups $G^\nu$ are defined as $\varprojlim_L\mathrm{Gal}(L/K)^\nu$ for $L/K$ finite Galois (e.g. as in [1]). This makes sense because the upper-numbering is ...
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  • 1,366
7 votes
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Is the set of hyperelliptic curves with a K-point closed?

In the "more sophisticated" direction, we can ask a similar question about the moduli stack $\mathscr{M}_g$ of hyperelliptic curves of genus $g$. If $K$ is a topological field, there is a ...
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6 votes

Finite extension of local fields

Yes. For an example, let $k$ be an uncountable algebraically closed field of characteristic $p>0$, and let $K = k((x))$, the field of fractions of the ring of formal power series $\mathcal{O}_K = k[...
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6 votes
Accepted

Theorem 7b of Serre's "Propriétés galoisiennes des points d'ordre fini des courbes elliptiques"

This is ultimately an application of Lang's vanishing theorem for degree-1 Galois cohomology of connected algebraic groups over finite fields (applied to tori). What follows may look complicated if ...
6 votes

Maximal separable extension of $\mathbb F_q((t))$

This is not correct. If it was the compositum then you would get that the absolute Galois group is a product of pro-$p$ group and pro-$p$' groups (i.e. inverse limit of prime to $p$ finite groups). ...
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6 votes

Argument of Zariski density to prove rationality of a regular map

This is valid in any characteristic (over an infinite field) and has nothing to do with completions, and is a "relative schematic density" result. It is a special case of EGA IV$_3$ 11.9.13, but it ...

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