86
votes
Accepted
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 ...
- 4,048
22
votes
Accepted
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 ...
- 181
15
votes
Accepted
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 }(...
- 11.1k
14
votes
Accepted
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 ...
- 40.3k
14
votes
Accepted
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 ...
- 11.4k
13
votes
Accepted
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 ...
- 574
13
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 ...
- 150k
12
votes
Accepted
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 ...
- 18.4k
11
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. ...
- 111
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. ...
- 20.6k
11
votes
Accepted
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 ...
- 11.2k
11
votes
Accepted
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 $\mathbb Q_p(\sqrt{\varepsilon p})$...
- 26.5k
11
votes
Accepted
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 ...
- 13k
10
votes
Accepted
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 ...
- 40.3k
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/\...
- 601
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 ...
- 355
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 ...
- 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 ...
- 4,048
8
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. ...
- 35.8k
7
votes
Accepted
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 ...
- 1,376
7
votes
Accepted
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 ...
- 19.2k
7
votes
Why does the field norm on the field extension $ \mathbb C/\mathbb R $ induce a vector space norm?
In fact, more is true: for any local field $K$, any degree $n$ field extension $L$ of $K$ and any absolute value $|\cdot|$ on $K$, $|N_K^L(\cdot)|^{1/n}$ is the unique absolute value on $L$
which ...
- 11.4k
7
votes
Accepted
A question about mod $p$ local Langlands for $\mathrm{GL}_{2}(\mathbb{Q}_{p})$
You seem to be expecting that mod $p$ local Langlands should satisfy the same compatibilities as "conventional" local Langlands (for smooth representations of $GL_2(\mathbf{Q}_p)$ and $WD(\...
- 35.8k
7
votes
Is every compact simply-connected reductive p-adic group perfect?
At your request, I post my comment as an answer: the answer to the Related question is "no", i.e., all simple, anisotropic groups over a non-Archimedean local field are of type $\mathsf A$; ...
- 10.6k
7
votes
Accepted
Looking for proof of Serre's mass formula
The article is not only available in Serre's Collected Papers: it first appeared in a journal, after all. Here's a scan from the Comptes Rendus archives: https://gallica.bnf.fr/ark:/12148/...
- 49.1k
7
votes
What are the jumps in the ramification filtration of the absolute Galois group of a local field?
The jumps are all nonnegative rational numbers.
To see that the jumps are only rational numbers, one can use the definition: An integral is involved, but this is an integral of piecewise linear ...
- 131k
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 ...
6
votes
Accepted
Characters of simply connected semsimple algebraic groups over local fields
As I have written in a comment, the answer is YES (any abstract homomorphism into an abelian group is trivial) when $G$ is an isotropic, simply connected, simple algebraic group over a nonarchmedean ...
- 13.4k
6
votes
Accepted
Hilbert Symbols, Norms, and p-adic roots of unity
I think I can construct an explicit counterexample with $a\in\mathbb{Q}_p$.
Choose a compatible sequence $\zeta_{p^m}$ of $p^m$th roots of unity in $\overline{\mathbb{Q}}_p$. Write $q=p^n$ with $n\...
- 40.3k
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