# Tag Info

## Hot answers tagged local-fields

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 ...
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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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### 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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• 601

### 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

### 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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### 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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### 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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### 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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### 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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### 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
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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 $WD(\... • 35.7k 7 votes ### Is every compact simply-connected reductive p-adic group perfect? At your request, I post my comment as an answer: the answer to the Related question is "no", i.e., all simple, anisotropic groups over a non-Archimedean local field are of type$\mathsf A$; ... • 10.6k 7 votes Accepted ### Looking for proof of Serre's mass formula The article is not only available in Serre's Collected Papers: it first appeared in a journal, after all. Here's a scan from the Comptes Rendus archives: https://gallica.bnf.fr/ark:/12148/... • 48.8k 7 votes ### What are the jumps in the ramification filtration of the absolute Galois group of a local field? The jumps are all nonnegative rational numbers. To see that the jumps are only rational numbers, one can use the definition: An integral is involved, but this is an integral of piecewise linear ... • 131k 6 votes ### Argument of Zariski density to prove rationality of a regular map This is valid in any characteristic (over an infinite field) and has nothing to do with completions, and is a "relative schematic density" result. It is a special case of EGA IV$_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\...
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