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5 votes
1 answer
516 views

Reference for Local class field theory via witt vectors

I would like to find some books or lecture notes on geometric local class field theory via Witt vectors. I can't find any good paper on this subject.All approaches in the books to local class field ...
3 votes
0 answers
347 views

Question about Corollary 7.3 from Silverman's The Arithmetic of Elliptic Curves

I am trying to understand an argument of Corollary 7.3 from Silverman's The Arithmetic of Elliptic Curves. I am stuck and I would appreciate any explanations. Let $E$ be an elliptic curve over $K$, ...
2 votes
0 answers
181 views

Reference request: Regarding the image of inertia group being a subgroup of Aut($\widetilde{E}$)

Let $E$ be an elliptic curve over $\mathbb{Q}_p$ with potential good reduction. I was told that if $F$ is the smallest Galois extension over $\mathbb{Q}_p$ such that $E$ has good reduction then the ...
6 votes
0 answers
370 views

What happens to Neron-Ogg-Shfarevich when characteristic of the residue field equals the prime at which Tate module is considered?

Neron-Ogg-Shafarevich criterion states that an elliptic curve $E$ over a local field $K$ has a good reduction if and only if the Tate module $T_{\ell}(E)$ is unramified for some prime $\ell$ which ...
13 votes
1 answer
400 views

What is the normal closure of $\mathbb{Q}_p \cap \bar{\mathbb{Q}}$ over $\mathbb{Q}$?

What is the normal closure of $\mathbb{Q}_p \cap \bar{\mathbb{Q}}$ over $\mathbb{Q}$? Is it $\bar{\mathbb{Q}}$?
5 votes
0 answers
208 views

Topology on two dimensional local fields

I posted my question here, but there is no reply yet. So, I guess I should post it on mathoverflow. I am reading the book of Schneider about Galois representation and $(\varphi, \Gamma)$-module, ...
6 votes
2 answers
819 views

Why is $K_{\upsilon}|K$ separable for a global field $K$?

I asked this question on math.stackexchange but maybe it fits here better. If not, I apologize in advance and will remove the question. Let $K$ be a global field and $\upsilon$ a prime of $K$. Then ...
3 votes
0 answers
375 views

compact Zariski-dense subgroups of p-adic groups

Given an algebraic group $G$ defined over a $\mathbb Q_p$. It has two topologies: one is induced by the $p$-adic metric, the other is the Zariski topology. Let $C$ be a compact (w.r.t. the $p$-adic ...
5 votes
1 answer
478 views

Artin map restricted to base field

Let $M/L/K$ be a tower of local fields such that $M/L$ is abelian with Galois group $G$. The Artin map $\psi_{M/L}$ restricted to $K^\times$ is a continuous map to $G$ and thus corresponds to some ...
10 votes
2 answers
265 views

Orbits of $GL(n, \mathcal{O})$ on pairs of linear subspaces over non-Archimedean local fields

Let $F$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers. Let $Gr_{i,n}$ denote the Grassmannian of $i$-dimensional linear subspaces in $F^n$. Can one describe ...
7 votes
0 answers
470 views

Explicit $H^2(K, \mu) = Q/Z$?

In the development of local class field theory, a very fundamental theorem is that, for every local field $K$ of characteristic zero, $H^2(K, \mu) \cong \mathbb{Q}/\mathbb{Z}$. $(*)$ Neukirch et al. ...
9 votes
1 answer
578 views

Rational points on varieties over local fields

In his expanded lecture notes Rational points on varieties, Bjorn Poonen writes the following: REMARK 2.5.3: There is an algorithm that, given a local field $k$ of characteristic $0$ and a $k$-...
58 votes
9 answers
16k views

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 ...
0 votes
0 answers
81 views

Index of a mysterious congruence subgroup

Let $F$ be a non-archimedean local number field, $\mathfrak{o}_F$ its ring of integers and $\mathfrak{p}_F$ its maximal ideal. Let $E$ be a quadratic unramified extension over $F$. Let $G$ be the ...
3 votes
1 answer
332 views

Inverse image of norm map on principal units for an unramified extension

For a local field $E$, denote by $U(E)$ the units of the corresponding valuation ring $\mathcal{O}_E$, and denote by $U_n(E)$ the prinicipal $n$-units, i.e. $U_n(E)=1+M_E^n$ where $M_E$ is the maximal ...
1 vote
0 answers
108 views

Characters of a quadratic extension and convergence

Let $F$ be a non-archimedean local field, $\chi$ a quasi-character of $F^\star$ and $\psi$ a positive character of $E^\star$. I would like to understand why the usual Rankin-Selberg zeta integrals ...
2 votes
0 answers
82 views

What is the classification of this group?

Let $K=\mathbb C((t))$ and $O=\mathbb C[[t]]$, and $n\geq 1$. Consider the matrix $$J_{2n}=\begin{pmatrix} 0& I_n \\ -I_n & 0\end{pmatrix},$$ And let $\Psi : K^{2n}\times K^{2n}\rightarrow K$ ...
1 vote
1 answer
190 views

Hilbert symbols vanishing

Let $p$ be an odd prime, and let $E/\mathbb{Q}_p$ be a finite extension that contains a primitive $p$-th root of unity $\zeta_p$ but not a primitve $p^2$-th root of unity $\zeta_{p^2}$. Let $a,b \in E^...
2 votes
0 answers
115 views

Converging sequence of base change

Here is a natural question that I hope will be of interest to some. Let $\mathbf{F}_p(\!(T)\!)$ be the field of formal Laurent series over $\mathbf{F}_p$. An automorphism of $\mathbf{F}_p(\!(T)\!)$ ...
2 votes
1 answer
414 views

Why is $\mathbb{Q}_p(p^{1/p^\infty})$ a complete topological field?

In Matthias Wulkau's exposition of Scholze's thesis, the term perfectoid field is defined as follows: Let $K$ be a field endowed with a non-archimedian absolute value $\lvert\cdot\rvert$, and let $\...
2 votes
1 answer
462 views

Complete fields with algebraically closed residue field

I am looking for a reference where the following result is proven: Let $k$ be an algebraically closed field. If $K$ is a complete and discretely valued field with residue field $k$. Then $K$ is one ...
4 votes
0 answers
415 views

Locally compact vector space over a finite field

In the wikipedia article titled "topological vector space", there is a line saying the following. "Let $K$ be a locally compact topological field, for example to real or complex numbers. A ...
9 votes
1 answer
695 views

Conductor as volume of the integers ring

I am working on Tate's thesis, and I have some problems with computations, yet the result seems to be a good natural motivation for introducing the arithmetic conductor of a character. Let $F$ be a ...
22 votes
3 answers
2k views

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

Let $K$ be a finite extension of $\mathbf{Q}_p$. Let $F_d$ be the unramified extension of $\mathbf{Q}_p$ of degree $d$. I would like to know whether there exists some $d \geq 1$ and some $L \subset K \...
10 votes
0 answers
409 views

Higher Adeles of a scheme and related topics

Let $X$ be a noetherian scheme. I will describe a construction of a simplicial ring which I think is called the Bellinson higher Adeles complex (or something similar). Consider the augmented ...
4 votes
1 answer
593 views

Hilbert Symbols, Norms, and p-adic roots of unity

Let $p$ be an odd prime number, let $\mathbb{Q}_p$ be the field of $p$-adic numbers, and let $\overline{\mathbb{Q}_p}$ be an algebraic closure of it. For a primitive $p$-th root of unity $\zeta_p \in ...
7 votes
1 answer
684 views

Type of place versus type of unitary group

Let $F$ be a totally real number field, $E$ a totally imaginary quadratic extension over $E$, and $V$ an hermitian $n$-dimensional vector space over $F$. I assume $n=2m$ is even. Let $U$ be a unitary ...
2 votes
0 answers
90 views

Uniformity in first order theory of p-adic fields of mixed characteristic

Perhaps the most successful attempt at analyzing first order theory of p-adic fields is through the use of RV language (aka. leading term structure). In this, quantifier elimination on the field sort ...
7 votes
1 answer
851 views

Understanding the structure of unitary groups

I would like to understand precisely the structure of unitary groups. Let $F$ be a global number field, $E$ a quadratic extension of $F$, and $U$ a unitary group on $E$ (i.e. the group of ...
19 votes
1 answer
4k views

A good book on adeles and ideles

Many results in number theory are stated either in a classical language or in an adelic one. I am often impressed of the efficiency and the satisfactory computational properties of the adelic setting, ...
2 votes
0 answers
109 views

What does equality modulo $p$ of $p$-adic linear groups imply?

$\newcommand{\GL}{\mathrm{GL}}\newcommand{\dbZ}{\mathbb{Z}}\newcommand{\dbF}{\mathbb{F}}\newcommand{\dbN}{\mathbb{N}}$ Hello. I have a small question regarding closed subgroup of $\GL_n(\dbZ_p)$, ...
4 votes
1 answer
265 views

Local triviality of Galois cohomology classes over $\mathbb{Q}$

Let $A$ be a $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$-module which is a finitely generated free $\mathbb{Z}$-module. I'm interested in the behaviour of cohomology classes in $$\mathrm{H}^1(\mathbb{...
13 votes
4 answers
2k views

Which groups are Galois over some p-adic field?

Suppose I have some finite $p$-group $G$, or a little extension of it. How do I know if there exists a prime $l$ and a finite extension $K$ of $\mathbb{Q}_l$ such that $G$ is the Galois group of ...
2 votes
1 answer
185 views

Fields of definition of parabolically induced representations of $\mathrm{SL}(2,q)$

Let $\alpha_0$ be the unique non-trivial character satisfying $\alpha_0^2=1$ of the split torus $\mathrm{T} \subset \mathrm{SL}(2,q)$ and denote by $\mathrm{R}(\alpha_0)$ the character of $\mathrm{SL}(...
10 votes
1 answer
654 views

Dyadic Hilbert symbols and higher unit groups

Let $F$ be a local dyadic number field, $\mathfrak{p}$ its maximal ideal, $(*,*)_F$ its quadratic Hilbert symbol and $e$ its ramification index (i.e. $\mathfrak{p}^e$ is exact divisor of $2$). Fix an ...
2 votes
0 answers
124 views

Property of a derivative in global field

Before posting I want to make it clear that I posted the same question in stack exchange awhile ago (https://math.stackexchange.com/questions/1533814/property-of-derivative-in-a-local-field) but didn'...
4 votes
1 answer
731 views

Reference for: Every local field can be realized as the completion of a global field

It is well known that every local field (i.e. nondiscrete topological field locally compact with respect to the topology) is the completion of some global field. I know the argument, a nice ...
8 votes
2 answers
1k views

Examples to keep in mind while reading the book 'The Admissible Dual...' by Bushnell and Kutzko and the importance of Interwining of representations

I am a beginner in the field of representation theory. I was reading the book 'The Admissible Dual of $GL(N)$ Via Compact Open Subgroups' by Bushnell and Kutzko. Let me first describe the book a ...
8 votes
1 answer
526 views

Integral representation of adjoint L-factor for GL(2)

My question is about a local computation in the paper of Gelbart and Jacquet, "A relation between automorphic representations of GL2 and GL3", from 1978. Let $\sigma$ be an irreducible smooth complex ...
9 votes
1 answer
4k views

Maximal tamely ramified extension of $\mathbf Q_p$

Is there an explicit description of the maximal tamely ramified extension of $\mathbf Q_p$?
3 votes
0 answers
437 views

Relation between ramification index and length of filtration of ramification groups

Given a complete valued field $K$ with a discrete value group $\mathbb{Z}$, consider a totally ramified finite Galois extension $L$ of $K$ with its Galois group $G$. Let $O_L$ be the valuation integer ...
6 votes
2 answers
798 views

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

For a field $F$ we denote by $F^{\mathrm{ab}}$ the compositum of all finite Galois abelian extensions of $F$. Is $\mathrm{Gal}(\mathbb{Q}_2(\sqrt[8]{3})^{\mathrm{ab}}/\mathbb{Q}_2(\sqrt[8]{3})) \...
10 votes
4 answers
2k views

Reference request: expository text on the structure of reductive groups over non-archimedean local fields

I am interested in an expository text in English, which summarizes the main results and aspects of the structure theory of reductive groups over local fields, in a hopefully not very technical manner (...
3 votes
0 answers
168 views

The left regular representation of the Jacobi groups over local fields of characteristic >2 is type I?

Let $K$ be a non-archimedean local field of characteristic $>2$. Consider the Jacobi group $G=H_{2n+1}(K)\rtimes Sp_{2n}(K)$, which is the semidirect product of the Heisenberg group $H_{2n+1}(K)$ ...
3 votes
0 answers
224 views

Metaplectic groups over non-archimedean local fields of characteristic>2

Let $K$ be a non-archimedean local field of characteristic $>2$. Consider the double cover metaplectic extension of symplectic groups $p: Mp_{2n}(K)\rightarrow Sp_{2n}(...
0 votes
0 answers
138 views

Orbits of some action of SL2 on Pontryagin dual of the field of formal Laurent series

Let $K=\mathbb{F}_2((t))$ be the field of formal Laurent series over the finite field $\mathbb{F}_2$. Now consider $K^3$ as an additive group and its dual group $\hat{K^3}$, which consists of all ...
1 vote
0 answers
64 views

Normgroup and the image of the Hilbert symbol are subgroups of index 2 in the principal units

Let $K$ be a local field over $\mathbb{Q}_2$ such that the extension $K(i)/K$ is ramified and let $U^1_{K(i)}$ and $U^1_K$ denote the groups of principal units in the fields $K(i)$ and $K$, ...
6 votes
1 answer
598 views

Clarification about Tits' article in the Corvallis

I am studying Tits' article in the Corvallis wherein he defines the apartment in the general case (not necessarily split). I wish to know what he means about the filtration of the groups $U_a(K)$ (...
2 votes
0 answers
110 views

Infinitesimally small elements in extensions of models of model-complete theories

Suppose that we have a first order language $\mathcal{L}$ that extends the language of rings. Let $T$ a be a topological $\mathcal{L}$-theory of fields in the sense of Pillay.. this means that not ...
2 votes
1 answer
123 views

F-points of product of closed subgroups vs. product of F-points, F a local field, reference?

Let $F$ be a finite extension of $\mathbb Q_p$, where p is an odd prime. Let $G$ be a connected reductive group defined over $F$. Let $M, H$ be closed $F$-subgroups of $G$ (in particular, I'm ...