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Questions tagged [local-fields]

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11
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0answers
161 views

sequences in non-abelian group cohomology

In general, if we have a (pro-)finite group $G$ and a sequence of (continuous) non-abelian $G$-modules $$1\rightarrow A\rightarrow B\rightarrow C\rightarrow 0,$$ such that the image of $A$ lies in the ...
-1
votes
0answers
48 views

Hilbert symbol computation

For the quadratic Hilbert symbol $(\;,\;)_F$ over a local field $F$, is the following true? If $(-1, ab)_F=-1$, then $(a, b)_F=1$ If $F$ is real, this is almost immediate. But is is also true for $p$...
6
votes
1answer
171 views

Integral Tate-Sen theory

Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $C=\widehat{\overline{K}}$ be the completion of the algebraic closure of $K$. Let $\mathscr{O}_C$ be the ring of integers in $C$, and let $G_K$ ...
2
votes
0answers
152 views

Is a reductive group scheme always parahoric?

Let $R$ be complete (or, more generally, Henselian) discrete valuation ring with fraction field $K$. Let $G$ be a reductive $R$-group scheme. Is $G$ a parahoric in the sense of Bruhat-Tits? If so, ...
1
vote
1answer
213 views

Under What assumptions on $p$, $\mathcal{O}_K^* \simeq \mathbb{Z}_p^{*} \oplus \mathbb{Z}_p^{*}$

Let $p$ be a fixed prime number and $\mathbb{Q}_p$ be the field of $p$-adic numbers and $K$ be an extension of degree $2$ of $\mathbb{Q}_p$. Let $\mathcal{O}_K$ be the ring of integers of $K$ and $\...
7
votes
1answer
236 views

analogue of Theorem of Mattuck for Abelian varieties over $\mathbf{F}_q(\!(t)\!)$

By a theorem of Mattuck [Abelian Varieties over $p$-Adic Ground Fields, Annals of Mathematics, Second Series, Vol. 62, No. 1 (Jul., 1955), pp. 92-119], for an Abelian variety $A$ of dimension $g$ over ...
7
votes
0answers
173 views

$p$-adic analog of independent vector fields on spheres

Let $F=\Bbb Q_p$, $V$ a $n$-dimensional vector space over $F$ equipped with a nondegenerate bilinear form $q$ and $S(V)=\{x \in V| q(x,x)=1\}$. A vector field on $S(V)$ is defined to be a continuous ...
1
vote
1answer
188 views

Factorisation of polynomials over finite field

Is there a method to factorise a polynomial, for $k \leq m$ and $a_i \in \mathbb{F}_p$, $$ 1 + t^k(1 + a_1 t + a_2 t + \ldots + a_m t^m)^k $$ as a product $$ (1 + t^k)^{x_1} \cdots (1 + t^l)^{x_l} \...
4
votes
0answers
71 views

Monoid cohomology of $\mathbb{N}$ for a linear algebraic group

Let $k$ be a finite field and $k_E:=k((X))$ denote the field of Laurent series over $k$. We define a Frobenius endomorphism on $k_E$ via $f(X)\mapsto f(X^p)$. We choose a lift $\varphi:k_E^{sep}\...
2
votes
1answer
157 views

Decomposition of $\widehat{k^{\times}}$ occuring in local class field theory

Let $k$ be a finite extension of $\mathbb{Q}_p$ very often we use the isomorphism that $Gal(\overline{k}/k)^{ab} \simeq \hat{(k^{\times})}$ given by local class field theory. My question would be do ...
1
vote
1answer
171 views

Subfields of higher local fields

Q1. Let $K$ be a local field with valuation $v$. Let us call $K'\subset K$ a nice local subfield if it is complete with respect to the induced from $K$ valuation. By local subfield I will mean a ...
5
votes
0answers
296 views

Torsion subgroup of the group of points of an elliptic curve over local field

Let $K$ be a local field with residue field $k$ and $E/K$ an elliptic curve. I'm interested for which $K$ and $E$ the group of torsion points on the curve is finite. I can prove that this group is ...
17
votes
2answers
545 views

Langlands correspondence for higher local fields?

Let $F$ be a one-dimensional local field. Then Langlands conjectures for $GL_n(F)$ say (among other things) that there is a unique bijection between the set of equivalence classes of irreducible ...
2
votes
1answer
76 views

Characters of the kernel of the norm map of an extension of local fields

Let $E$ be a quadratic extension of a local nonarchimedean field $F$ of characteristic zero (and odd residual characteristic). Let $\sigma$ be a generator of the Galois group $G = Gal(E/F)$. I'm ...
6
votes
1answer
138 views

Is the intersection of ramification groups in upper numbering of a $p$-adic local field trivial?

Let $K$ be a $p$-adic local field, for example $\mathbb{Q}_p$. Let $G$ be the absolute Galois group of $K$, and let $G^v$($v\ge -1$) be the ramification groups in upper numbering, then is it true ...
8
votes
0answers
203 views

Finding a cyclic cubic extension of a field

Let $K$ be a field and let $E/K$ be a Galois extension of degree 6 with $\text{Gal}(E/K) = S_3$, the symmetric group on 3 letters. Pick two different transpositions $s_1, s_2$ in $S_3$ (hence $s_1s_2$ ...
5
votes
0answers
110 views

Arithmetic of Cuspidal Reps. Fundamental non split stratum and simple stratum

I started to read Colin Bushnell's notes on this title. The last theorem in the 3rd section claims that if $\pi$ is an irr. smooth rep. of $GL_N(F)$ containing a non-split fundamental stratum, then $\...
5
votes
1answer
283 views

Is there any explicit description of the maximal totally ramified extension of $\mathbb{Q}_p$?

It is well known that the maximal unramified extension of $\mathbb{Q}_p$ can be extended by adding the roots of unity of order prime to $p$. Is there any explicit description of the maximal totally ...
10
votes
0answers
442 views

A formal group scheme in explicit local class field theory

Let $K$ be a nonarchimedean local field with residue field $k$ of characteristic $q = p^N$, and pick a uniformizer $\pi\in \mathscr{O}_K$. Recall that explicit local class field theory, à la Lubin--...
2
votes
0answers
207 views

The Breuil-Mezard Conjecture and Generalizations (Survey)

What's the current state of the Breuil-Mezard conjecture? Has the original version (from the 2002 paper) been solved in its entirety? What are some of the new directions being explored?
3
votes
0answers
132 views

Is the special case of Abhyankar's lemma is also considered as such?

Consider the following statement: Assume $E$ and $F$ are unramified (over some fixed prime) finite separable extensions of a field $K$. Then $EF$ is also unramified. I always thought that it is ...
3
votes
0answers
247 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
0answers
108 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
0answers
160 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 ...
5
votes
0answers
162 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, ...
3
votes
0answers
206 views

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

I would like to ask for the paper of Kato: A generalization of local class field theory by using K -groups I, J. Fac. Sci. Univ. Tokyo Sec. IA 26 No.2, 1979, 303–376. I could not find it. Could anyone ...
5
votes
2answers
323 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
0answers
134 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 ...
10
votes
2answers
214 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
0answers
184 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. ...
7
votes
1answer
286 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 $k$-...
0
votes
0answers
76 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 ...
1
vote
1answer
112 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
0answers
68 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
0answers
79 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
1answer
151 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
0answers
111 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 $\mathbf{F}_p(\!(T)\!)$ ...
1
vote
1answer
320 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
1answer
199 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 ...
5
votes
1answer
241 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 ...
4
votes
0answers
168 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
1answer
482 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 ...
13
votes
1answer
273 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 $\bar{\mathbb{Q}}$?
20
votes
3answers
843 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 \...
9
votes
0answers
327 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
1answer
319 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 ...
5
votes
1answer
278 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
0answers
73 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 ...
6
votes
1answer
325 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 ...
2
votes
0answers
101 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 $\GL_n(\dbZ_p)$, ...