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Questions tagged [galois-representations]

Questions relating to the representations of the absolute Galois group $\mathrm{Gal}(\overline K/K)$ of a number field or of a local field.

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The splitting of Galois representations

Suppose $X$ is a smooth projective variety defined over a number field $K$, then the etale cohomology $H^i_{et}(X,\mathbb{Q}_\ell)$ defines a continuous representation of the absolute Galois group $\...
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irreduciblity of $\ell$-adic representation attach to the elliptic curve over $\mathbf{Q}$ with complex multiplication

Currently I am reading the book, Fermat’s Last Theorem written by Darmon, Diamond and Taylor. (You can find this pdf online http://www.math.mcgill.ca/darmon/pub/Articles/Expository/05.DDT/paper.pdf) ...
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A question about Galois characters

Let $F$ be a number field and $\chi:\mathrm{Gal}(\overline{\mathbb{Q}}/F)\to\overline{\mathbb{Q}_\ell}^{\times}$ ($\ell$ a prime) a Galois character. My question is: Can we find a finite extension $K/...
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Why does the Frobenius-semisimplicity of a Weil representation not depend on the choice of the Frobenius element?

Definition: Let $K$ be a (non-Archimedean) local field and $k$ its residue field. A Frobenius element of the absolute Galois group $G_K$ is any element of $G_K$ which is a lift of the Frobenius ...
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Characterization for the continuity of Weil representations

Let $K$ be a non-Archimedean local field and $W_K$ be the Weil group of $K$. We consider a representation $\rho: W_K \to \operatorname{GL}_n(\mathbb{C})$ between two topological groups. Here, $\...
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Are there counterexamples of isogeny elliptic curves with non-isomorphic integral Tate modules?

Let $K$ be a field and $G_K$ be its absolute Galois group. Let $E_1,E_2$ be two elliptic curves over $K$. Assume that there exists an isogeny $f:E_1\rightarrow E_2$. Let $p$ be a prime number. Then $f$...
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How to compute the $p$-adic period of $\mathbb G_m$?

I am reading one paper by Colmez about periods of abelian varieties with complex multiplication. As a motivation, he says we can compute the periods of $\mathbb G_m$: I know the example for the ...
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Uniqueness of Algebraic Belyi Pairs from Dessin

From a Dessin d'Enfant on a surfaces $X$ we get a meromorphic function $f:X\rightarrow\mathbb{CP}^{1}$ such that the only critical values are $\{0,1,\infty\}$ i.e a Belyi pair $(X,f)$. Belyi's ...
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Closed points and rational points

If $X$ is a $k$-scheme of finite type with $k$ a field, there is a surjective map $\gamma$ from the set of $\overline{k}$-rational points of $X \times_{\mathrm{Spec}(k)}\mathrm{Spec}(\overline{k})$, ...
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Why can we always restrict an Galois representation so that it becomes unramified?

Let $K$ be a local field and $\rho: G_K \to \operatorname{GL}_n(\mathbb{C})$ be a Galois representation where $G_K$ denotes the absolute Galois group of $K$. We call a Galois representation $\rho$ ...
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Do the local polynomials of Weil representations coincide if they are Artin representation (factor through a finite quotient)?

Let $K$ be a local field, $G_K$ its absolute Galois group, $I_K$ the inertia subgroup of $G_K$, $\operatorname{Frob}_K \in G_K$ be a Frobenius element, i.e. any element of $G_K$ acting as $x \...
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Why are all Frobenius elements conjugated?

Let $K$ be a local field, $k$ its residue field and $G_K$, $G_k$ be the absolute Galois groups of $K$ and $k$, respectively. A Frobenius element is an element $\operatorname{Frob}_K \in G_K$ such ...
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What are Robba rings and why are they important?

If we study $p$-adic Galois representations we come across the so called "Robba rings", defined as the ring of Laurent series which converge in some annulus $\{z| \,\,R<|z|<1\}$ for some ...
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What is the order of an unramified character?

Let $K$ be a local field and $G_K$ be its absolute Galois group. A character is a group homomorphism $\phi: G_K \to \mathbb{C}^*$ with finite image, i.e. $|\phi(G_K)| < \infty$. It is unramified if ...
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Correspondence for Artin characters

Let $K$ be a local field and $G_K$ be the absolute Galois group of $K$. An Artin character is any continuous group homomorphism $G_K \to \mathbb{C}^*$ whose image is finite. Correspondence: Now I ...
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topology on the ring of Witt vectors in the theory of period rings of Fontaine

For a $p$-adic field $K$ with perfect residue field $k$, we know the standard construction of the ring $R$. I will recall it briefly. It is $\varprojlim_{x \rightsquigarrow x^p} O_{C_K}/pO_{C_K}$, ...
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Surjective restriction map

I am quoting this from Lior -Soroker's blog on Embedding problems: Let $K$ be a field. Assume that $E$ is a finitely generated regular extension of $K$ and that $F/E$ is a finite Galois extension ...
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Notation for the local polynomial of a Weil representation

Let $\rho$ be a Weil representation over a local field $K$. In this paper, I found the following defintion of the local polynomial: $$ P(\rho,T) = \det(1-\operatorname{Frob}_K^{-1}T | \rho^{I_K}). $$ ...
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Constuct the element of Galois group of rational number

Is there a element of absolute Galois group of $\mathbb{Q}$ that is an extension of some 'easy map' like $\sigma : \mathbb{Q}(\sqrt{2}) \rightarrow \mathbb{Q}(\sqrt{2})$, $\sigma(x+y\sqrt2)=x-y\sqrt2$....
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Artin $L$-functions and abelianization

Let $L/K$ be a finite Galois extension of global fields, with $G=\mathrm{Gal}(L/K)$. Let $H \leq G$ and $\chi:H \to \mathbb{C}$ a non-trivial irreducible character of $H$. Then we can define the (...
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Hilbert 90 and K-forms

Studying the proof of Hilbert's 90 theorem modern version, I went through this lemma:given a Galois finite extension $K \subset L$ and an $L$ algebra $A$,we define the $(A,K)$ forms as the $K$ ...
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Why do these elliptic curves of conductor $11$ have different representations modulo $5$?

Consider the elliptic curves $$E:y^2 + y = x^{3} - x^{2} - 7820 x - 263580,$$ $$E':y^2 + y = x^{3} - x^{2} - 10 x - 20,$$ and $$E'':y^2 + y = x^{3} - x^{2}$$ over $\mathbf{Q}$. All three of these ...
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Explicit formula to compute the conductor of Etale cohomology?

I think I am stuck at a calculation problem. Suppose there is an elliptic surface $X$ defined over $Q$, take $\overline{X}:=X\times_Q \overline{Q}$, and denote by $\varphi: G_{Q}\to Aut(H^2_{et}(\...
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Artin Representations in MAGMA

Here's a toy example of the difficulty I'm having... Let $K/\mathbb{Q}$ be a Galois extension of degree 3. There are two non-trivial irreducible characters on Gal$(K/\mathbb{Q}),$ call them $\chi_1$ ...
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Galois representation on Tate module of a twist of an elliptic curve

Let $E/K$ be an elliptic curve with a twist $E'/K$. Let $f: E_{\overline K} \to E'_{\overline K}$ be an isomorphism. Let $m(\sigma) = f^{-1}\circ f^\sigma$ be the 1-cocycle corresponding to this twist....
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condition of potentially good reduction of representations

Let $l, p$ be distinct primes and let $K$ be a local field of mixed characteristic $(0,p)$. Let $G=G(K^s/K)$, $I_K$ be the inertia subgroup and $V$ a finite dimensional vector space over $\mathbb Q_l$ ...
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Topology on two dimensional local fields

I am reading the book of Schneider about Galois representation and $(\varphi, \Gamma)$-module, Section 1.7, but I don't understand his proof on Lemma 1.7.6. In this section, he introduced the weak ...
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Chebotarev's density theorem and density of Frobenius elements

Why does the Chebotarev density theorem imply that if $K$ is a number field with only a finite set of primes that ramify in an algebraic closure, then the Frobenius elements corresponding to the ...
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Galois representation and simple algebras

I'm reading the book of Hida about Galois representation. He wants to introduct semisimple algebras. let $\rho:G\longrightarrow GL(n,E)$ a linear representation (G is a group and E a field). Let $R=R(\...
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Galois representations viewed as the fundamental group of $2,3,5,… \infty$

In a paper of T. Saito http://www.ms.u-tokyo.ac.jp/~t-saito/pp/GR2.pdf he said that the Galois group $\text{Gal}(\mathbb{Q})$ could be seen as the fundamental group of the set $2,3,5,... \infty$. ...
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What is the Weil group of a global field $K$?

The question is in the title. For context, I know some things about the local Weil group. I know that the abelianization is isomorphic to the multiplicative group $K^{\times}$, and I know that it is ...
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Inverse Galois Problem: the two-fold central extension of $Sp_6(\mathbb F_2) \cong [W(E_7),W(E_7)]$

I came across the following problem when I was trying to construct a certain type of homomorphisms from $\Gamma_{\mathbb Q}$ to $E^{sc}_7(\mathbb F_p)$ for any prime $p$: Is the double cover of $Sp_6(...
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What is a semistable representation?

This feels like a silly question, but it keeps coming up in seminars and even after much searching I still have no idea what it means. I would like to see a definition and some examples of things that ...
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How to prove Tate module of the Tate Curve is not crystalline

Let $K/\mathbb{Q}_p$ be a finite extension and let $q \in K^{\times}$ be such that $|q| <1$. Let $E_q:= \bar{K}^{\times}/q^{\mathbb{Z}}$ be the Tate curve where $q^{\mathbb{Z}}:= \{q^n| n\in \...
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Reference to proof of Weil-Langlands theorem

Could anyone please provide a reference for a proof of this theorem? It was mentioned on p48 of this article without proof and I couldn't find a proof of it anywhere. The conditions in the theorem ...
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Galois Representation and Number Theory

Galois representation is supposedly of much interest in number theory; many say that number theory is about `understanding' $\text{Gal}(\bar {\mathbb Q} |\mathbb {Q} )$ . I'd like to know how this is ...
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Tate module of an elliptic curve is Hodge-Tate

Let $E$ be an elliptic curve over $\mathbb{Q}_p$, and let $T_p(E)$ be its $p$-adic Tate module. Is there a simple way of seeing that $V_p(E):= T_p(E)\otimes_{\mathbb{Z}_p}\mathbb{Q}_p$ is a Hodge-Tate ...
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Definition of Galois representation unramified at a prime.

In J. Weinstein's note: Reciprocity laws and Galois representations: recent breakthroughs, section 3.3, a Galois representation that is unramified at a prime $\mathfrak{p}$ is defined in a different ...
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$f\in K[t]$ is irreducible if and only if $G$ $(\Gamma(L:K))$ acts transitively on the roots of $f$. [duplicate]

Let $K$ be a field, $f\in K[t]$ separable over $K$, $L$ the splitting field of $f$ over $K$, and $G=\Gamma(L:K)$. I need to prove that $f$ is irreducible if and only if $G$ acts transitively on the ...
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Restriction of a group representation and coefficient ring

I am working in the context of Mazur's deformation rings, namely let $p$ be a prime number and $R$ be a complete local noetherian ring with maximal ideal $\mathfrak{m}$ and a isomorphism $R / \...
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Define the image of Frobenius automorphism in Galois representations of finite fields

I am reading Fontaine and Yi's note "Theory of $p$ adic Galois Representations, http://www.math.u-psud.fr/~fontaine/galoisrep.pdf I got confused in section 1.2.1. Say $K$ is a finite field of ...
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Freeness of Tate module of the multiplicative group

I am reading Fontaine and Yi's note "Theory of $p$ adic Galois Representations, http://www.math.u-psud.fr/~fontaine/galoisrep.pdf In section 1.1.4, (1), page 4, let $K$ be a field and $K^s$ an ...
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Irreducible Galois representation and lattices

Let $\scr{G}$ be the absolute Galois group of $\mathbb{Q}_p$. Let $V$ be a $n$-dimensional $\mathbb{Q}_p$ vector space where $\scr{G}$ acts. Consider the representation $$\rho:\scr{G} \rightarrow \...
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Absolute irreducibility for schurs lemma over local field

I have a representation $\rho: G \rightarrow GL(V)$ where $G$ is a Galois group of an extension of $p$-adic fields ($p \neq 3$) and $V$ is a vector space over $\mathbb{Q}_3$. Now I know that this ...
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the corresponding modular form of an elliptic curve

The modularity theorem reveals the relationship between elliptic curves and modular forms. Is there a series of steps or an algorithm such that we can obtain the corresponding modular form, when given ...
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Topology of a module of finite type

In 'Theory of p-adic Galois representations' (Fontaine-Ouyang), given a topological commutative ring B equipped with a continuous compatible action of a topological group G the authors define: A $\...
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Fix field of a certain galois group action

Let $E= \mathbb{F}_p(\!(u)\!)$, $E^s$ a separable closure of $E$ and write $G_E= \mathrm{Gal}(E^s/E)$ for the absolute Galois group of $E$. Take a lift of the $u$-adic valuation on $E$ to $E^s$ and ...
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Why study Gal($\overline{\mathbb{Q}}|\mathbb{Q})$? [duplicate]

I was told by one of my teachers that Galois Representation is a major tool to study Gal($\overline{\mathbb{Q}}|\mathbb{Q}$). If we know this Galois group, what knid of questions can we answer ? In ...
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Why study Galois Representations?

In Mathematics, often a theory becomes popular because it tells us something new or gives different proof for already established facts. For example, I have read that algebraic number theory is ...
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177 views

Reference book for Galois Representations

I want to study about Galois Representation. All the books I have seen have titles like [some topic] and Galois Representation, where some topic=modular forms or automorphic forms. Is there any ...