Questions tagged [galois-representations]
The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring, but can also be used as a synonym for G-module. The study of Galois modules for extensions of local or global fields is an important tool in number theory.
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Ramification of mod $\ell$ representation of elliptic curves [closed]
Let $E$ be an elliptic curve over $\mathbb{Q}$, and let $p,\ell$ be two prime numbers.
Consider the mod $\ell$ representation $$\rho:Gal(\mathbb{\overline{Q}}/\mathbb{Q})\to Aut(E[\ell])= GL(2,\ell).$$...
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Bounding dimension of $H^1(G_{\mathbb{Q}}, (V_pE)^{\otimes n})$
Let $E$ be an elliptic curve over $\mathbb{Q}$, let $p$ be a prime of good reduction, $T_pE$ is its $p$-adic Tate module, $V_pE = T_pE\otimes \mathbb{Q}_p$, and $(V_pE)^{\otimes n}$ its $n$'th tensor ...
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Are there Maass forms where the expected Galois representation is $\ell$-adic?
Recall that by theorems of Deligne and Deligne--Serre, there is the following dichotomy:
Modular forms on the upper half plane of level $N$ and weight $k\geq 2$ correspond to representations $\rho:\...
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Elliptic curves and images of decompositions group exceptional?
Given an elliptic curve $E$ and its mod $p$ Galois representation $\bar{\rho}_{E,p}$, I am wondering what are the possibilities for $\bar{\rho}_{E,p}(G_l)$, where $G_l:=$Gal($\overline{\mathbb{Q}_l}/\...
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Taylor-Wiles systems for higher dimensional deformation rings
Let $R$ be a deformation ring and $M$ be a finitely generated $R$-module.
A strategy for proving the theorems $R=T$ is to associate with $(R,M)$ a Taylor-Wiles system denoted $(R_{Q},M_{Q})$. Here I'm ...
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Is the completed tensor product (over a complete dvr) of two reduced complete Noetherian local rings again reduced?
To be more specific, Let $\mathcal{O}$ be a finite extension of $\mathbb{Z}_{p}$. Let $A=\mathcal{O}[[X_{1},\ldots, X_{n}]]/\left( f_{1},\ldots,f_{r}\right) $ and $B=\mathcal{O}[[Y_{1},\ldots, Y_{m}]]/...
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Analytic properties of $L$-functions attached to a compatible system of $\ell$-adic Galois representations
Let $F$ and $E$ be number fields, $G_F$ be the absolute Galois group of $F$, and $S$ be a finite set of primes of $F$. For $\lambda$ a prime of $E$ we denote by $\ell$ its residual characteristic. We ...
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Compatibility of system of $\ell$-adic representations associated to Voevodsky motives
Let $M$ be an object of Voevodsky's category $DM_{gm}(K,\mathbb{Q})$ for a number field $K$. For each prime number $\ell$, there is an $\ell$-adic realization $M_{\ell}$ in the bounded derived ...
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Is the weight-monodromy conjecture known for unramified representations?
Let $X$ be a smooth proper variety over a number field $K$, $v$ a place of $K$ lying over a prime number $p \neq \ell$, and $V := H^n(X_{\overline{K}};\mathbb{Q}_{\ell})$. Suppose $V$ is unramified at ...
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Local systems on $\mathbb P^1$ and on the formal punctured disc
Consider the projective curve $\mathbb P^1$ over a finite field $k$.
Consider $\ell$-adic local systems $E$ on $\mathbb P^1\backslash \{0,\infty\}$ such that
a) $E$ is tame at $\infty$
b) The ...
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Norm 1 elements of an unramified quadratic extension of a local field
Let $E$ be an unramified quadratic extension of a local field $F$, with $p$ odd. Let $E^1$ denote the set of norm $1$ elements of $E$. What can be said about the following index:
$$
{\rm 1.}\ \ \ \ [ ...
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A computation of nearby cycles
I'm currently reading P.Scholze's paper "THE LANGLANDS-KOTTWITZ APPROACH FOR THE MODULAR
CURVE". In Lemma 7.7, he showed a (maybe simple) nearby cycle computation, which I can't follow.
Now ...
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Families of Galois representations over disks
Edit on Nov. 20, 2023. This question is answered below in the case that $0<r_i<1$. And indeed it is shown in the answers to not be an interesting question in that case. So please take all $r_i=1$...
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List of techniques that have been used to prove topological properties of locus in the deformation ring or the Hecke algebra
My question is maybe going to be a bit vague. My apologies if so.
The setting:
Let $\overline{\rho}$ be a residual representation and $R$ be a deformation ring of $\overline{\rho}$.
Let $\mathbb{T}$ ...
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Calculate $D_{\mathrm{cris}}(V)$ for a crystalline representation
$\newcommand{\cris}{\mathrm{cris}}$In my setting, $K/\mathbb Q_p$ is finite and unramified, and $V$ is a $2$-dimensional crystalline representation of $G_K$. Then we have $D_{\cris}(V)$, which is $2$-...
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Conductor of determinant of a 2-dimensional Galois representation divides conductor of representation
I am studying Serre's paper "Sur les représentations modulaires de degré 2 de $\mathrm{Gal(\overline{\mathbb Q}/\mathbb Q)}$" and I'm stuck trying to prove that $N(\det\rho)$ divides $N(\rho)...
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Irreducibility of Tate module (as a Galois representation) of elliptic curves with good reduction
This question is following the previous question.
Definitions:
Suppose $F$ is an unramified finite extension of $\mathbb Q_p$ and $E$ is an elliptic curve defined over $F$ with good reduction. Denote ...
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Calculation of Frobenius on de Rham cohomology of elliptic curves with good reduction
I'm reading "An introduction to the theory of $p$-adic representations" by Laurent Berger. In the page 14 it says:
Suppose $F$ is an unramified finite extension of $\mathbb Q_p$ and $E$ is ...
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Is the Frobenius semisimple on the de-Rham cohomology?
Suppose $K$ is a unramified finite extension of $\mathbb Q_p$, and $X$ is a projective smooth curve defined over $K$. By $p$-adic Hodge theory we know $D_{cris}(H_{et}^i(X,\mathbb Q_p))=H_{dR}^i(X)$. ...
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Is there a bound on the number of $p$-adic semisimple representations?
Faltings proved the following:
Fix integers $w, d \geqslant 0$, and fix a number field $K$ and a finite set $S$ of primes of $\mathcal{O}_K$. There are, up to conjugation, only finitely many ...
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Kernel of restriction map in Galois cohomology
Let $S$ be the algebraic group $SL_2/\mathbb{Q}_p$ with a $G=G_{\mathbb{Q}}$ action, (acts by conjugation with a representation $\rho: G\longrightarrow GL_2$.)
Let $G_p$ be the decomposition group at ...
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Description of $\operatorname{Gal}(K(E[n])/K)$ as a subgroup of $\operatorname{GL}_2(\mathbb{Z}/n\mathbb{Z})$ for a CM elliptic curve $E$
I am looking for a specific description of the Galois groups $\operatorname{Gal}(K(E[n])/K)$ as a subgroup of $\operatorname{GL}_2(\mathbb{Z}/n\mathbb{Z})$ for an elliptic curve $E$ with complex ...
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Two different local Langlands parameters for quadratic extension
Let $E/F$ be a quadratic extension of nonarchimedean local field, and let $G$ be a reductive group over $E$, and $G(E)$ the $E$-points of $G$. (For my question, one may just assume $G=\operatorname{GL}...
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Galois weight of a Serre twisted pure Galois representation
Let $V$ be a pure, 2-dimensional weight $k$ $p$-adic global Galois representation for $G$, the absolute Galois group of the rationals.
Let $c\in Z^1(G,\text{SL}_2)$ be a cocycle and consider the Serre ...
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A variant of the inverse Galois problem
In Theorem I of Construction of maximal unramified p-extensions with prescribed Galois groups, it's proved that
for any prime $p$ and any given finite $p$-group $G$, there exists a number field $F$ ...
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$\pi$-adic Galois representations of attached to newforms at $p \nmid N$ are crystalline
Is [Scholl, Motives for modular forms, Theorem 1.2.4 (ii)] proven for any $p$ independent of the weight?
Concretely, let $f$ be a normalized eigenform of weight $w$. Let $p$ be a prime not dividing ...
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Reference showing no proper subgroups of p-adic orthogonal groups surject onto mod p orthogonal groups
I am looking for a reference for the following statement:
Let $O$ be an orthogonal group associated to a nondegenerate quadratic form of rank $r$ over the p-adic integers $\mathbb Z_p$. Suppose $r$ is ...
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Image of $\gamma-1$ on etale $(\varphi,\Gamma)$-modules
Let $p\geq 3$ be a prime, $D$ be an etale $(\varphi,\Gamma)$-modules over the classical period ring $A_{\mathbb{Q}_p}=\mathbb{Z}_p[\![T]\!][1/T]^{\widehat{\phantom{xx}}}_p$ and $\gamma$ be a ...
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Galois cohomology of Tate modules
Let $E,E'$ be a pair of elliptic curves defined over $\mathbb{Z}$. Let $T_p[E], T_p[E']$ be their associated ($p$-adic) Tate modules. These are Galois representations for the absolute Galois group of $...
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Local Tate duality for F-vector space
A version of local Tate duality stated: Let $K$ be a finite extension of $\mathbb Q_p$, $A$ be a finite $G_K=Gal(\overline K/K)$ module. Then for $0\le i\le 2$, the cup product induces a perfect ...
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Inertia group representation from $p^{n}$-torsion of ordinary elliptic curve
Let $K$ be a complete local field. Suppose that $K$ is an unramified extension of $\mathbb{Q}_{p}$ and let $E$ be an elliptic curve over $K$ with good ordinary reduction. Let $G_{K}=\text{Gal}(\...
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What is the smallest group not known to be a Galois group over $\mathbb{Q}$?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$What is the smallest group not known to be a Galois group over $\mathbb{Q}$?
Variants have been asked here before (e.g. Which small finite ...
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Elliptic curve with CM and image of Galois representation in normalizer of nonsplit Cartan
I am trying to understand the following. Let $E/\mathbb{Q}$ be an elliptic curve with complex multiplication given by the ring of integers $\mathcal{O}_K$. We are given a fixed rational prime $p$ ...
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About simple motives
I'm reading through Jannsen's paper Motives, numerical equivalence, and semi-simplicity and I'd like to pose two questions.
Suppose all motives are $F$-linear, for some characteristic zero field $F$, ...
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Representability of $\operatorname{Hom}(G_{\mathbb{Q}}, \operatorname{GL}_2)$
Let $G_{\mathbb{Q}}$ be the absolute Galois group of the rationals, and let $F: \mathrm{Aff}/\textbf{Q}_p\longrightarrow \mathrm{Sets}$ be the functor which associates to every affine $\mathbb{Q}_p$ ...
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Understand the $p$-adic local Langlands correspondence with examples
Let $\rho:G_{\mathbb{Q}}\rightarrow \mathbf{Gl}_{n}(\mathbb{Q}_{p})$. I would like to understand in depth why the local Langlands correspondence for $\rho_{|\mathbb{Q}_{p}}$ must consider $p$-adic ...
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Effect of the surjectivity of Galois representation
Let $K$ be any arbitrary number field and $E$ be any elliptic curve over it. For any integer $m,$ consider the well-known Galois representation
$$\rho:\text{Gal}(\overline{K}/K)\to \text{GL}_2(\mathbb{...
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Double centralizer theorem for ($\ell$-adic) Lie algebras
$\DeclareMathOperator\GL{GL}$I'm reading the proof of Serre's open image theorem from his book "Abelian $\ell$-adic representations and elliptic curves". This is chapter IV Section 2.2 of ...
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Decomposition of the Galois group of the $m$-th division field of an elliptic curve with CM into a direct product of Galois groups
Let $E/\mathbb{Q}$ be an elliptic curve with CM from an imaginary quadratic field $K$. Let $K(E[m])$ denote $m$-th division field (number field obtained by adjoining the coordinates of the $m$-torsion ...
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Reference request: Newton above Hodge
Let $K$ be a p-adic field, and let $\mathcal{O}$ be the ring of integers inside $K$ with residue field $k$. Let $\mathcal{X}$ be a smooth proper formal scheme over $\mathcal{O}$ (with topology given ...
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How does the cohomology of the Lubin-Tate/Drinfeld tower fit into categorical p-adic local Langlands?
In conjecture 6.1.14 of this article, Emerton-Gee-Hellmann formulate the p-adic local Langlands conjecture, which posits the existence of a fully faithful functor from (the appropriate derived ...
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Why does Deligne's construction of the Galois representation attached to the new cuspidal forms require that the Kuga-Sato manifold be regular?
The origin of this question is related to the construction of Galois representations of Deligne attached to $f$ a new cuspidal form (of weight $k\geq 2$). To do this, we consider the fiber product $k$-...
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A question on a paper of K. Murty
Let $f=\sum_{n\ge 1}a_nq^n$ be a normalized Hecke eigenform which is not of CM-type, of weight $k\ge 2$ for the congruence subgroup $\Gamma_0(N)$. Let $a\in\mathbb{Z}$ and define
$$
\pi_f(x,a):=\#\{p\...
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How to construct this non-geometric mod $p$ Galois representation?
Let $ K $ be a totally real cubic field. I am concerned with the following kind of Inverse Galois Problem:
Is there a prime $ p>5 $ and a continuous representation $ \bar{\rho}:G_{K}\to {\rm
GL}_{...
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On the determinant of an odd, continuous Galois representation
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Frob{Frob}$In his paper, Duke paper, Serre consider continuous, odd Galois representation
$\rho: G_{\mathbb{Q}}\longrightarrow \GL_{n}(\overline{\...
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Infinitely ramified $p$-adic representations as limits of finitely ramified $p$-adic representations
Let $ \rho:G_{\mathbb{Q}}\to {\rm GL}_{2}(\mathbb{Q}_{p}) $ be a continuous $ p $-adic representation of the absolute Galois group $ G_{\mathbb{Q}} $ of the rational number field $ \mathbb{Q} $. Let $ ...
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Purity for proper varieties
Let $X$ be a proper, geometrically connected, geometrically integral variety over $\mathbf{F}_q$. There exists a finite field extension $k/\mathbf{F}_q$ of degree $d$ and an alteration $X'\to X_k$ ...
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In practice, how explicitly can we describe a Galois representation?
In most parts of representation theory, we naturally want to describe a given representation as explicitly as possible. This seems to me a feasible project only if, as a first step, the group itself ...
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Crystalline when restricted to inertial subgroup
$\newcommand{\ur}{\mathrm{ur}}\newcommand{\cris}{\mathrm{cris}}$Let $K$ be a finite extension of $\mathbb{Q}_p$, $G_K=\operatorname{Gal}(\overline{K}/K)$ and $I_K \subset G_K$ its inertial subgroup. ...
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Trianguline representation
I have a problem in understanding the concept of trianguline representation. Maybe someone can enlighten me.
Let $K$ be a finite extension of $\mathbb{Q}_p$ and $V$ be a $p$-adic representation of $...
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