# 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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### Galois theory of ramified coverings vs classical Galois theory

That's an exact copy of my former MSE question I asked a couple of weeks ago and unfortunately not got the answer I was looking for.
The question adresses reuns' answer in this thread: Algebraic ...

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### Necessary and sufficient condition for a prime to be represented by an arbitrary positive definite binary quadratic form $ax^2+bxy+cy^2$

Given an arbitrary (but fixed) positive definite primitive integral binary quadratic form $g(x, y)=ax^2+bxy+cy^2$, and let $m$ be an arbitrary integer. We will denote the discriminant of $g$ by $D=D_g=...

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### The Fontaine Mazur conjecture for $\text{GL}_1$ over $\mathbf{Q}$

The Fontaine-Mazur cojectures says that if $\chi : \text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \to \mathbf{Q}_p^\times$ is a character which is unramified almost everywhere and de Rham at $p$ then it ...

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### Freeness of completed homology over universal deformation ring

In Theorem 7.4 of the paper "patching and the $p$-adic Langlands program for $\mathrm{GL}2(\mathbf{Q}_p)$" (arXiv link), it is proved that in the minimally ramified case, the completed homology of ...

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### Where general mixed Galois representations are defined?

I am interested in etale cohomology of varieties, and respectively, in mixed $\mathbb Q_{\ell}$-adic Galois representations over finitely generated fields. What is the canonical reference for this ...

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### How does Langlands define Artin L-functions?

Let $\rho: \operatorname{Gal}(K/F) \rightarrow \operatorname{GL}_n(\mathbb C)$ be a representation for an unramified extension $K/F$ of $p$-adic fields. Let $\operatorname{Frob}_{K/F}$ be the (...

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### The profinite topology on the Mordell Weil group

In this lecture of Serre on his open image theorem, around 6 minutes, Serre mentions the following theorem of Tate:
Let $A/k$ be an abelian variety over a number field and consider the Mordell-Weil ...

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### galois deformation ring with type is union of irreducible components

Notation:
$K$ finite extension of $\mathbb{Q}_p$, $G_K$ absolute Galois group of $K$,
$E$ is finite extension of $\mathbb{Q}_p$ (coefficient field), $O_E$ is ring of integer in $E$.
In this paper of ...

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### Kottwitz global gerbes

I've been trying to understand global gerbes constructed by Kottwitz in $B(G)$ for all local and global fields (arXiv:1401.5728). Scholze explained in $p$-adic geometry (arXiv:1712.03708) why I would ...

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### L-function in p-adic spaces

I've been learning more about different $p$-adic geometries, namely Berkovich spaces, Huber's Adic spaces and ridgid analytic spaces. In arithmetic geometry, it is often very interesting to assoicate ...

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### Galois action on $\overline{k}$-valued points extends to action on $k$-scheme $X$

Let $X$ be $k$ variety or more genrally a $k$-scheme. Denote the algebraic closure of $k$ by $\overline{k}$. it's a fact that $X(\overline{k}):=Hom(\operatorname{Spec} \ \overline{k}, X)$ as set is ...

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### Modularity of elliptic curves with only minimal lifting

I have been trying to understand a bit of the basics of deformations of Galois representations. One point which leaves me curious now is that proving modularity lifting with arbitrary ramification on ...

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### How geometry changes up to Hermitian inner product on Line bundle (Kodaira embedding)

Riemann metric $g \colon= \Sigma g_{ij} dx_i \otimes dx_j$ on a Kähler manifold $M$ will define the length of a line on $M$, i.e. intrinsic geometry. The line bundle $L$ on $M$ is equipped with a ...

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### Determinant of a special matrix in characteristic $p$

Let $K$ be a field of characteristic $p > 0$. Choose $p^i$ numbers of elements $c_1,\ldots,c_{p^i} \in K$ and consider the determinant $D$ of the following matrix$\colon$
\begin{pmatrix}\label{...

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### Independence of $p$ of Hodge-Tate weights

Let $X$ be a smooth and proper variety over $\mathbb{Q}$. Then for each prime $p$ we have the representation $R_p=H^i_{et}(X\times \overline{\mathbb{Q}_p}, \mathbb{Q}_p)$ of $\mathrm{Gal}(\overline{\...

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### Irreducible local Galois representation with arbitrary Hodge-Tate weights

Let $p$ be a prime and $M$ be a finite multiset of non-negative integers. Does there exist a continuous irreducible de Rham representation $\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)\to GL_n(\...

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### Irreducible global Galois representation with weights 0, 1, 3?

Fix a prime number $p$. Can there exist a continuous irreducible representation $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \mathrm{GL}_3(\mathbb{Q}_p)$ that is unramified at almost all primes, ...

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### Isomorphism between finite algebras over ${\Bbb Z}_p$

Let $\pi \colon R \twoheadrightarrow {\Bbb T}$ be a surjective ring homomorphism between finite algebras over ${\Bbb Z}_p$. Further, we suppose the following three conditions$\colon$
$R$ is a ...

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### Galois representations with trivial determinant that do not factor through a number field

In arithmetic geometry one often encounters continuous representations $\rho:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \operatorname{GL}_n(\mathbb{Q}_l)$ for some $n\geq 1$ and some prime ...

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### Properties of Mod $\ell^m$ Galois representation associated to modular form

(Sorry for my poor english..)
Let $F(z)\in S_{2k}(SL_2(\mathbb{Z})$) be a newform and $\ell$ be a prime larger than $3$. Let $K$ be a some number field and $v$ be a prime of $K$ over $\ell$. Let $K_v$...

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### Is the Fargues–Fontaine curve proper?

It is well known that Fontaine's curve $X=\bigoplus_{k\geq0}B_{\text{cris}}^{+,\varphi=p^k}$ is a Noetherian irreducible complete scheme of dimension $1$. And completeness means that the degree ...

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### An example that a $p$ adic Galois representation is crystalline but not $B_e$ admissible

$B_e=B_{\text{cris}}^{\phi=1}$, so if a $p$-adic Galois representation $V$ is $B_e$ admissible, then it is crystalline, so I want to know an example that $V$ is crystalline but not $B_e$ admissible.
...

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### $B_{\mathrm{dR}}=B_{\mathrm{cris}}+{B_{\mathrm{dR}}^+}$?

$B_{\mathrm{cris}}\subseteq B_{\mathrm{dR}}$ and $B_{\mathrm{dR}}^+$ are well-known period rings in $p$-adic Hodge. I know $B_{\mathrm{dR}}=B_{\mathrm{dR}}^+[\frac{1}{t}]$ and $\frac{1}{t}\in B_{\...

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### Would it be a little but good exercise to construct or find out Breuil modules?

My question is about p-adic Hodge-Tate theory and p-adic Galois representation.
One of the important semi-linear object in p-adic Galois representation is the $\text{Breuil Module}$. There are ...

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### Restrictions on the Galois representations coming from singular varieties

Fix a prime number $p$. Choose an algebraic closure $\mathbb{Q}_p\to \overline{\mathbb{Q}_p}$. Given a proper geometrically irreducible scheme $X$ over $\mathbb{Q}_p$ and a non-negative integer $i$, ...

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### On a certain radical of the formal power series ring $K[[X_1,X_2,\ldots,X_{\infty}]]$

Let $K$ be a field of characteristic $p > 2$ and $A_{\infty} \colon= K[[X_1,X_2,\ldots,X_{\infty}]]$ be an infinitely-many-variable formal power series ring over $K$ (the symbol $X_{\infty}$ is to ...

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### Adjoint Selmer groups and Deformation rings

Let $\rho:\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow \operatorname{GL}_2(\bar{\mathbb{Z}}_p)$ be a $p$-adic Galois representation associated to a $p$-ordinary Hecke eigencuspform, let $...

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### Proving automorphy of the Galois representations of number fields without considering the residual representation

All the papers proving automorphy of the representations of Galois groups of number fields that I have come across seem to first reduce the representation modulo a prime, prove the automorphy of the ...

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### Globalizable Galois representations

Let $\rho$ be a $p$-adic representation of $G=\text{Gal}(\bar{\mathbb{Q}_p}/\mathbb{Q}_p)$.
When does $\rho$ extend to a representation of the global galois group? What can be said about the locus ...

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### Generalized Hodge-Tate weights of an arbitrary p-adic Galois representation

Let $V$ be a continuous representation of the absolute Galois group of $\mathbb{Q}_p$ with coefficients in $\mathbb{Q}_p$. The theory of Sen attaches to $V$ generalized Hodge-Tate weights which are ...

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### Rigidity of universal abelian variety

When $\phi \colon A^{\mathrm{univ}} \to \overline{A_g}$ denotes the universal Abelian variety over a compactified Siegel moduli space $\overline{A_g}$, does there exist non-trivial automorphism $\...

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### Motivations of families of modular forms, elliptic curves and Galois representations?

I want to know some reference, why do some number theorists study the families of the elliptic curves, modular forms or Galois representations? As far as I know, I always consider the Galois ...

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### Intersection of a certain linear ideals of $K[[X_1,\ldots,X_{np}]]$ for ${\mathrm{ch}}(K) = p > 0$

Suppose ${\mathrm{ch}}(K) = p > 0$ and we consider the formal power series ring $K[[X_1,\ldots,X_{np}]]$ over $K$ in $np$ variables $X_1,\ldots, X_{np}$. Let $\Lambda$ be the set defined as follows$...

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### Field of definition of compatible system of Galois representations

Let $K,F$ be number fields and suppose that there is a compatible system of Galois representations
$$(\rho_{\lambda} : \text{Gal}(\overline{K}/K) \longrightarrow \text{GL}_n(\overline{F}_\lambda) )$$
...

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### View Dirichlet character as a character of Galois group

In Jaclyn Lang's article "On the image of the Galois representation associated to non-CM Hida family" section 2, the Dirichlet character $\chi$ module $N$ is also viewed as a character $\chi\colon\...

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### Failure of local Fontaine Mazur

This question unfortunately has a very similar name to this one, but I what want to ask here is different.
Let $K$ be a finite extension of $\mathbb{Q}_p$. It seems to be well known that the local ...

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### Topology on $p$-adic period rings in an article by Fontaine, part II

This is a follow-up to this question. See that question for background and relevant notation.
In the answer to that question, it is claimed, if I understand the answer correctly, that a basis of ...

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### Power series ring $\Theta[[X_1,\ldots,X_d]]$ and prime ideals

Let $\Theta$ be a domain. We shall choose $d$ elements $\theta_1,\ldots,\theta_d \in \Theta$ such that any chosen $j$ elements $\theta_{i_1},\ldots,\theta_{i_j}$ form a prime ideal $(\theta_{i_1},\...

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### Stacks project for Galois representations and automorphic forms

Is there anything like Stacks project for Galois representations and automorphic forms? I am not asking people to start something like Stacks project, just asking if something like Stacks project ...

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### Level Lowering Galois representations over Totally real fields

Let $F$ be a totally real number field and $\mathbb{F}$ a finite field. Let $\bar{\rho}:\text{Gal}(\bar{F}/F)\rightarrow \text{GL}_2(\mathbb{F})$ an irreducible Galois representation arising from a ...

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### Derived weight filtration on motivic Galois representations

Thanks to modern techniques (such as the pro-etale site), we can now understand etale (co)homology of varieties and motives as "genuinely" derived (e.g. DG) Galois-equivariant objects. I'm looking for ...

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### Fontaine-Fargues curve and period rings and untilt

When I read the paper "THE FARGUES–FONTAINE CURVE AND DIAMONDS" of Matthew Morrow, I have a question on page 11.
Question: The arthur said that the de Rham and crystalline period rings implicitly ...

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### How to prove the p-adic Galois representations atteched to the Tate module of an abelian variety is de Rham directly?

Recently I read a thesis p-adic Galois representations and elliptic curves. Using Tate's curve, the author proved the p-adic Galois representations atteched to the Tate module of an elliptic curve is ...

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### Classify 2-dim p-adic galois representations

Recently I have known how to classify 1-dim p adic Galois representations $\phi$. The p-adic Galois representations mean that a representation $G_K$ on a p-adic field $E$, where $K$ is also a p-adic ...

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### Isomorphism of the $\ell$-adic Tate module of an elliptic curve with CM

Let $E$ be an elliptic curve over $K$ (totally real number field) with complex multiplication by the field $L$. Let $\psi$ be the Grössencharacter associated to $E$, assume that $\psi$ of type $(-r,0)$...

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### Comparison of weight-monodromy filtrations

Setup:
Let $R$ be a finitely generated subring of $\mathbb{C}$. Let $X \rightarrow \mathbb{A}^1_R$ be a proper morphism of $R$-varieties, smooth except over a rational point $s \in \mathbb{A}^1_R$ ...

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### Classification of finite flat group schemes over integers?

One can classify (commutative) finite flat group schemes (with order of $p$-powers) over $\mathbb Z_p$ using semi-linear algebraic datas such as Breuil-Kisin modules. And we can fix the special fiber ...

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### On the product in the power series ring

Let $A_n \colon= K[[X_1,\ldots,X_n,Y_1,\ldots,Y_n]]$ be a power series ring over a field $K$ in $2n$ variables and ${\frak m}_{A_n}$ be the unique maximal ideal of $A_n$.
Suppose we have two ...

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### Intrinsicness of Hodge-theoretic properties of Galois representations in a general reductive group

In the paper "The conjectural connections between automorphic representations and Galois representations" by Buzzard and Gee, it is said
"We say that
$\rho$ is crystalline/de Rham/Hodge–Tate if ...

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### Lifting Galois representations

Let $G$ be the absolute Galois group of a finite extension of $\mathbb{Q}$ or $\mathbb{Q}_p$. Let $k$ be a perfect field of characteristic $l>0$ (possibly $l=p$).
If we have a homomorphism $G\...