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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Normalizing factor in Reciprocity of traces of Frobenius with solutions of equations mod p

One kind of Reciprocity tells us that we can count solutions to polynomial equations over finite fields and relate them to traces of Galois Representations with some normalization factor. For ...
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Inverse limit of Noetherian rings

Let $R_i$ be a noetherian regular local ring of Krull dimension being finite. Suppose we are given a surjective homomorphism $\phi_{i,j} \colon R_{j} \twoheadrightarrow R_i$ for each $i,j$ with $j >...
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Stalks of the sheaf

Let $X$ be a scheme. For $m < \infty$, given the surjection ${\cal O}_X^{\oplus m} \twoheadrightarrow {\cal F}$ between sheaves on $X$, where ${\cal O}_X$ is the structural sheaf of $X$. Choose a ...
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On semi-complete ring K[X_1,X_2,…,X_∞]] and Popescu theorem

Let $P_n \colon= K[X_1,...,X_n]$ be a $n$-variables polynomial ring. We define 'semi-complete' polynomial ring $P_{\infty}$ by the following$\colon$ $P_{\infty} = K[X_1,...,X_\infty]] \colon = \...
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Everywhere holomorphic functions over C [closed]

Let $f, g$ be two everywhere holomorphic functions over ${\Bbb C}$. We consider the local representation of $f, g$ at the origin of ${\Bbb C}$, i.e. $z = 0$. That is, we can consider $f, g \in {\Bbb C}...
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1answer
141 views

Complete subring of F_p[[X]]

Pointed out on famous disbelief, I know now that there is an embedding $\iota_n \colon {\Bbb F}_p[[T_1,...,T_n]] \hookrightarrow {\Bbb F}_p[[X,Y]]$ for any $n \leq \infty$. Then I would like to ask ...
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Popescu-Neron Desingularization for K[[T_1,…,T_∞]]

Let $K[[T_1,...,T_n]]$ be a finitely many variables formal power series ring over a field $K$. Dorin Popescu proved that there are smooth algebras $A_{\lambda}$'s which are of finite type over $K$ ...
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Coherence of subrings of K[[X,Y]]

Let $K[[X,Y]]$ be a two-variables formal power series ring over a field $K$. Consider a sub-ring $\iota \colon A \subset K[[X,Y]]$. Q. Is A coherent? $\quad$ Or is it automatic that $\iota$ is ...
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Is there a yoga of effectivity for motives and their realizations?

Inside the 'usual' categories of motives (Chow, Voevodsky ones) there are the categories of effective motives. Similarly, there are effective Hodge structures; they seem to be closely related with ...
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790 views

Even Galois representations “mod p”

Consider an irreducible $\mathrm{mod}$ $p$ representation: $$\rho: \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_2(\bar{\mathbb{F}}_p)$$ If $\rho$ is odd, it was conjectured by Serre in ...
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Characterization of Singular locus

Let $A$ be a complete regular local ring over a field $k$ and $B$ be a complete normal local ring over a field $k$. We assume that the Krull-dimension of $A$ is greater than 1. We consider the ...
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1answer
279 views

Frobenius at ramified primes

Let $E$ be an elliptic curve defined over $\mathbf{Q}$, fix an odd prime $p>3$, let $T_p$ denote the $p$-adic Tate module of $E$, and let $V_p = T_p \otimes \mathbf{Q}_p$. If the action of $G_\...
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218 views

On the coherence of formal power series ring

Let $A = {\Bbb F}_p[[X_1,X_2,...]]$ be the ring of formal power series with infinitely many variables over the finite field ${\Bbb F}_p.$ $A$ consists of such formal sum elements as $\sum c_{e_1,.....
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791 views

Why does $H^i(X_{ét},\mathbb{Q}_p)$ have a Hodge-Tate structure?

Let $X$ be a variety over a $p$-adic field $K$. Is there a simple or intuitive explanation of why the $G_K$ representation $H^i(X_{ét},\mathbb{Q}_p)$ is Hodge-Tate? More precisely, why do the powers ...
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1answer
157 views

Applications of Level Lowering

What are some applications/consequences of level lowering of Galois representations? I understand the application of Ribet's theorem in the proof of Fermat's last theorem but I am wondering what other ...
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Elliptic curves and the $\ell$-adic image of the decomposition group

Let $E$ be an elliptic curve over $\mathbb{Q}_\ell$ and consider the image of the $\ell$-adic representation. Is there a description of this image similar to Serre's description of the image of the ...
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131 views

Is Howe's construction of tame supercuspidal representations independent of additive character?

Let $F$ be a $p$-adic field. In "Tamely ramified supercuspidal representations of $Gl_n$" (Am. J. Math 73 (1977)), Howe constructs a supercuspidal representation $\pi_{\psi}$ of $GL_n(F)$ from the ...
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271 views

Are prime ideals of finite height in the powers series ring in infinitely variables finitely generated?

Let $A:= {\mathbb F}_p[[X_1,...,X_∞]]$ be the infinitely many variables formal power series ring over ${\mathbb F}_p$, which is UFD. Consider an arbitrary prime ideal $P$ of $A$ such that the height ...
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156 views

Number of minimal primes for UFD

Let $R$ be a UFD which is NOT noetherian. It is well-known that $R$ is a Krull ring. Let $I$ be an ideal of $R$ such that the height of $I$ is $d$ which is finite. Question. Is the number of minimal ...
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259 views

Weight filtration on certain Galois representations

Let $G$ be the absolute Galois group of a number field $K$. Let $\ell$ be a prime number. There are representations $\mathbb{Z}_\ell(n)$ of $G$ on the group of $\ell$-adic integers given by the ...
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Hodge-Tate weights of etale cohomology

Let $K/\mathbb Q_p$ be a local field, $X/K$ a proper scheme with semi-stable reduction. Question: What is the possible range of Hodge-Tate weights of the etale cohomology $H^i(X_{\overline K}, \...
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extending local systems on a neighbourhood

Let $Y$ an affine finite type scheme over an algebraically closed field $k$. Let $S$ be a closed subscheme of $Y$ and $Y'$ the henselization of $Y$ along $S$. If we have a $\mathbb{Z}_{\ell}$ local ...
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163 views

l-adic local system. on hensel schemes

Let $k$ be a field, $\ell$ a prime different from the characteristic. If I take $S$ a closed subscheme of $Y$, which is a $k$-scheme of finite type, is it true that any $\mathbb{Z}_{\ell}$-local ...
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336 views

Hodge–Tate structures of modular forms

The title refers to the paper of Faltings: Hodge-Tate structures and modular forms. Math. Ann. 278 (1987), no. 1-4, 133–149. The main theorem in the paper says that the associated Galois rep to a ...
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Status of Fontaine-Mazur conjecture

In the language of Richard Taylor's 2004 (extended) ICM article (''Galois Representations'', Annales de la faculté des sciences de Toulouse (2004) Tome XIII, no. 1, 73-119), the conjecture is the ...
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204 views

Mazur's Galois Deformations paper for non-residually irreducible case

In Barry Mazur's paper introducing Galois deformations, he hints at having a general theory for representations which are not residually Schur, but with more complicated statements. Does anyone know ...
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429 views

Galois representations for the curve $y^{2} = x^{3} - x$

Let $E / \mathbb{Q}$ be the elliptic curve given by $y^{2} = x^{3} - x$. I would like to know explicitly what the field of all $2$-power torsion looks like, as well as the image in $\mathrm{GL}(T_{2}(...
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404 views

realizing uniform boundedness of Galois representations associated to elliptic curves

This is less of a direct question and more of an argument that I've been worried about for a while and want to check (apologies for the length and if my writing is unclear). Suppose I have an ...
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1answer
111 views

How much extra ramification in a residual representation

Suppose $\rho:G _{\mathbb{Q}} \rightarrow GL_n(\mathbb{Q}_p)$ is a Galois rep. It has a uniquely defined (up to semisimplification residual rep $\bar{\rho}$. $\bar{\rho}$ is unramified where $\rho$ is....
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How to calculate log or exp of a value in GF(2^n) using log/exp table of GF((2^k)^m) where n=k*m?

Consider Galois fields $\mathbb{F}_{2^n}$ and $\mathbb{F}_{2^k}$, where $n=km$, and $\mathbb{F}_{2^k}$ is a ground field of $\mathbb{F}_{2^n}$. I’d appreciate pointers to papers or suggestions on: ...
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Minimal discriminant of an elliptic curve in terms of its Galois representation

From the Galois representation of an elliptic curve $E$ we can read the conductor of $E$, and further some information about the minimal discriminant. So is there any more information about the ...
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Residual Representation of a Motive

Suppose we have $M$ a hypergeometric motive, and $\rho$ its associated Galois rep over $\mathbb{Q}_{l}$. Is there any easy/concrete way to find $\bar{\rho}$, the residual representation at a prime (in ...
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What are the local Langlands conjectures nowadays, for connected reductive groups over a $p$-adic field?

Let me stress that I am only interested in $p$-adic fields in this question, for reasons that will become clear later. Let me also stress that in some sense I am basically assuming that the reader ...
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2answers
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non-continuous inverse Galois problem

Let $G=Gal(\bar{\mathbf{Q}}/\mathbf{Q})$ be the absolute Galois group over $\mathbf{Q}$. Q1: Is it possible to find a (necessarily non-closed) normal subgroup $K\leq G$ such that $G/K$ is free of ...
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Propagation of modularity and the Artin conjecture

The (still incomplete) solution of the Artin conjecture on dimension $\leq2$ has been a massive research effort that has spanned (knowingly or not) around a century. A very natural question is, what ...
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379 views

When is the image of a 2-dim l-adic representation associated to a modular form open

I know the following theorems by Serre: 1, The 2-dim l-adic representation associated to a non-CM elliptic curve is open. 2, The 2-dim l-adic representation associated the weight-12 cusp form $\...
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When do the Galois reps of modular forms have open image?

Suppose f is a newform (with coefficients generating some number field E), and $\rho_{f,\lambda}: {\rm Gal}(\overline{\mathbb{Q}} / \mathbb{Q}) \to {\rm GL}_2(E_\lambda)$ the associated Galois rep (...
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1answer
283 views

Why is every l-adic Galois representation conjugate to one over the l-adic integers? [closed]

Why is every l-adic Galois representation $$G_{\mathbb{Q}_p}\rightarrow GL_n(\mathbb{Q}_{l})$$ conjugate to one over the l-adic integers? $$G_{\mathbb{Q}_p}\rightarrow GL_n(\mathbb{Z}_{l})$$
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Type of a modular form

Let $f$ be an arbitrary weight 1 newform. We know by Serre-Deligne that there is an odd 2-dimensional irreducible Artin representation $\rho$ such that $L_f(s)=L(\rho,s)$. I was wondering how much ...
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1answer
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Converse to Modularity II: Maass cusp forms

(This comes from this other question. You can find more details there) The following bijection is now a theorem: Odd irreducible 2-dim Galois repn $\longleftrightarrow$ weight 1 newforms note:...
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Serre's surjective theorem importance

I'm studying Serre's paper in wich he shows the following theorem: Let K be a number field, $E$ an elliptic curve over K without CM. Then the representation $$\rho_{\ell}:\mathrm{Gal}(\bar K/K)\...
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Status of conjectures in Serre's 1969 expose on Galois representations on l-adic cohomology

In [S]: Serre, Jean-Pierre. Facteurs locaux des fonctions zeta des varietes algebriques (definitions et conjectures), Seminaire Delange-Pisot-Poitou, 1969-70 Serre presents nine conjectures *C*$_1$...
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Iwasawa main conjectures vs Bloch-Kato conjectures

Let $p$ be a prime, $K$ be a number field, $S$ a finite set of finite places of $K$ containing the set $S_p$ of places above $p$ and the places at infinity, $G:=G_{K,S}$ the Galois group of the ...
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1answer
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Smoothness of Hecke algebras

First I will introduce some notation and definitions. Fix a level $N$ (take $N=1$ if it makes things easier) and a prime $p$. Let $k$ be a finite field of characteristic $p$ and let $\mathcal{C}$ be ...
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To what extent are modular parametrizations expected to generalize?

By the Modularity Theorem (a.k.a. the Shimura--Taniyama--Weil Conjecture), if $E$ is an elliptic curve over $\textbf{Q}$ with conductor $N$, then there exists a “modular parametrization” $\psi: X_0(N) ...
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2answers
599 views

Explicit Chebotarev and Langlands - irreducibility of X^5-X-1 mod primes

Is there an explicit infinite set of primes, modulo which $X^5 - X - 1$ is irreducible? Since our polynomial's Galois group over $\mathbb{Q}$ is $S_5$, Chebotarev's density theorem implies that ...
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1answer
196 views

The infinity-type of automorphic representations in the Langlands correspondence

Let $K$ be a number field, $\rho\colon \mathrm{Gal}_K\to \mathrm{GL}_n(\overline{\mathbf{Q}_p})$ a geometric (i.e.: unramified a.e., de Rham above $p$) irreducible Galois representation. One piece of ...
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1answer
313 views

Computing an eigencuspform in $S_2(\Gamma_0(1776))$

Consider $$\bar{\rho}:G_{\mathbb Q}\longrightarrow\operatorname{GL}_2(\mathbb F_7)$$ the residual 7-adic Galois representation attached to the elliptic curve $y^2=x^3+x^2-4x-4$ of conductor 48. Then $...
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1answer
481 views

Which degree does a motivic Galois representation show up in?

Consider a representation $\rho: \operatorname{Gal} (\overline{\mathbb Q} | \mathbb Q ) \to GL_n ( \overline{\mathbb Q}_\ell)$ that is a subrepresentation of $H^i(X, \overline{\mathbb Q}_\ell (j))$ ...
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2answers
380 views

Lifting projective Galois representation with condition

Let $\bar{\rho}: G_K\to PGL_n(\mathbb{C})$ be projective representation of the absolute Galois group of a number field $K$ and $\varphi\in Aut(G_K)$. A theorem of Tate tells us that we can always ...