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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Eigenvarieties and functoriality
In Langlands' review of Hida's book "$p$-adic automorphic forms on Shimura varieties", he discusses a nexus of 4 areas of modern number theory: automorphic representations, motives, spaces ...
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Is the weight in Serre's conjecture "minimal"?
Serre's conjecture says that given any odd, irreducible, continuous representation $\rho:G_{\mathbb{Q}}\rightarrow GL_2(\overline{\mathbb{F}_p})$ there is some eigenform $f$ of weight $k(\rho)$, level ...
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Modularity of higher genus curves
The modularity conjecture for elliptic curves over number fields is well known, and indeed, is a theorem for all elliptic curves over $\mathbb{Q}$, and at least potentially, over any CM field.
What ...
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Mumford-Tate conjecture for mixed Tate motives
Let $X$ be a (not necessarily smooth or proper) variety over a number field $k$. Suppose we are given
A subquotient $V_{dR}$ of the algebraic de Rham cohomology $H_{dR}^i(X)$ (defined in the non-...
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Effective weight-monodromy conjecture
$\DeclareMathOperator\Gr{Gr}$Let $G$ be the absolute Galois group of a finite extension of $\mathbb{Q}_p$ with inertia subgroup $I$, and let $V$ be an $\ell$-adic representation of $G$. Grothendieck's ...
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Automorphic forms and coherent cohomology
Why is it (and what does it mean) that automorphic forms do not contribute in the coherent cohomology of Siegel modular varieties parametrizing abelian varieties of dimension $d>2$ (see section 7 ...
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Finiteness of points over the cyclotomic extension for modular forms
Let $\rho(f):G_\mathbb{Q} \rightarrow GL_2(K_f)$ be the Galois representation attached to some cuspidal modular form $f$ where $K_f$ is a finite extension of $\mathbb{Q}_p$.
Let $V_f$ be the vector ...
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The modularity theorem as a special case of the Bloch-Kato conjecture
In the homepage for the CRM's special semester this year, I found the interesting statement that the modularity theorem (formerly the Taniyama-Shimura-Weil conjecture) is a special case of the Bloch-...
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Modularity switching for primes $p>7$
In Freitas, Le Hung, and Siksek's 2014 paper proving that elliptic curves over totally real quadratic fields are modular they prove (and use) the following result (Theorems 3 and 4): let $\bar \rho_{E,...
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Is the cohomology of rigid varieties semisimple?
Let $X$ be a smooth projective geometrically connected scheme over $\mathbb{Q}_p$. Assume that $H^1(X, T_{X/\mathbb{Q}_p})=0$.
Is the Galois representation $H^*(X_{\overline{\mathbb{Q}_p}}, \mathbb{Q}...
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Roadmap for studying Galois deformation theory/modularity theorems from a modern perspective
I am a graduate student with some background in Galois deformation theory. I am familiar with the basics (the existence of a universal deformation space with prescribed conditions) and with some ...
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Which cases of Beilinson-Bloch-Kato for elliptic motives are known?
Let $V$ be a semisimple geometric Galois representation of a number field. Then the Bloch-Kato conjectures state that
$$
\operatorname{ord}_{s=0}{L(V^*(1),s)} = \operatorname{dim}{H^1_f(G_k,V)}-\...
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Artin reciprocity via Shimura varieties
The point of Shimura varieties, as far as I've understood it, is that for a given Shimura datum $(G,D)$, there exist models, by which I mean that for congruence subgroups $\Gamma$ there exists a ...
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$\mathrm{mod}\:p$ Galois representation with respect to Zariski topology
Let $G$ be the absolute Galois group of some number field. Can there be a semisimple continuous representation $G\to GL_n(\overline{\mathbb{F}_p})$ (the latter has Zariski topology) with infinite ...
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Regular representations of Galois groups
Suppose $\mathcal{G}_k$ is the absolute Galois group of a number field $k$.
$\mathcal{G}_k$ is a topological group, with profinite topology. How does the theory of harmonic analysis of regular ...
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Uncountably many non-isomorphic Tate modules
Do there exist uncountably many abelian surfaces with good reduction over $\mathbb{Q}_p$ with pairwise non-isomorphic rational $p$-adic Tate modules?
If we took $l$-adic Tate modules there would be ...
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What unramified Galois representations come from geometry?
I think we don't know what crystalline representations come from geometry. What about the unramified ones? Specifically let $\phi:\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)\to GL_n(\mathbb{Q}...
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Galois representation absolutely irreducible after restricting to open subgroup of finite index
Let $E$ and $F$ be finite extensions of $\mathbb{Q}_p$. Let $\phi:\mathrm{Gal}(\overline{E}/E)\to GL_n(F)$ be an absolutely irreducible continuous representation. Assume that the restriction of $\phi$ ...
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Galois representation with infinite image but finite image everywhere locally
Fix a prime $l$. Let $\phi:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to GL_n(\mathbb{Q}_l)$ be a semisimple continuous representation. Assume $\phi$ has finite image when restricted to $\mathrm{...
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$l$-adic Galois representations factor through a common finite quotient
Let $X$ be a smooth projective geometrically connected variety over $\mathbb{Q}$. Assume that for some $m>0$ we have $h^{i, 2m-i}(X)=0$ unless $i=m$.
Does there exist a number field $E$ such that ...
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$p$-adic Galois representation and Étale homology
Let $X$ be a smooth proper scheme over some $p$-adic field $K$. The "usual" way to get a Galois representation out of this is to consider the étale cohomology (either $p$ or $\ell$-adic). ...
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Bloch–Kato–Selmer group of a one-dimensional representation
Let $L/\mathbb{Q}$ be a finite extension and let $V$ be a one-dimensional $L$-linear representation of $G_{\mathbb{Q}}$ which is given by $\chi\rho^*\kappa^n_{\text{cyc}}:G_{\mathbb{Q}}\rightarrow L^\...
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Algorithmically recover the $l'$-adic Galois representation from the $l$-adic one (assuming the Tate conjecture)
Let $E$ be a number field. For any finite Galois extension $E\subset F$ there is a continuous homomorphism $\pi_F:\mathrm{Gal}(\overline{E}/E)\to \mathrm{Gal}(F/E)$.
Let $X$ be a smooth projective ...
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Is semi-simplicity of Galois representations local?
Let $\rho:G_{\mathbb{Q}}\rightarrow \text{Gl}(V)$ be a finite dimensional $\ell$-adic Galois representation. Then for each prime, by pre-composing $\rho$ with the natural inclusion $G_{\mathbb{Q}_p}\...
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Outer Galois representations and Tate modules of Jacobian varieties
Let $X$ be a proper smooth curve over a field $k$. Then we have an exact sequence of profinite groups
\begin{equation*}
1 \to \pi_1(X_{\overline k}) \to \pi_1(X) \to G_k \to 1,
\end{equation*}
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"Universal coefficent theorem" for pro-étale cohomology
In algebraic topology, for any space with finite homology type, the universal coefficient theorem states that for any abelian group $G$, we have
$$H^n(X,G)\cong \left( H^n(X,\mathbb{Z})\otimes G\right)...
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Galois representations and pro-étale Site
On a scheme, we can define the pro-étale site. This is an improvement over the étale site in that we can define the $\ell$-adic cohomology as the sheaf cohomology of the constant sheaf $\underline{\...
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Galois action of Weil restriction
Let $K/\mathbb{Q}$ be a quadratic field. Let $E$ be an elliptic curve defined over $K$ but not over $\mathbb{Q}$, and let $\bar{E}$ be the Galois conjugate of $E$. Then by the descent theory (for ...
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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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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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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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Misunderstanding in the hypotheses of Schlessinger's criterion
In studying deformation theory of Galois representations, I've come surely to an error, relating Schlessinger's criterion.
Let's fix a representation $\bar{\rho}$ of a group $G$ and let $D_{\bar{\...
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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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The difficulties in proving modularity lifting theorems over non-totally real fields
First of all, let me apologize in advance for the terseness of this question.
It seems that by now there are well-developed techniques (the "Taylor-Wiles-Kisin" method) for proving modularity lifting ...
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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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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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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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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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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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