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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Is a pro-algebraic group over $\mathbb{Q}_p$ with Galois action the inverse limit of Galois-equivariant quotients?
Let $\mathcal{G}$ be a pro-algebraic group over $\mathbb{Q}_p$ with a continuous action of $G_K$ for a field $K$ (if $\mathcal{G}$ were an abelian unipotent group, this is precisely a $p$-adic Galois ...
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Galois representations attached to discrete automorphic representations
Let $F$ be a totally real field. Let $G$ be a (split) connected reductive group over $F$. Let $\pi$ be an irreducible automorphic representation of $G$.
Recall in the work of Buzzard and Gee "The ...
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Descent of isogenies between p-divisible groups
Let $\mathcal{G}$ be a $p$-divisible group over $K$, which is a finite extension of $\mathbb{Q}_p$. Let $\rho: \text{Gal}(\bar{K}/K)\rightarrow \text{GL}(T_p\mathcal{G})$ be the associated Galois ...
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Galois representations with semisimple residue representation
$\DeclareMathOperator\GL{GL}$Let $\mathbb{Z}_p$ be the ring of integers of $p$-adic numbers $\mathbb{Q}_p$, $G$ a profinite group (e.g. Galois group of local field or global field) and $\rho:G\to \...
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When is the image of $\operatorname{Gal}(\bar K/K)$ open in $\operatorname{Aut}(V)$, where $V$ is the vector space coming from a $p$-adic Tate module?
Let $K$ be a finite extension of $\mathbb{Q}_p$ with ring of integers $A$. Let $F$ be a $p$-divisible group and $T$ be the Tate module. Consider the vector space $V=T \otimes_{\mathbb{Q}_p} C$, where $...
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Reference request: étale local system on $\Gamma\backslash\mathcal H$ for $\Gamma$ non-congruence
Suppose $\mathcal H$ is the upper half plane, and $\Gamma\subset \operatorname{PSL}_2(\mathbb Z)$ is a finite index subgroup. Then by Belyi's theorem we know that $\Gamma\backslash\mathcal H$ is an ...
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mod $p$ local Galois representation attached to elliptic curves
In the paper, lemma 4.4. The author gives the form of the representation of $G_p$ on $E[p]$ of the form
$$\begin{pmatrix} \varepsilon\chi & *\\0 & \chi^{-1} \end{pmatrix}.$$
Do they assumed ...
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On local Galois deformation rings
Let $p,\ell$ be two different primes. Let $K$ be a finite field extension of $\mathbb{Q}_{\ell}$ and $ \bar{\rho}:G_{K}\to {\rm GL}_{n}(\mathbb{F}_p) $ be a continuous mod $p$ representation of the ...
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Field of definition of automorphic Galois representation
Let $\pi$ be a regular, cuspidal, algebraic automorphic representation of $GL_n(\mathbb{A}_K)$ for a totally real field $K$. Then for every embedding $\lambda$ of $E=\mathbb{Q}(\pi)$ in $\overline{\...
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Symmetric power lift of modular forms
Let $f_1$ and $f_2$ be two cuspforms of weights $k_1$ and $k_2$ and nebentypus $\epsilon_1$ and $\epsilon_2$ respectively such that $f_1 \neq f_2 \otimes \chi$ for some Dirichlet character $\chi$ of ...
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Nonabelian reciprocity law
I heard the following relation in a talk by Peter Scholze. Could someone explain "in a simple way" what is the precise relation between the polynomial $x^4-7x^2-3x+1 $ and the integral ...
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A specific $2$-dimensional Galois representation of $G_{\mathbb{Q}_2}$ and its Langlands correspondence
I am interested in understanding a situation in (classical, not $p$-adic) local Langlands for $\mathrm{GL}_p(\mathbb{Q}_p)$. An example of
it is as follows: Let $F=\mathbb{Q}_2$ and $E$ be the ...
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Semistability of the $\ell$-adic representation of variety with semistable reduction
The question is in the title, but here's some quick background.
It's easy to show (assuming smooth-proper base change) that the $\ell$-adic cohomology of a variety over the fraction field of a DVR ...
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Galois action on the cohomology of a curve over a local field with bad reduction
Let $C/\mathbb Q_p$ (or a p-adic local field more generally) be a smooth projective curve with split semistable reduction over $\mathbb Z_p$. What can we say about the action of the Galois group $\...
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Finiteness and bounds for elliptic curves realizing a given galois representation
Let $\rho: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \text{GL}_2(\mathbb{Z}_p)$ be a continuous, irreducible Galois representation. Consider the set $\mathcal{L}_\rho$ of all elliptic curves $...
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Question about trianguline representations
Following the notation in https://arxiv.org/abs/1011.3447 a representation $V$ is split trianguline iff $D(V)$ has a basis in which the matrices of $\varphi$ and of all the elements of $\Gamma$ are ...
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Existence of even mod $ p $ Galois representations with full image
My question is that:
For any prime $p>7$, is there a mod $ p $ representation $ \bar{\rho}:G_{\mathbb{Q}}\to \operatorname{GL}_{2}(\mathbb{F}_{q}) $ of the absolute Galois group $ G_{\mathbb{Q}} $ ...
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Crystalline at $\ell=p$ implies unramified at $\ell\neq p$
Let $X$ be a smooth, projective variety defined over some $p$-adic field $K$. Is it true that if the etale cohomology $H^i_{et}(X_{\overline{K}},\mathbb{Q}_\ell)$ is crystalline at $\ell=p$, then $H^...
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Evidence for the equivariant BSD conjecture with higher multiplicity
Let $E/\mathbb{Q}$ be an elliptic curve and let $\rho$ be an irreducible Artin representation. Let $K_\rho/\mathbb{Q}$ be the smallest Galois extension such that $\rho$ factors through $\mathrm{Gal}(...
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Semisimplicity of induced representation of a irreducible representation
This question occurs when I read this one.
Suppose $k$ is a field of char $0$ (not necessarily complex numbers, in my case it's $\mathbb Q_p$), and $G$ is a profinite group (in my case, $\operatorname{...
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Are there known effective bounds on the number of semisimple Galois representations?
In continuation to my question here, are there known effective bounds on the total number of semisimple $p$-adic Galois representations unramified outside a finite set of primes $S$, of dimension $d$, ...
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Reference for Faltings' proof on finiteness of semisimple $d$-dimensional $p$-adic Galois representations
I'm looking for a reference to Faltings' proof concerning the finiteness of $d$-dimensional semisimple $p$-adic Galois representations. Specifically, the result states that there are only finitely ...
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Prime ideal of $A[X_1,...,X_d]$
Let $A$ be a UFD domain, i.e. integral and for any height one prime
${\frak p}$ of $A$, we have ${\frak p} = (u_{\frak p})$ for some $u_{\frak p} \in A$.
Once and for all, we fix the algebraic ...
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Question About Page 11 of Cornell, Silverman, Stevens, "Modular Forms and Fermat's Last Theorem"
Looking at page 11 of the text, consider a Galois representation $\rho: G_{\mathbb Q} \to \operatorname{GL}_2(A)$, where $A$ is a coefficient ring (i.e. complete Noetherian local ring with finite ...
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Depth of the filtration of higher ramification groups in the ramified case in Serre's modularity conjecture
I am studying Serre's paper "Sur les représentations modulaires de degré 2 de $\mathrm{Gal(\overline{\mathbb Q}/\mathbb Q)}$" and I have some questions about Serre's definition of "peu ...
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Automorphy of the twisted representation
The Artin reciprocity says that if
$$
\chi: \operatorname{Gal}(K/\mathbb Q) \to \mathbb C
$$
is a 1-dimensional representation of a finite Galois extension $K/ \mathbb Q$, then it corresponds to a ...
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Rational isogenies of prime degree $p\in\{11,17,19,37,43,67,163\}$
Let $p\in\{11,17,19,37,43,67,163\}$ be a prime number. In [1], B. Mazur proves that there are only finite number of elliptic curves $E$ [over $\mathbb{Q}$] having an isogeny of degree $p$.
Here is my ...
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Vector bundles on pro-etale topology over a field
Suppose $K$ is a finite extension of $\mathbb Q_p$. Consider the one-point adic space $X=\operatorname{Spa}K$, and let $C=\hat {\bar K}$, $G=\operatorname{Gal}(\bar K/K)$. I heard that the category of ...
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Action of complex conjugation on etale cohomology
Let $X$ be a genus $g$ smooth projective curve, defined over $\mathbb{Q}$, and let $\overline{X}$ denote the base change of $X$ to $\overline{\mathbb{Q}}$.
It is well known that $H^1_{\text{ét}}(\...
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Lifting mod $p$ representations of arithmetic fundamental groups of a non-affine scheme over a finite field of characteristic $p$
Let $X$ be a geometrically irreducible scheme (not necessarily affine) over $\mathbb{F}_{p}$ and let $ \pi_{1}(X) $ be the arithmetic etale fundamental group of $ X $. Let $ \overline{\mathbb{F}}_{p} $...
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Absolute Bloch-Kato Cohomology
The étale cohomology $R\Gamma_{\mathrm{ét}}(X;\mathbb{Z}_p(n))$ of a scheme $X/K$ can be computed by a Hochschild-Serre spectral sequence with terms of the form $H^i(K;H^j(X_{\overline{K}};\mathbb{Z}...
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Finite dimensionality of Galois cohomology
Let $K_S$ denote the maximal extension of $\mathbb{Q}$, unramified outside a finite set of primes $S$, and let $G_S$ denote the Galois group of $K_S/\mathbb{Q}$.
It is known that for any finitely ...
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Unramified lisse $\overline{\mathbb{Q}}_{\ell}$-sheaves
Let $X$ be a connected noetherian scheme and $\ell$ a prime invertible on $X$. Let $D \subset X$ be a regular effective Cartier divisor (or more generally a normal crossings divisor, I suppose). Write ...
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Local property of residual representations attached to elliptic curves over rational numbers
I found the following claim - without reference - in the (famous) book ''Modular Forms and Fermat’s Last Theorem'':
Let $E$ be a semistable elliptic curve over $\mathbb{Q}$. Let $\Delta_E$ be the ...
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Global minimal discriminants of elliptic curves and Galois representations
Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $\Delta$ be the global minimal discriminant. By A. Wiles et al. $E$ is modular so that there is a corresponding modular form $f$. I am wondering if ...
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Why locally algebraic characters of $\text{Gal}(\overline{\mathbb Q}/\mathbb Q)$ are associated to $A_0$ Grossencharacters/algebraic Hecke characters?
$\DeclareMathOperator\Res{Res}\DeclareMathOperator\Gal{Gal}$I am trying to understand lemma 3.1 of "Abelian Varieties over \mathbb Q and modular forms" of Ribet. ArXiv link
Just so everyone ...
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Local units in a family of $S_4$-extensions
Let $a \in \mathbb{Z}$ and consider the polynomial $f(X)=X^4+aX+1$; we assume that $a$ is chosen such that $f$ is irreducible and that the discriminant $4^4-27a^4=-p$ for some prime $p$ (for example $...
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Absolute irreducibility implies free action on framed universal deformation ring
Let $\overline{\rho}: G\longrightarrow \text{GL}_n(\mathbb{F}_p)$ be a residual representation of the Galois group of a number field. Let $R_{\overline{\rho}}^{\square,\text{univ}}$ be the universal ...
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Applications of Langlands for GLn explicit reciprocity laws other than elliptic curves
Is there any explicit application of Langlands conjecture for $\mathrm{GL}(n)$
for $n\ge 3$, to get some reciprocity laws for higher dimensional varieties or higher genus curves?
I've never found such ...
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The criterion for dimensional conjecture for universal Galois deformation rings
I’m writing to ask a question about Mazur’s dimensional conjecture in Lemma 7.5 of the paper [Galatius S, Venkatesh A. Derived Galois deformation rings. Advances in Mathematics. 2018 Mar 17;327:470-...
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Differential structures on unit-root Frobenius modules
Let $\mathcal{E}$ be the ring of series $f(T)=\sum_{n \in \mathbb{Z}} a_nT^n$, $a_n \in \mathbb{Q}_p$ such that the $\{a_n\}$ are bounded and $a_{-n}\rightarrow 0$ as $n \rightarrow +\infty$. This is ...
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Existence of odd mod $p$ Galois representations whose image is $p'$-group
Let $K$ be a number field and let $G_K$ be the absolute Galois group of $K$. Let $p$ be an odd prime and $\mathbb{F}_p$ be a finite field of order $p$. Can we always find a continuous representation $\...
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Multiplicities of Galois representations in the semisimplification of the reduction of a Tate module
Let $C$ be a smooth proper curve, of genus $g$ over a number field $K$. Let $v$ be a prime of good reduction for $C$ above $p>2$, and let $T_pJ$ denote the $p$-adic Tate module of $J$, the Jacobian ...
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A question related to Kirillov model
I am reading Jackson - The theory of admissible representations of $\operatorname{GL}(2, F)$ and am not able to understand the following map related to Kirillov model. This result appears on p. 54:
I ...
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Prime to $p$ monodromy of local system on rigid variety
Suppose $F$ is a finite extension of $\mathbb Q_p$, and $X$ is a rigid variety over $F$. I saw in proposition 3.7 of Oswal, Shankar, Zhu, and Patel - A $p$-adic analogue of Borel's theorem: "Let $...
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On the irreducible submodules of adjoint representations $\text{ad}^{0}$
Let $k$ be a finite field of characteristic $p$. Let $H$ be a subgroup of $\rm{GL}_{n}(k)$ of order prime to $p$ where $n\geq2$. Assume that the representation $H\hookrightarrow \rm{GL}_{n}(k)$ is ...
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Discrepancy between $\dim H^2(G, \mathrm{ad}(\bar \rho))$ and the number of relations in a minimal presentation of the universal deformation ring $R$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\ad{ad}\DeclareMathOperator\gen{gen}$Let $p$ be a prime and $G$ be a profinite group such that the pro-$p$ completion of every open subgroup is ...
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Subgroups of ${\rm GL}_{2}(\mathbb{Z}/2^{n} \mathbb{Z})$
Assume that $n \geq 3$. Is there a subgroup $H \leq {\rm GL}_{2}(\mathbb{Z}/2^{n} \mathbb{Z})$ whose order is a power of $2$ so that
$\bullet$ $\det : H \to (\mathbb{Z}/2^{n} \mathbb{Z})^{\times}$ is ...
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Prime splitting in the division field of an elliptic curve
Let $E/\mathbb{Q}$ be an elliptic curve with good reduction at two distinct primes $p, \ell$. Suppose the mod $\ell$ Galois representation associated to $E$ is surjective. Let $K=\mathbb{Q}(E[\ell])$ ...
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Understanding absolute Galois group from its representations
Background. A major theme of modern number theory is to study the absolute Galois group $\text{Gal}(\overline{\mathbb Q} / \mathbb Q)$. Galois representation theory attempts to understand $\text{Gal}(\...
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