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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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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Finitely ramified global Galois representation is trivial on a subgroup of finite index in \mathcal{O}_K^*?
In Lawrence-Venkatesh, they state before lemma 2.8 that if a continuous $\eta:{\rm Gal}(\bar{K}/K)\to\mathbb{Q}_p^*$ is finitely ramified, where $K$ is a number field, then the induced homomorphism $\...
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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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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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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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