# 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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### Power series rings and the formal generic fibre

Let $S = K[[S_1,\ldots,S_n]]$ and consider $d$ elements
\begin{equation*}
f_1,\ldots,f_d \in S[[X_1,\ldots,X_d]]
\end{equation*}
and the prime ideal ${\frak P} \colon\!= (f_1,\ldots,f_d)$ generated ...

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### Power series ring $R[[X_1,\ldots,X_d]]$ over a domain $R$

Let $R$ be a domain and
\begin{align*}
T \,\colon= R[[X_1,\ldots,X_d]].
\end{align*}
Suppose that we have $d$ elements $f_1,\ldots,f_d \in T$ and let us consider an ideal $J$ of $T$ such that $(f_1,\...

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### Deforming Modular Symbols

This is probably a silly question, as I don't really know a whole lot about modular symbols over arbitrary rings.
How do modular symbols over a finite field square with Katz modular forms? If they ...

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### Étale fibration for $K[[X_1,…,X_n]]$

Let us consider a formal power series ring $A_n \colon= K[[X_1,\ldots,X_n]]$ with $0 \ll n < \infty$ and we shall consider a prime ideal ${\frak P}$ of $A_n$ such that $1 < {\mathrm{ht}}({\frak ...

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### Primes of the power series rings

Let $A_n \colon= K[[X_1,\ldots,X_n]]$ be a $n$-variable formal power series ring. By setting $X_n \mapsto 0$, we obtain a natural surjection
\begin{equation*}
\psi_{n,n-1} \colon A_n \...

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### Gauss lemma for a complete Noetherian domain

Suppose that $R$ is a Noetherian complete domain over a field $K$.
Suppose that a monic polynomial $f(X) \in R[X]$ (i.e., the highest degree $X^e$ in $f$ has the coefficient $1$), satisfies the ...

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### Hodge-Tate weights of cohomological cuspidal automorphic representation

Let $\Pi$ be an algebraic cuspidal automorphic representation for $GL_{n}/\mathbb{Q}$ cohomological with respect to a dominant integral weight $\mu \in X^{*}(T)$ ($T \subset GL_{n}$ being the standard ...

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### Nearby cycles and extension by zero

Let $f: X\to \text{Spec}(R)$ be a proper and smooth morphism, with $R$ a strictly henselian dvr. Call $s = \overline{s}$ the closed point and $\eta$ the geometric point of $\text{Spec}(R)$.
Call $i_s ...

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### On the exponent of a certain matrix $A$ in characteristic $p > 0$

Let $A$ be a square matrix in characteristic $p > 0$ with both column and row having length $(1 + p^0 + p + \cdots + p^i)$, where $i \geq 0$.
Suppose that further the $(m,n)$-component $a_{m,n}$ ...

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### Berthelot’s comparison theorem and functoriality

Let $A$ be a noetherian $p$-adically complete ring with an ideal $I$ equipped with a PD structure and such that $p$ is nilpotent on $A/I$.
Let $S = \text{Spec}(A)$, $S_0 = \text{Spec}(A/I)$, $Y\to S$ ...

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### Classification of mod p Galois Representations for l not equal to p

Let $l\neq p$ be primes and let $\text{G}_l:=\text{G}_{\mathbb{Q}_l}$. Let $k$ be a finite field of characteristic $p$ and $\bar{\rho}:\text{G}_l\rightarrow \text{GL}_2(k)$ a local Galois ...

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### Some basic questions on crystalline cohomology

Let $X_0$ be a smooth projective variety over $\mathbf{F}_q$ and ${X}$ its base change to an algebraic closure $k$ of $\mathbf{F}_q$.
Crystalline cohomology $H^*_{\rm cris}(X) := H^*((X/W(k))_{\rm ...

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### How to compute Galois representations from etale cohomology groups of a generalized flag variety?

Let $G$ be a connected reductive group over a number field $K$, $P$ be a parabolic subgroup of $G$ defined over $K$, $X=G/P$ be the generalized flag variety which is a smooth projective variety over $...

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### Condition on a Fontaine Laffaille module which prescribes the image of the associated Galois representation

The Setup:
Let $m\geq 1$ be an integer, $\mathbb{F}$ be a finite field of characteristic $p$ and $W(\mathbb{F})$ the ring of Witt-vectors with residue field $\mathbb{F}$ and $\sigma:W(\mathbb{F})\...

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### Functoriality for $\ell$-adic cohomology - a question

This should a be basic enough question, but I’m a little confused.
In proving that $H^*(X,\mathbf{Q}_{\ell})$ is functorial (in the sense of Weil cohomology theories: see axiom D2 here) as $X$ ranges ...