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You can find answers to your questions in: Deligne, Pierre: Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques. Automorphic forms, representations and L …

Here I give details of the reduction in LSpice's comment. I write it as an answer rather than a string of comments in order to have an editable text. The reduction goes as follows. According to Will's …

Let $k$ be a $p$-adic field and $G$ be a connected non-abelian reductive algebraic group over $k$. I am asking for a proof of the following lemma: Lemma. Assuming that $p$ is "good" for $G$, there ex …

$\newcommand{\Kbar}{{\overline K}} \newcommand{\Q}{{\mathbb Q}} $I consider Example 5.7 of Sansuc's paper Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres of …

Let $T\subset {\rm GL}(N,{\mathbb Q})$ be an $n$-dimensional ${\mathbb Q}$-torus given by its Lie algebra $\mathfrak{t}\subset \mathfrak{gl}(N,{\mathbb Q})$. I want to compute, in some sense explicit …

$\newcommand{\Q}{{\mathbb Q}} $Let $f\in \Q[x]$ be a polynomial, and let $L/\Q$ be the finite Galois extension obtaining by adjoining to $\Q$ all roots of $f$. Magma knows how to compute $\Gamma:={\r …

There is quite a lot of work on computing Galois groups of splitting fields of polynomials over $\Bbb Q$. Magma is quite good at it. In this answer to Decomposition groups for the Galois module $\mu_8 …

In addition to the answer of Gro-Tsen: Let $G=\operatorname{SU}(3)$. Then $H^1({\mathbb R},G)$ classifies matrices of Hermitian forms in 3 variables with determinant 1. There are two equivalence class …

Let $K$ be a number field, and let $S=S_f\cup S_{\mathbb R}\cup S_{\mathbb C}$ be a finite set of places of $K$, where $S_f$ denotes the set of finite places in $S$, $S_{\mathbb R}$ denotes the set of …

Let $K$ be an arbitrary field, and let $K^s$ be a fixed separable closure of $K$. Let $F/K$ a be a finite Galois extension in $K^s$. Let $n>0$ be a natural number. Let $A$ be a central simple algebra …

1. Let $G$ be a finite group of order $n$. I need an estimate for the number $c$ of conjugacy classes of subgroups $D\subseteq G$. Note that any subgroup of $G$ contains $1_G$, and so the set of all s …

Let $F$ be a global field without real places (that is, a function field or a totally imaginary number field). Let $K/F$ be a cyclic extension of degree $n$. Let $S$ be a ${\rm Gal}(K/F)$-invariant fi …

Let $L_0$ be a global field without real places, that is, a global function field or a totally imaginary number field, and let $V_f(L_0)$ denote the set of finite (that is, non-archimedean) places of …

Let $G$ be a group and $M$ be a $G$-module, that is, an abelian group written additively on which $G$ acts: $$ (g,m)\mapsto g m.$$ We consider the group of coinvariants $$ M_G:=G/\langle g m -m\ |\ g\ …

Let $K$ be a number field and $S$ be a finite set of non-archimedean places of $K$. Let $n>1$ be a natural number. Question. Does there exist a normal extension $L/K$ of degree $n$ such that $L\otime …