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$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Gal{Gal}\newcommand{\Z}{\mathbb{Z}}$In Grothendieck's Esquisse he claims that the action of $$\Gal(\mathbb Q)\to\text{Out}(\pi_1(M_{1,1})=\text{Out}(\...

Let $X$ be a smooth projective variety over $\mathbf{F}_q$ and $\overline{X}$ its base change to $\overline{\mathbf{F}_q}$. By Deligne’s Weil I, the eigenvalues of the geometric Frobenius acting on $...

In the paper "The conjectural connections between automorphic representations and Galois representations" by Buzzard and Gee, it is said "We say that $\rho$ is crystalline/de Rham/Hodge–Tate if ...

A field is called a $ C_{k}$- field if every form i.e. every homogeneous polynomial of degree $ d $ in $ n > d^{k} $ indeterminates has a nontrivial zero. We know that any finite field is a $ C_{...

Let ${\Bbb Q}(\zeta_{\infty})$ be the field obtained by adjoining all roots of unity. We define Cl(${\Bbb Q}(\zeta_{\infty})$)$\colon= \underset{m > 1}{\varinjlim}~{\mathrm{Cl}}({\Bbb Z}[\zeta_m])...