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The motivation for this question is to understand a recent theorem of Francis Brown which implies that all periods of mixed Tate motives over $\mathbb{Z}$ lie in $\mathcal{Z}[\frac{1}{2\pi i}]$, where …

Choose a smooth projective $X/R$ of positive dimension, and pick a set-theoretic splitting $\varphi:X(\kappa(s))\to X(R)$ of the reduction map $\pi:X(R)\to X(\kappa(s))$. Take $a=\varphi\circ \pi$, an …

If $k$ is a number field, each embedding $\sigma:k\hookrightarrow\mathbb{C}$ determines a Weil cohomology theory $H^*_{B,\sigma}$ on smooth projective $k$-varieties given by taking the topological coh …

It is my understanding that in dimension 0, the theory of motives should just be Galois theory for fields. I am hoping to find a reference or two to help me get some things straightened out. One can …

Elements of $\operatorname{Lie}(\operatorname{Aut}(X))$ are not $k$-algebra maps, but rather maps over the ring $k[\epsilon]/(\epsilon^2)$ that reduce to the identity $k$-algebra map under $\epsilon\m …

Let $X$ be a smooth projective variety over $\mathbb{Q}$. The theory of motives predicts that for each cohomology theory, there should be a distinguished Zariski closed subgroup of $GL(H^k_{\bullet}(X …

Tate twists do indeed shift the Hodge filtration. Since the de Rham realization of the Tate motive $\mathbb{Q}(1)$ is concentrated in the degree $-1$ part of the Hodge filtration, we have $F^n M(m)_{d …

The matrix entry $*$ is $\log(u^{1-p})/p$ (where $\log$ is the $p$-adic logarithm). This can be obtained from the description of the inverse of Frobenius given in sections 2.9-2.10 of [1]. You can fin …

Here is a counterexample. Fix a field $k$, and let $Y$ be built from two copies of the affine nodal curve $y^2=x^3+x^2$, glued together on the complement of the singular point. In other words $Y$ is a …

A counterexample is $X=\mathbb{A}^2\backslash\{y=0\}$, $A=\mathbb{C}[x,y,y^{-1}]$, and $\mathfrak{g}$ the span of the derivation $D(x)=1$, $D(y)=y$. Now $\mathfrak{g}$ is $1$-dimensional and $X$ is $2 …

Since $K$ is generated by $p$ elements, $\mathbb{C}(x_0,\ldots,x_{q-1})$ cannot be an algebraic extension of $K$ if $q>p$. I claim that $\mathbb{C}(x_0,\ldots,x_{q-1})$ is a finite Galois extension of …

The answer to part 1 is yes. Given $K/\mathbb{Q}_p$, let $\alpha\in K$ be a primitive element, with minimal monic polynomial $f(x)=x^n+\sum_{i=1}^n a_i x^{n-i}$, $a_i\in\mathbb{Q}_p$. So we have $K\co …

Suppose $X$ is a smooth variety defined over $\mathbb{Q}$. There are (at least) two automorphisms of cohomology groups of $X$ that are called "Frobenius", and I would like to understand how they are r …

The answer to the both questions is no. Consider $X=\mathbb{A}^1$, $k=\mathbb{Q}$, and $K=\mathbb{C}$ (any transcendental extension will do here). Let $\eta=\{\pi\}$. The cycles on $X_K$ of the form $ …

Let $k$ be a field, and suppose $G$ is a group-scheme over $k$ (I am happy to assume that $k=\mathbb{Q}$ and that $G$ is affine). A $G$-torsor over $k$ is a non-empty $k$-scheme $T$ equipped with an a …