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Periods arise from the comparison between Betti and de Rham cohomology for an algebraic variety. The Period Conjecture, due to Grothendieck, is a transcendence conjecture for periods which says that e …

Let's say $k=\mathbb{C}$ (although something like this should work over any algebraically closed field). Let $V_1=\mathbb{P}^1-\{0,1,\infty,\pi\}$ and $V_2 = \mathbb{P}^1-\{0,1,\infty,e\}$. One can se …

Here's an argument that for $A$ a finite set of algebraic numbers, $d_A(n)$ decays at most exponentially. Suppose $A$ is contained in some number field $K$. For $x\in K^\times$, there's a product for …

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 …

For a variety $X$ defined over $\mathbb{Q}$, there's a (functorial) comparison isomorphism $$ H^i_{dR}(X)\otimes\mathbb{C}\to H^i_B(X,\mathbb{Q})\otimes\mathbb{C}. $$ If we pick $\mathbb{Q}$-bases for …

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 …

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 …

Write $X=\mathrm{Spec}\, k$, which is a $0$-dimensional variety. Motives of $0$-dimensional varieties are called Artin motives, and they are pure. The Betti realization is the Betti cohomology of $X$, …

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 …

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 …

The affine group scheme $G$ you describe is not finite type. It is possible to describe $G$ explicitly. For $R$ a $k$-algebra, the $R$-points of $G$ are the tensor functorial automorphisms of $\mathr …

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 …

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 $I'\subset B\otimes_A B$ be the ideal generated by the elements $b_i\otimes 1-1\otimes b_i$, and define $$ R=\{b\in B:b\otimes 1-1\otimes b\in I'\}. $$ It’s not hard to check that $R$ is an $A$-su …

We should be able to construct such a real number $x$. Let $\epsilon\in(0,1/2)$ be fixed. If $S\subset \mathbb{R}_{>0}$ and $r>0$, we write $S^r$ for the set of positive $r$-th powers of elements of $ …