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Fix an algebraically closed field $k$ and a smooth projective (geometrically) integral genus $\geq 2$ curve $C$ over $k$. Denote by $J_C$ the Jacobian variety of $C$. Chapter VI, Proposition 11 of Ser …

I have a feeling this question will be left hanging on StackExchange, so I've decided to just post it here, let me know if it's not suitable. First of all, I'm trying to show that given an immersion o …

Let $X$ be a smooth geometrically connected variety over a number field $k$ and let $X(\mathbb{A}_k)$ be the space of adèlic points of $X$, i.e., for some finite set $S$ containing the infinite places …

Let $X \rightarrow Y$ be an immersion of smooth geometrically connected varieties over a number field $k$. Let $G$ be a finite étale non-abelian group scheme. Are there known cases where we have an is …

In the literature, there are a number of results called Rosenlicht's lemma, but I am talking about the following one: Let $T$ be a torus over a, in my case, number field $k$. Denote by $\bar{k}[T]$ th …

Fix a number field $k$ and consider a smooth projective geometrically connected $k$-curve $C$, with a $k$-rational divisor class of degree one inducing the embedding into its Jacobian $J$. For any den …

Fix $k$ a number field, and let $C$ be a smooth geometrically integral affine curve over $k$. We can "associate" to $C$ a semi-abelian variety in the following way: One knows that $C$ is a finite numb …

I have some doubts about what an abelian covering is, and I'll try my best to articulate them. In Serre's Algebraic groups and class fields Chapter VI.2, he fixed a base field $k$ with algebraic closu …

In this MathOverflow post, the smooth projective curve $C$ was defined over $\mathbb{C}$ and we have an isomorphism of de Rham cohomology groups $$H^1(C, \mathbb{C}) \cong H^1(J_C, \mathbb{C}),$$ wher …

Let $k$ be a number field and $X$ a smooth geometrically connected variety over $k$. We denote by $H(k,X)$ the set of sections $G_k \rightarrow \pi_1(X)$, where $G_k$ is the absolute Galois group of $ …

Let $k$ be a number field. The section conjecture predicts that for a (smooth geometrically connected) hyperbolic curve over $k$, the profinite Kummer map $\kappa :X(k) \rightarrow \mathscr{J}_{\pi_1( …

Let $X$ be a smooth affine curve over a number field $k$, and let $C$ be its smooth compactification. Let $i:X(\mathbb{A}_k) \rightarrow C(\mathbb{A}_k)$ be the induced morphism of adelic points by th …

Are there known cases of a morphism of smooth geometrically connected curves $f: X \rightarrow Y$ over a number field $k$ (to be specific) that would give rise to a surjective restriction map $f^*:\ma …

Let $C$ be a smooth projective curve of genus $\geq 1$ over a number field $k$ with a $k$-rational divisor of degree $1$ inducing the embedding $C \hookrightarrow J$, where $J$ is the Jacobian variety …

I asked this question a few days ago on math.stackexchange with no success, and it doesn't seem like there'll be any. So I thought I'll repost it here. A recent big result proved by $\mathrm{\check{C} …