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The classical book of Walker on algebraic curves described a method of computing the genus of curves with non-ordinary singularities by transforming such a curve into a birational one with only ordina …

I'm not sure if this is the right place for a question like this. In Diophantine approximation, on a complex variety $X$ there is a notion of a Weil function for a Cartier divisor $D$ on $X$ which is …

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 …

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 …

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 …

The book Algebraic groups and class fields by Serre explains a lot about the construction of the generalized Jacobian $J_\mathfrak{m}$ of a smooth projective curve $X$ with respect to the modulus $\ma …

Let $C$ be a smooth projective geometrically connected curve of genus 2 defined over a number field $k$. Here are some definitions: The index $I$ of a curve $C$ is the greatest common divisor of all e …

Let $X$ be a smooth proper geometrically connected curve over a number field $k$, and let $k(X)$ denote its field of rational functions, i.e., its function field. Then the (cohomological) Brauer group …

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 …

As the title suggests, does anyone have a reference for the proof of this fact? Actually, I can't remember where I've seen it before, or if I even remembered the statement correctly. Here are some con …

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 …

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 …

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 $k$ 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( …

This is a question for those familiar with the section conjecture, so I'll do away with the definition of a ramification map in this case. Here is the definition of a ramification map from an etale co …