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Motivation: Let $\ell$ be an odd prime. There is a conductor-preserving correspondence between primitive Dirichlet characters of order $\ell$ and cyclic, degree $\ell$ number fields $K/\mathbb{Q}$. T …

Setup. Let $k$ be an algebraically closed field of characteristic zero. Let $X/k$ be a normal variety, and let $Y/k$ be a proper variety. It is well-known that the indeterminacy locus of a rational ma …

Generic polynomials, which are recalled below, play an important role in the constructive aspects of the inverse Galois problem. Definition. Let $P(\mathbf{t},X)$ be a monic polynomial in $\mathbb{Q …

For rigid spaces, you can find a reference in Theorem 5.2.1, part 1 of Conrad's Irreducible components of rigid spaces. The statement for Berkovich spaces can be found in Proposition 3.4.6 of Berkovic …

Setup. Let $k$ be an algebraically closed field of characteristic zero, and let $G/k$ be a semi-abelian variety i.e., $G$ is a commutative algebraic group which is an extension of an abelian variety $ …

Since this answer might be of use to other people in the future, I wanted to add an answer to this question. Brian Conrad provided me with a proof of this fact, which is written up Section 3 of arXiv …

Let $(X,d)$ be a compact, connected, locally connected, locally compact metric space. A result of Bing and Moise (independently) states that $(X,d)$ admits a topology preserving convex metric i.e., …