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Notation: Let $K/\mathbb{Q}$ be a quadratic number field; let $p\geq 3$ be a rational prime and let $\mathfrak{p}$ denote a prime lying above $p$; let $K_{\mathfrak{p}}$ denote the completion of $K$ w …

There is a new manuscript on the arXiv by Giulio Bresciani, A higher dimensional Hilbert irreducibility theorem, arXiv:2101.01090, which shows that assuming the weak Bombieri--Lang conjecture, there c …

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 …

The below Magma code determines the size of $J_C(\mathbb{F}_p)$ for various primes $p$, and finally compute the GCD of their orders, which gives you a bound on the size of $J_C(\mathbb{Q})$. For a dis …

Roughly speaking, the Kodaira dimension is an invariant of a variety that corresponds to curvature. One can show that curves of genus $\geq 2$ have Kodaira dimension 1 using Riemann-Roch. In Corollary …

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 $p > 2$ be a prime, let $K$ be the completion of the maximal unramified extension of $\mathbb{Q}_p$, and fix an algebraic closure $\overline{K}$ of $K$. Let $A/K$ be an abelian variety of d …