5 problems

Let $E_1$ and $E_2$ be non-CM elliptic curves over $\mathbb{Q}$. For $x\geq 1$, define … where $N_{E_i}$ denotes the conductor of $E_i$ for $i=1,2$. Coprimality conjecture. As…

Let $X$ be a smooth, irreducible complex algebraic variety and let $f\in\mathcal{O}_X(X)$ be nonzero. Choose a model $(X_A,f_A)$ over a finitely generated ${\mathbf Z}$-subalgebra…

In the variation setup, let $E$ be a numerically flat vector bundle on a smooth complex projective variety, with model $\mathcal E$ over $S$. Let $\Sigma^{\mathrm{net}}$ be the set…

Keep the mixed-characteristic model $\pi:\mathcal X\to S$ and the vector bundle $E$ from the variation setup, and assume that $E$ is numerically flat. A bundle is étale trivializab…

Let $X$ be a smooth complex projective variety, let $\mathcal O_X(1)$ be ample, and let $E$ be a slope-semistable locally free sheaf. Choose a finitely generated model…