14 problems

Mustată–Srinivas conjecture. There are infinitely many primes $p$ such that

Let $\frak{a}$ be an ideal in $\mathbb{Z}[X_1,\ldots,X_n]$, contained in $(X_1,\ldots,X_n)$, and let $\frak{a}_p$ be the localization at $(X_1,\ldots,X_n)$ of its image in…

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 $R$ be a singularity in characteristic zero and let $I$ be an ideal. For reductions modulo primes $p$, write $R/p$ for the reduction of $R$ and $IR/p$ for the induced ideal. Th…

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…

Let $X_{\bC}$ be a normal affine algebraic variety over $\bC$, let $\Delta\geq 0$ be an $\R$-divisor, and let $H$ be a Cartier divisor. Choose a model…

Elliptic-prime conjecture. For any polynomial $f_{cmathbb{Q}}$ over $cmathbb{Q}$, there exist infinitely many primes $p$ such that

Let $X/{\mathbb C}$ be a smooth, proper variety, and consider primes of good reduction for $X$. A prime is an ordinary reduction prime if the reduction of $X$ is ordinary. Serre's…

Let $X/{\mathbb C}$ be a smooth, proper variety, and consider primes $p$ of good reduction for $X$. A prime is a Hodge-Witt reduction prime if the reduction of $X$ at $p$ is Hodge-…

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…

Log canonical versus dense $F$-pure type conjecture. The pair $(Y,\Delta)$ is log canonical if and only if $(Y,\Delta)$ has dense $F$-pure type.

Let $R$ be a normal ring essentially of finite type over a field of characteristic zero, let $X:=\operatorname{Spec} R$, and let $\Delta$ be an effective $\mathbb{Q}$-divisor on…