6 problems

Let $L_0$ be a field and let $L_1$ be a nontrivial finitely generated extension of $L_0$. Suppose there exists a valuation $v$ on $L_1$ with residue field $L_0$. Weak resolution of…

Large-prime existence conjecture. For all sufficiently large primes $p$, there exist an elliptic curve $E/\mathbb{Q}$ and an imaginary quadratic field $K_0$ such that the triple…

Denef–Lipshitz conjecture. For every number field $L$, the extension

Let $D\subseteq \mathbb{C}(z)$ be a Diophantine set. For each integer $n\geq 0$, write $\mathbb{C}[z]_n=\mathbb{C}^{n+1}$ for the polynomials of degree at most $n$. Kollár's conjec…

Let $S\subseteq\mathbb{N}$ be a simple set, meaning that $\mathbb{N}\setminus S$ is infinite and contains no infinite recursively enumerable subset. Let $U\subseteq\mathbb{Q}$ be t…

Let $\mathbb{Z}$ denote the ring of integers and $\mathbb{Q}$ the field of rational numbers. A subset of $\mathbb{Q}$ is Diophantine if it is defined by the existence of solutions…