3 problems

Realization conjecture. All 54 reduction types of the genus $2$ classification are realised over $\mathbb{Q}$.

For a prime $p>2$, let $P_1(p)$ be the probability that a uniformly randomly chosen integer in the Hasse–Weil interval $[(\sqrt p-1)^4,(\sqrt p+1)^4]$ is prime, and let $P_2(p)$ be…

Let ${\mathcal C}_N$ be the family of genus-$2$ curves defined by integral binary sextics with coefficients bounded in absolute value by $N$. For a rational point $P$ on such a cur…