2 problems

Positive-density conjecture. The set of primes $p$ of good reduction for $C$ such that $\bar{\omega}_p$ does not vanish on $C(\mathbb F_p)$ has positive density, unless $J$ splits…

Let $X$) be a smooth projective curve of genus $g\geq2$ whose Jacobian has Mordell–Weil rank at most $g-2$. Let $U_1^p$ denote the abelianisation of the fundamental group of…