# Current status of uniform boundness of rational points on higher genus curves

We know Lang conjecture can imply uniform boundless of rational points on higher genus (smooth projective) curves over a fixed number field by works of Mazur and others.

How is the conjecture of uniform boundness now? Are there some new results?

• Michael Stoll proved this assuming the Zilber-Pink conjecture (ems-ph.org/journals/…), and Katz-Rabinoff-Zuerick-Brown proved this for all curves satisfying $r \leq g-3$, where $r$ is the Mordell-Weil rank of the curve and $g$ is the genus (projecteuclid.org/euclid.dmj/1476450482) – Stanley Yao Xiao Mar 23 at 4:45
• @StanleyYaoXiao What I actually prove assuming Zilber-Pink is that there is a uniform bound in terms of $g$ and $r$ for the number of (geometric) points on a curve of genus $g$ mapping into a subgroup of rank $r$ of (the geometric points of) its Jacobian. So if you assume Zilber-Pink and a uniform bound on the Mordell-Weil rank of Jacobians of curves of genus $g$, uniform boundedness of the number of rational points follows. – Michael Stoll Mar 23 at 13:12
• (continued:) What Katz, Rabinoff and Zureick-Brown do (extending results for hyperelliptic curves in my paper) is to prove an unconditional result like this for $p$-adic points and assuming that $r \le g-3$. The bound depends on $g$ and the $p$-adic field considered. – Michael Stoll Mar 23 at 13:15
• – Michael Stoll Mar 23 at 13:21
• @MichaelStoll Thank you a lot ! – zzy Mar 24 at 0:00