Joint IAS/Princeton University Number Theory Seminar

We show that there is a bound $N(d,g,r)$ for the number of $K$-rational points on hyperelliptic curves $C$ of genus $g$ when the degree of the number field $K$ is $d$ and the Mordell-Weil rank $r$ of the Jacobian of $C$ is at most $g - 3$. The proof uses an extension of the method of Chabauty-Coleman, based on $p$-adic integration on ($p$-adic) disks and annuli covering the $p$-adic points of the curve. We also deduce a uniform version of the result (due to Poonen and the speaker) that "most" hyperelliptic curves of odd degree over $\mathbb Q$ have only one rational point, where "uniform" refers to families of curves defined by congruence conditions.