Uniform bounds for the number of rational points on curves of small Mordell-Weil rank

Michael Stoll, Bayreuth University
Fine Hall 214

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 Q have only one rational point, where "uniform" refers to families of curves defined by congruence conditions.