Uniform bounds for the number of rational points on curves of small MordellWeil rank
Uniform bounds for the number of rational points on curves of small MordellWeil rank

Michael Stoll, Bayreuth University
Fine Hall 214
We show that there is a bound N(d,g,r) for the number of Krational points on hyperelliptic curves C of genus g when the degree of the number field K is d and the MordellWeil rank r of the Jacobian of C is at most g3. The proof uses an extension of the method of ChabautyColeman, based on padic integration on (padic) disks and annuli covering the padic 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.