Séminaire d'arithmétique à Lyon

A Chabauty-Coleman bound for surfaces in Abelian Varieties

by Jerson Caro (Pontificia Universidad Catolica de Chile)

M7.411 (ENS Lyon, UMPA)




Coleman's explicit version of Chabauty's p-adic approach to Mordell's conjecture gives a concrete bound for the number of rational points of a curve C of genus g >1 over the rationals, provided that the rank of its Jacobian is r< g. Namely, if p >2g is a prime of good reduction for C, the number of rational points of C is bounded by the number of mod-p points of C plus a contribution coming from the canonical sheaf of C. 

There have been several striking developments around this result, but at present such a precise bound has remained out of reach for higher dimensional varieties. In this talk we will outline the proof of an analogous result for surfaces of general type contained in abelian varieties of Mordell-Weil rank 0 or 1, under some assumptions on the reduction type at p. For this, we develop a method based on overdetermined w-integrality in positive characteristic. This is joint work with Hector Pasten.