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

We show that there is a bound depending only on g and [K:Q] for the number of K-rational points on a hyperelliptic curve C of genus g over a number field K such that the Mordell-Weil rank r of its Jacobian is at most g-3. If K = Q, an explicit bound is 8 r g + 33 (g - 1) + 1. The proof is based on Chabauty's method; the new ingredient is an estimate for the number of zeros of a logarithm in a p-adic `annulus' on the curve, which generalizes the standard bound on disks. The key observation is that for a p-adic field k, the set of k-points on C can be covered by a collection of disks and annuli whose number is bounded in terms of g (and k). We also show, strengthening a recent result by Poonen and the author, that the lower density of hyperelliptic curves of odd degree over Q whose only rational point is the point at infinity tends to 1 uniformly over families defined by congruence conditions, as the genus g tends to infinity.Comment: 32 pages. v6: Some restructuring of the part of the argument relating to annuli in hyperelliptic curves (some section numbers have changed), various other improvements throughou

The mixed Ax-Lindemann theorem and its applications to the Zilber-Pink conjecture

The Zilber-Pink conjecture is a common generalization of the Andre-Oort and the Mordell-Lang conjectures. In this dissertation, we study its sub-conjectures: Andre-Oort, which predicts that a subvariety of a mixed Shimura variety having dense intersection with the set of special points is special; and Andre-Pink-Zannier which predicts that a subvariety of a mixed Shimura variety having dense intersection with a generalized Hecke orbit is weakly special. One of the main results of this dissertation is to prove the Ax-Lindemann theorem, a generalization of the functional analogue of the classical Lindemann-Weierstrass theorem, in its most general form. Another main result is to prove the Andre-Oort conjecture for a large class of mixed Shimura varieties: unconditionally for any product of the Poincare bundles over A6 and under GRH for all mixed Shimura varieties of abelian type. As for Andre-Pink-Zannier, we prove several cases when the ambient mixed Shimura variety is the universal family of abelian varieties: for the generalized Hecke orbit of a special point; for any subvariety contained in an abelian scheme over a curve and the generalized Hecke orbit of a torsion point of a fiber; for curves and the generalized Hecke orbit of an algebraic point.UBL - phd migration 201