Density of integral points on algebraic varieties

Let K be a number field, S a finite set of valuations of K, including the archimedean valuations, and OS the ring of S-integers. LetX be an algebraic variety defined over K and D a divisor on X. We will use X and D to denote models over Spec(OS). We will say that integral points on (X,D) (see Section 2 for a precise definition) are potentially dense if they are Zariski dense on some model (X ,D), after a finite extension of the ground field and after enlarging S. A central problem in arithmetic geometry is to find conditions insuring potential density (or nondensity) of integral points. This question motivates many interesting and concrete problems in classical number theory, transcendence theory and algebraic geometry, some of which will be presented below. If we think about general reasons for the density of points the first idea would be to look for the presence of a large automorphism group. There are many beautiful examples both for rational and integral points, like K3 surfaces given by a bihomogeneous (2, 2, 2) form in P×P×P or the classical Markov equation x+ y+ z = 3xyz. However, large automorphism groups are “sporadic” they are hard to find and usually, they are not well behaved in families. There is one notable exception namely automorphisms of algebraic groups, like tori and abelian varieties. Thus it is not a surprise that the main geometric reason for the abundance of rational points on varieties treated in the recent papers [11], [3], [12] is the presence of elliptic or, more generally, abelian fibrations with multisections having a dense set of rational points and subject to some nondegeneracy conditions. Most of the effort goes into ensuring these conditions.