As mentioned above, we define the Lang dimension of a variety X to be , and Lang's conjecture implies that if K is a number field, and if has positive Lang dimension, then is not Zariski - dense in X. In [H1], J. Harris proposed a complementary statement:
It is illuminating to consider the motivating case of an elliptic surface of positive rank.
Let be a pencil of cubics through 9 rational points in . By choosing the base points in general position we can guarantee that the pencil has 12 irreducible singular fibers which are nodal rational curves. The Mordell - Weil group of has rank 8. The relative dualizing sheaf where is a fiber. Let be a map of degree at least 3. Let be the pull-back of along f. Then , therefore and X has Kodaira dimension 1. The Iitaka fibration is simply . The elliptic surface X still has a Mordell - Weil group of rank 8 of sections. By applying these sections to rational points on we see that the set of rational points is dense in X.
It is not hard to modify this example to obtain a varying family of elliptic surfaces which has a dense collection of sections. Let B be a curve and let be a family of rational functions on which varies in moduli (such families exists as soon as the degree is at least 3). Let Y be the pullback of X to . Then is a family of elliptic surfaces, of variation , and relative kodaira dimension 1. By composing sections of E with g and arbitrary rational maps , we see that p has a dense collection of sections.
Harris's weak conjecture for elliptic surfaces is attributed to Manin. At least over , it is closely related to a conjecture of Mazur [Maz]. In case of surfaces over , it has been related to the conjecture of Birch and Swinnerton-Dyer: let be an elliptic surface defined over . In [Man], E. Manduchi shows that under certain assumptions on the behavior of the j function, the set of points in where the fiber has root number -1 is dense (in the classical topology). According to the conjecture of Birch and Swinnerton-Dyer, the root number gives the parity of the Mordell-Weil rank. It is likely that some of Manduchi's conditions (at least condition (1) in Theorem 1 of [Man]) can be relaxed once one passes to a number field.
What can be said in case ? In [H2], Harris proposed the following definition:
Harris proceeded to propose the following:
I do not know whether or not Harris himself believes this conjecture. This does not really matter. What is appealing in this conjecture, apart from it's ``tightness'', is that any evidence, either for or against it, is likely to be of much interest.
For lack of any better results, we just note that proposition 1.8 directly implies the following:
ACKNOWLEDGEMENTS I would like to thank D. Bertrand, J. Harris, J. Kollár, K. Matsuki, D. Rohrlich and J. F. Voloch for discussions related to this note.