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.
