Multiplicative relations on curves and other varieties
(Un exposé de David Masser au séminaire de
théorie des nombres de Chevaleret le 14 mars 2005)
Résumé :
Consider an algebraic curve on which none of the affine coordinates
$x_1,...,x_n$ vanishes identically. The intersection with a fixed
multiplicative relation $x_1^{a_1}...x_n^{a_n}=1$ is usually a finite set.
If the exponents are allowed to vary, then we can easily get sets as large
as we like, but it was observed by Bombieri, Zannier, and the speaker that
these are severely restricted. For example when the curve is defined over
the field of algebraic numbers, then the intersection points usually have
absolute Weil height which is uniformly bounded above. Further the points on
the curve for which there are two independent relations usually form a
finite set. We describe some recent progress on such problems, mostly for
varieties other than curves. In particular the natural analogues of the
above statements hold for a plane; but only after removing some ``anomalous
curves" from the plane.