Mathematics – Algebraic Geometry
Scientific paper
1995-05-21
Mathematics
Algebraic Geometry
7 pages. AMSLaTeX, dvi file available at http://math.bu.edu/INDIVIDUAL/abrmovic/conjh.dvi
Scientific paper
The purpose of this note is to wish a happy birthday to Professor Lucia Caporaso.* We prove that Conjecture H of Caporaso et. al. ([CHarM], sec. 6) together with Lang's conjecture implies the uniformity of rational points on varieties of general type, as predicted in [CHarM]; a few applications in arithmetic and geometry are stated. Let X be a variety of general type defined over a number field K. It was conjectured by S. Lang that the set of rational points X(K) is not Zariski dense in X. In the paper [CHarM] of L. Caporaso, J. Harris and B. Mazur it is shown that the above conjecture of Lang implies the existence of a uniform bound on the number of K-rational points of all curves of fixed genus g over K. The paper [CHarM] has immediately created a chasm among arithmetic geometers. This chasm, which often runs right in the middle of the personalities involved, divides between loyal believers of Lang's conjecture, who marvel in this powerful implication, and the disbelievers, who try (so far in vain) to use this implication to derive counterexamples to the conjecture. In this paper we will attempt to deepen this chasm, using the techniques of [CHarM] and continuing [aleph], by proving more implications, some of which very strong, of various conjectures of Lang. Along the way we will often use a conjecture donned by Caporaso et. al. Conjecture H (see again [CHarM], sec. 6) about Higher dimensional varieties, which is regarded very plausible among experts of higher dimensional algebraic geometry. In particular, we will show
No associations
LandOfFree
Lang's conjectures, Conjecture H, and uniformity does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Lang's conjectures, Conjecture H, and uniformity, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Lang's conjectures, Conjecture H, and uniformity will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-288251