## Diophantine approximation on abelian varieties.(English)Zbl 0734.14007

The author proves two theorems regarding rational points on abelian varieties, both of which answer long-standing conjectures of S. Lang.
For the first theorem, let $$X$$ be a subvariety of an abelian variety $$A$$; assume that both varieties are defined over a number field $$k$$. If $$X\times_ k\bar k$$ contains no nontrivial translated abelian subvariety of $$A\times_ k\bar k$$, then the author shows that the set $$X(k)$$ of $$k$$-rational points on $$X$$ is finite. In particular, this holds in the situation where $$X$$ is a curve of genus $$>1$$ embedded in its Jacobian. Thus the first theorem generalizes Mordell’s conjecture [proved by the author, Invent. Math. 73, 349–366 (1983; Zbl 0588.14026)]. In more recent work (unpublished), the author has extended this proof to eliminate the restriction on translated abelian subvarietes of $$A$$ (the conclusion needs to be modified accordingly, though).
The second theorem in this paper shows that if $$E$$ is an ample divisor on an abelian variety $$A$$, then $$A\setminus E$$ has only finitely many integral points over any given number ring. As in the proof of Siegel’s theorem using Roth’s theorem, this proof reduces the problem to a statement in diophantine approximations.
Both proofs use Thue’s method, as later refined by Siegel, Dyson, and Roth. - In the immediately preceding paper by the reviewer [Ann. Math. (2) 133, 509–548 (1991; Zbl 0774.14019)], this method is extended to cover rational points. The author further refines and simplifies the method of that paper; in particular he replaces the use of the Gillet-Soulé Riemann-Roch theorem with a simpler argument using Siegel’s lemma. - For the proof of his second theorem, the author extends Arakelov theory by defining and proving some lemmas on the height of a subvariety in projective space. [The definition had already been made by P. Philippon, Publ. Math., Inst. Hautes Étud. Sci. 64, 5–52 (1986; Zbl 0615.10044); see also J.-B. Bost, H. Gillet and C. Soulé, C. R. Acad. Sci., Paris, Sér. I 312, No. 11, 845–848 (1991; Zbl 0756.14012).]

### MSC:

 14G05 Rational points 11J99 Diophantine approximation, transcendental number theory 14G40 Arithmetic varieties and schemes; Arakelov theory; heights

### Citations:

Zbl 0588.14026; Zbl 0615.10044; Zbl 0774.14019; Zbl 0756.14012
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