Diophantine approximation on abelian varieties.

*(English)*Zbl 0734.14007The 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 In more recent work (unpublished), the author has extended this proof to eliminate the restriction on translated abelian subvarietes of \(A\) (the conclusion needs to be modified accordingly,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).]

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).]

Reviewer: Paul Vojta (Berkeley)

##### MSC:

14G05 | Rational points |

11J99 | Diophantine approximation, transcendental number theory |

14G40 | Arithmetic varieties and schemes; Arakelov theory; heights |