Heights of algebraic points on subvarieties of abelian varieties. (English) Zbl 0897.11020

Bombieri, E.; Zannier, U.

Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 23, No. 4, 779-792 (1996).

Let \(A\) be an abelian variety defined over \(\overline{\mathbb{Q}}\). Let \(\widehat h\) be the Néron-Tate height associated to a given very ample symmetric line bundle \({\mathcal L}\) on \(A\) and \(d\) the associated semi-distance. Further, let \(X\) be a closed, geometrically irreducible subvariety of \(A\) defined over \(\overline{\mathbb{Q}}\) and \(X^0\) the set obtained by removing from \(X\) all translates of positive-dimensional abelian subvarieties of \(A\) which are contained in \(X\). Recently, L. Szpiro, E. Ullmo and S. Zhang [Invent. Math. 127, 337-347 (1997)] proved the conjecture proposed by Bogolomov that the set of algebraic points in \(X^0\) is discrete with respect to the semi-distance \(d(P,Q)\). In the present paper, the authors prove a more precise quantitative version of this result, but only for abelian varieties satisfying certain strong conditions. Modulo basic theory of abelian varieties, the proof of the authors is elementary.
More precisely, let \(A\) be defined over a number field \(K\). Assume that either (A1) \(A\) has complex multiplication or (A2) there exists an infinite set \(\Sigma\) of primes such that for \(p\in\Sigma\) and some positive integer \(r\), the isogeny \([p^r]\) on the reduction \(A/\wp\) of \(A\) modulo some prime ideal \(\wp\) on \(K\) dividing \(p\) is equal to the Frobenius map on \(A/\wp\). Let a projective embedding \(j: A\hookrightarrow {\mathbb{P}}^m\) be given, defined by some basis of global sections of the line bundle \({\mathcal L}\) chosen above, and suppose \(X\) has degree \(d\) with respect to \(j\). Then there exist effectively computable constants \(\gamma\) and \(N\), depending on \(j\), \(d\), \(K\), the field of complex multiplication if (A1) holds and the set \(\Sigma\) if (A2) holds, such that for every \(P\in A(\overline{\mathbb{Q}})\), there are at most \(N\) points \(Q\in A(\overline{\mathbb{Q}})\) with \(d(P,Q)\leq\gamma\). The hard core of the proof consists of elementary congruence arguments, but for these to work it is necessary to assume either (A1) or (A2). It should be mentioned that there are analogous results with analogous proofs for algebraic points on linear tori \({\mathbb{G}}_m^n\). The first to prove an analogue of Bogolomov’s conjecture for linear tori was S. Zhang. His results were refined later by several people, among them Bombieri and Zannier and then Schmidt [see the following review Zbl 0897.11021].

Reviewer: J.-H.Evertse (Leiden)

MSC:

11G35	Varieties over global fields
14G40	Arithmetic varieties and schemes; Arakelov theory; heights
14K15	Arithmetic ground fields for abelian varieties
14G05	Rational points
11H99	Geometry of numbers

Keywords:

heights; algebraic points; abelian varieties over a number field

Citations:

Zbl 0897.11021

References:

[1]	D. Abramovich , Subvarieties of semi-abelian varieties , Compositio Math. 90 ( 1994 ), 37 - 57 . Numdam | MR 1266493 | Zbl 0814.14041 · Zbl 0814.14041
[2]	D. Bertrand - P. Philippon , Sous groupes de groupes algébriques commutatifs , Illillois J. Math. 32 ( 1988 ), 263 - 280 . MR 945863 | Zbl 0618.14020 · Zbl 0618.14020
[3]	E. Bombieri - U. Zannier , Algebraic points on subvarieties of G nm , Internat Math. Res. Notices 7 ( 1995 ), 333 - 347 . MR 1350686 | Zbl 0848.11030 · Zbl 0848.11030 · doi:10.1155/S1073792895000250
[4]	J.F. Burnol , Weierstrass points on arithmetic surfaces , Inv. Math. 107 ( 1992 ), 421 - 432 . MR 1144430 | Zbl 0723.14019 · Zbl 0723.14019 · doi:10.1007/BF01231896
[5]	N. Elkies , The existence of infinitely many supersingular primes for every elliptic curve over Q , Inv. Math. 89 ( 1987 ), 561 - 567 . MR 903384 | Zbl 0631.14024 · Zbl 0631.14024 · doi:10.1007/BF01388985
[6]	W. Fulton , Introduction to Intersection Theory in Algebraic Geometry , Regional Conference Series in Mathematics, ISSN 0160-7642; n. 54 , A.M.S ., 1984 . MR 735435 | Zbl 0913.14001 · Zbl 0913.14001
[7]	P. Griffiths - J. Harris , Principles of Algebraic Geometry , Wiley , 1978 . MR 507725 | Zbl 0408.14001 · Zbl 0408.14001
[8]	M. Hindry , Autour d’une conjecture de S.Lang , Inv. Math. 94 ( 1988 ), 575 - 603 . MR 969244 | Zbl 0638.14026 · Zbl 0638.14026 · doi:10.1007/BF01394276
[9]	S. Lang , Complex Multiplication , Springer, Verlag , 1983 . MR 713612 | Zbl 0536.14029 · Zbl 0536.14029
[10]	M. Mcquillan , Division points on semi-abelian varieties , Inv. Math. 120 ( 1995 ), 143 - 159 . MR 1323985 | Zbl 0848.14022 · Zbl 0848.14022 · doi:10.1007/BF01241125
[11]	M. Raynaud , Sous variétés d’une variété abelienne et points de torsion , In: ” Arithmetic & Geometry ”, Coates & Helgason eds., Vol. 1 , Birkhauser 1983 . MR 717600 | Zbl 0581.14031 · Zbl 0581.14031
[12]	W.M. Schmidt , Heights of algebraic points lying on curves or hypersurfaces , to appear in Proc. Amer. Math. Soc . ( 1995 ). MR 1343724 | Zbl 0867.11046 · Zbl 0867.11046 · doi:10.1090/S0002-9939-96-03519-8
[13]	W.M. Schmidt , Heights ofpoints on subvarieties of Gn m , preprint 1995 . MR 1330740
[14]	L. Szpiro , Sur les proprietes numériques du dualisant relatif d’une surface arithmétique, The Grothendieck Festschrift Volume III, Progress in Mathematics 88 , Birkhäuser 1990 . MR 1106917 | Zbl 0759.14018 · Zbl 0759.14018
[15]	S. Zhang , Positive line bundles on arithmetic surfaces , Ann. of Math. 136 ( 1992 ), 569 - 587 . MR 1189866 | Zbl 0788.14017 · Zbl 0788.14017 · doi:10.2307/2946601
[16]	S. Zhang , Positive line bundles on arithmetic varieties , J. Amer. Math. Soc. 8 ( 1995 ), 187 - 221 . MR 1254133 | Zbl 0861.14018 · Zbl 0861.14018 · doi:10.2307/2152886
[17]	S. Zhang , Small points and adelic metrics , J. Algebraic Geom. 4 ( 1995 ), 281 - 300 . MR 1311351 | Zbl 0861.14019 · Zbl 0861.14019