Courbes sur une variété abélienne et points de torsion. (French) Zbl 0564.14020

Raynaud, M.

Invent. Math. 71, 207-233 (1983).

Let A be an abelian variety over \({\mathbb{C}}\) and let \(X\subset A\) be an algebraic curve; denote by T the sub-group of torsion points of A(\({\mathbb{C}})\); if X is not an elliptic curve (hence \(genus(X)\geq 2\) in this case), then \(T\cap X({\mathbb{C}})\) is a finite set. This is the main theorem proved in this paper. Note that this was called ”the Manin- Mumford conjecture”. It is interesting what Manin wrote (in 1966) when talking about such problems: ”As Lang remarked, Mordell’s conjecture is equivalent to the assertion that a curve in an abelian variety has only a finite number of points in common with any subgroup of finite type of variety. This formulation contains no explicit references to the ground field. Nevertheless, it seems probable that the proof of Mordell’s conjecture cannot be achieved without using the arithmetic (in the wide sense of the word) of the ground field. If that is so, then our result can be regarded as a reduction of the general problem to the case when the ground field is an algebraic number field: a case which has so far remained inviolate to known methods” [cf. Yu. I. Manin, Am. Math. Soc., Translat., II. Ser. 50, 189-234 (1966), translation from Izv. Akad. Nauk SSSR, Ser. Mat. 27, 1395-1440 (1963; Zbl 0166.169)]. Note that Bogomolov proved that the n-primary part \(T(n^{\infty})\cap X({\mathbb{C}})\) is finite for every integer \(n>1\) (F. A. Bogomolov, C. R. Acad. Sci. Paris, Sér. A 290, 701-703 (1980; Zbl 0457.14020)]. Note that together with the proof of the Mordell conjecture given by Faltings [G. Faltings, ”Endlichkeitssätze für abelsche Varietäten über Zahlkörpern”, Invent. Math. 73, 349-366 (1983); Erratum, Invent. Math. 75, 381 (1984)], this result by the author proves a conjecture by S. Lang [cf. S. Lang, Ann. Mat. Pura Appl., IV. Ser. 70, 229-234 (1965; Zbl 0151.274)].
For the proof the author first remarks that it suffices to prove the theorem in case \(X\subset A\) is defined over a number field (under specialization \(X\cap T\) ”cannot become smaller”). Note that an analogous statement over the algebraic closure of a finite field is not correct. Thus it is understandable that in the proof of the theorem a careful study is made what happens under reduction modulo \(p^ 2\), where p is a carefully chosen prime number. This study of the torsion points in the local case is the main (and technically difficult) point of the proof. - We like to stress that the problem studied can be formulated purely over \({\mathbb{C}}\), but at the moment we have no proof which does not use arithmetic properties. Does there exist a proof using analytic tools? - For a related theorem, see VI.4 in the book edited by G. Faltings and G. Wüstholz, ”Rational points” (Seminar Bonn/Wuppertal, 1983/84).
This is a nice paper, the methods of proof are ingeneous and convincing.

Reviewer: F.Oort

MSC:

14K15	Arithmetic ground fields for abelian varieties
14G05	Rational points
14G25	Global ground fields in algebraic geometry
14H45	Special algebraic curves and curves of low genus

Keywords:

algebraic curve on abelian variety; Mordell conjecture; rational points on algebraic curves; Manin-Mumford conjecture; torsion points

Citations:

Zbl 0178.551; Zbl 0166.169; Zbl 0457.14020; Zbl 0151.274

References:

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