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Torsion of abelian varieties over GL(2)-extensions of number fields. (English) Zbl 0661.14032

Let K be a number field and A an abelian variety over K (which we assume for the moment to be absolutely simple). Suppose that L/K is a Galois extension such that Gal(L/K) is an \(\ell\)-adic Lie group with Lie algebra \({\mathfrak g}\). Consider the following question: Fixing the (isomorphism class of the) Lie algebra \({\mathfrak g}\), which abelian varieties A can possible have an infinite number of torsion points rational over the field L?
For example, if \({\mathfrak g}\) is \({\mathbb{Q}}_{\ell}\) then from work of Serre [see also K. Wingberg, Math. Ann. 279, 9-24 (1987; Zbl 0657.14024)] one knows that A must be of CM-type, and indeed the same conclusion holds if A has an infinite number of torsion points rational over the maximal abelian extension field of K.
In the present paper, the author treats the “next” case, which is of interest not only because of the role it plays in the above general question but also because of its relationship to classical modular forms. He considers the case where \({\mathfrak g}\) is the Lie algebra either of \(GL_ 2({\mathbb{Z}}_ p)\) or of the units in a quaternion algebra over \({\mathbb{Z}}_ p\). He proves that, in these cases, if A has an infinite number of torsion points rational over L, then End(A)\(\otimes {\mathbb{Q}}\) contains a number field of degree equal to the dimension of A. In fact he assumes something a bit weaker than absolute irreducibility of A to obtain this conclusion: he needs only that A is irreducible over the field L.
Reviewer: Yu.Zarkhin

MSC:

14K05 Algebraic theory of abelian varieties

Citations:

Zbl 0657.14024
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References:

[1] Bogomolov, F.A.: Sur l’algébraicité des représentationsl-adiques. C.R. Acad. Sci. Paris290, 701-704 (1980)
[2] Coleman, R.F.: Ramified torsion points on curves. Duke Math. J.54, 615-640 (1987) · Zbl 0626.14022 · doi:10.1215/S0012-7094-87-05425-1
[3] Faltings, G.: Endlichkeitssätze für Abelsche Varietäten über Zahlkörpern. Invent. Math.73, 349-366 (1983) · Zbl 0588.14026 · doi:10.1007/BF01388432
[4] Imai, H.: A remark on the rational points of abelian varieties with values in cyclotomic ? p -extensions. Proc. Japan Acad.51, 12-16 (1975) · Zbl 0323.14010 · doi:10.3792/pja/1195518722
[5] Kurchanov, P.: Elliptic curves of infinite rank over non-cyclotomic ?-extensions. Math. USSR Sb.19, 320-324 (1973) · Zbl 0273.14009 · doi:10.1070/SM1973v019n02ABEH001762
[6] Manin, Yu.I.: Cyclotomic fields and modular curves. Russ. Math. Surv.26, 7-78 (1971) · Zbl 0266.14012 · doi:10.1070/RM1971v026n06ABEH001272
[7] Mazur, B.:Rational points of abelian varieties with values in towers of number fields. Invent. Math.18, 183-266 (1972) · Zbl 0245.14015 · doi:10.1007/BF01389815
[8] Mumford, D.: Abelian varieties. Oxford: Oxford University Press 1974 · Zbl 0326.14012
[9] Ribet, K.A.: Torsion points of abelian varieties in cyclomotic extensions, pp. 315-319. Appendix to N.M. Katz, S. Lang: Finiteness theorems in geometric classfield theory. Enseign. Math.27, 285-319 (1981)
[10] Ribet, K.A.: Galois actions on division points of abelian varieties with many real multiplications. Am. J. Math.98, 751-804 (1976) · Zbl 0348.14022 · doi:10.2307/2373815
[11] Serre, J.-P.:Abelianl-adic representations and elliptic curves. New York Amsterdam: Benjamin 1968
[12] Serre, J.-P.: Propriétés galoisiennes des points d’ordre finis des courbes elliptiques. Invent. Math.15 259-331 (1972) · Zbl 0235.14012 · doi:10.1007/BF01405086
[13] Serre, J.-P.: Représentations linéaires des groupes finis. Hermann: Paris 1978
[14] Serre, J.-P.: Résumé de cours de 1984-1985. Ann. Collége de France, Paris 1985
[15] Serre, J.-P.: Résumé de cours de 1985-1986, Ann. Collége de France, Paris 1986
[16] Serre, J.-P.: Représentationsl-adiques. Kyoto Int. Symposium on Algebraic Number Theory, Japan Soc for the Promotion of Science, 177-193 (1977)
[17] Shimura, G.: Introduction to the arithmetic theory of automorphic functions. Publ. Math. Soc. Japan11 (1971) · Zbl 0221.10029
[18] Zarhin, Yu.G.: Endomorphisms and torsion of Abelian varieties. Duke Math. J.54, 131-145 (1987) · Zbl 0632.14035 · doi:10.1215/S0012-7094-87-05410-X
[19] Zarhin, Yu.G.: A finiteness theorem for unpolarized Abelian varieties over number fields with prescribed places of bad reduction. Invent. Math.79, 309-321 (1985) · Zbl 0557.14024 · doi:10.1007/BF01388976
[20] Zarhin, Yu.G.: Abelian varieties,l-adic representations and SL2. Math. USSR Izv14, 275-288 (1980) · Zbl 0451.14015 · doi:10.1070/IM1980v014n02ABEH001105
[21] Wingberg, K.: On the rational points of Abelian varieties over ? p -extensions of number fields. Math. Ann.279, 9-24 (1987) · Zbl 0657.14024 · doi:10.1007/BF01456190
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