Abstract
A classical theorem by K. Ribet asserts that an abelian variety defined over the maximal cyclotomic extension K of a number field has only finitely many torsion points. We show that this statement can be viewed as a particular case of a much more general one, namely that the absolute Galois group of K acts with finitely many fixed points on the étale cohomology with
Funding statement: The second author was partially supported by NKFI grant No. K112735.
Acknowledgements
Many thanks to Anna Cadoret, Jean-Louis Colliot-Thélène, Wayne Raskind and Douglas Ulmer. We are also very grateful to the referee for his insightful comments which in particular enabled us to remove an unnecessary restriction in the statement of Theorem 1.9.
References
[1] P. Berthelot, Altérations des variétés algébriques (d’après A. J. de Jong), Séminaire Bourbaki, Vol. 1995/96, Exposés 805–819, Astérisque 241, Société Mathématique de France, Paris (1992), Exposé 815, 273–311. Search in Google Scholar
[2] A. Cadoret and A. Tamagawa, Core representations and families of abelian varieties with a common isogeny factor, to appear. Search in Google Scholar
[3] X. Caruso, Conjecture de l’inertie modérée de Serre, Invent. Math. 171 (2008), 629–699. 10.1007/s00222-007-0091-9Search in Google Scholar
[4]
J.-L. Colliot-Thélène,
Hilbert’s Theorem 90 for
[5] J.-L. Colliot-Thélène, Cycles algébriques de torsion et K-théorie algébrique, Arithmetic algebraic geometry (Trento 1991), Lecture Notes in Math. 1553, Springer, Berlin (1993), 1–49. Search in Google Scholar
[6]
J.-L. Colliot-Thélène and W. Raskind,
[7] J.-L. Colliot-Thélène and W. Raskind, Groupe de Chow de codimension deux des variétés définies sur un corps de nombres: Un théorème de finitude pour la torsion, Invent. Math. 105 (1991), 221–245. 10.1007/BF01232266Search in Google Scholar
[8] J.-L. Colliot-Thélène, J.-J. Sansuc and C. Soulé, Torsion dans le groupe de Chow de codimension deux, Duke Math. J. 50 (1983), 763–801. 10.1215/S0012-7094-83-05038-XSearch in Google Scholar
[9] P. Deligne and N. M. Katz, Groupes de monodromie en géométrie algébrique (SGA7 II), Lecture Notes in Math. 340, Springer, Berlin 1973. 10.1007/BFb0060505Search in Google Scholar
[10] P. Deligne and N. M. Katz, Preuve des conjectures de Tate et de Shafarevitch, Séminaire Bourbaki, Vol. 1983/1984, Exposé 616, Astérisque 121–122, Société Mathématique de France, Paris (1985), 25–41. Search in Google Scholar
[11] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), 349–366. 10.1007/BF01388432Search in Google Scholar
[12] G. Faltings, p-adic Hodge theory, J. Amer. Math. Soc. 1 (1988), 255–299. Search in Google Scholar
[13] J.-M. Fontaine and G. Laffaille, Construction de représentations p-adiques, Ann. Sci. Éc. Norm. Supér. (4) 15 (1982), 547–608. 10.24033/asens.1437Search in Google Scholar
[14] J.-M. Fontaine and W. Messing, p-adic periods and p-adic étale cohomology, Contemp. Math. 67 (1987), 179–207. 10.1090/conm/067/902593Search in Google Scholar
[15]
O. Gabber,
Sur la torsion dans la cohomologie
[16] A. Grothendieck, Géométrie formelle et géométrie algébrique, Séminaire Bourbaki 5 (1958–1960), 193–220. 10.1007/BF02684778Search in Google Scholar
[17] A. Grothendieck, Le groupe de Brauer II: Théories cohomologiques, Séminaire Bourbaki 9 (1964–1966), 287–307. Search in Google Scholar
[18] R. Hartshorne, Algebraic geometry, Grad. Texts in Math. 52, Springer, New York 1977. 10.1007/978-1-4757-3849-0Search in Google Scholar
[19] L. Illusie, Grothendieck’s existence theorem in formal geometry, Fundamental algebraic geometry: Grothendieck’s FGA explained, Math. Surveys Monogr. 123, American Mathematical Society, Providence (2005), 181–233. 10.1090/surv/123/08Search in Google Scholar
[20]
H. Imai,
A remark on the rational points of abelian varieties with values in cyclotomic
[21] B. Kahn, Algebraic K-theory, algebraic cycles and arithmetic geometry, Handbook of K-theory. Vol. 1, Springer, Berlin (2004), 351–428. 10.1007/978-3-540-27855-9_9Search in Google Scholar
[22] S. Lang, Fundamentals of Diophantine geometry, Springer, New York 1983. 10.1007/978-1-4757-1810-2Search in Google Scholar
[23] B. Mazur, Rational points of abelian varieties with values in towers of number fields, Invent. Math. 18 (1972), 183–266. 10.1007/BF01389815Search in Google Scholar
[24] J. S. Milne, Etale cohomology, Princeton University Press, Princeton 1980. 10.1515/9781400883981Search in Google Scholar
[25] M. B. Nathanson, Generalized additive bases, König’s lemma, and the Erdős–Turán conjecture, J. Number Theory 106 (2004), 70–78. 10.1016/j.jnt.2003.12.012Search in Google Scholar
[26] K. Ribet, Torsion points on abelian varieties in cyclotomic extensions. Appendix to: N. M. Katz and S. Lang, Finiteness theorems in geometric class field theory, Enseign. Math. 27 (1981), 285–319. Search in Google Scholar
[27] M. Rosen and S. Wong, The rank of abelian varieties over infinite Galois extensions, J. Number Theory 92 (2002), 182–196. 10.1006/jnth.2001.2692Search in Google Scholar
[28]
P. Salberger,
Torsion cycles of codimension 2 and
[29] J.-P. Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259–331. 10.1007/978-3-642-39816-2_94Search in Google Scholar
[30] J.-P. Serre, Cohomologie galoisienne, 5th ed., Springer, Berlin 1994. 10.1007/BFb0108758Search in Google Scholar
[31]
J.-P. Serre,
Abelian
[32] J.-P. Serre and J. Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), 492–517. 10.2307/1970722Search in Google Scholar
[33]
A. A. Suslin,
Torsion in
[34]
J. Tate,
Relations between
[35] Y. G. Zarhin, Weights of simple Lie algebras in the cohomology of algebraic varieties, Math. USSR Izv. 24 (1985), 245–281. 10.1070/IM1985v024n02ABEH001230Search in Google Scholar
[36] Y. G. Zarhin, Endomorphisms and torsion of abelian varieties, Duke Math. J. 54 (1987), 131–145. 10.1215/S0012-7094-87-05410-XSearch in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston
