Duke Mathematical Journal

Formal groups and the isogeny theorem

Philippe Graftieaux

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this paper, we prove an isogeny criterion for abelian varieties that involves conditions on the formal groups of the varieties (see Theorem 1.1). In the particular case of abelian varieties over ℚ with real multiplication, we easily deduce from our criterion a new proof of the Tate conjecture which is independent of G. Faltings's work [11], as well as a bound for the minimal degree of an isogeny between two isogenous abelian varieties, as in the paper of D. Masser and G. Wüstholz [17]. To this end, we use C. Deninger and E. Nart's result giving the link between the L-functions and the formal groups of such varieties (see [9]). Our method generalizes D. and G. Chudnovsky's transcendental proof of the isogeny theorem for elliptic curves over ℚ [6, Prop. 2.3] to the case of abelian varieties, with a systematic use of the Arakelov formalism of J.-B. Bost (see [1]).

Article information

Duke Math. J., Volume 106, Number 1 (2001), 81-121.

First available in Project Euclid: 13 August 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14K02: Isogeny
Secondary: 14L05: Formal groups, $p$-divisible groups [See also 55N22]


Graftieaux, Philippe. Formal groups and the isogeny theorem. Duke Math. J. 106 (2001), no. 1, 81--121. doi:10.1215/S0012-7094-01-10614-5. https://projecteuclid.org/euclid.dmj/1092403890

Export citation


  • \lccJ.-B. Bost, Périodes et isogénies des variétés abéliennes sur les corps de nombres (d'après D. Masser et G. Wüstholz), Astérisque 237 (1996), 4, 115–161, Séminaire Bourbaki 1994/95, exp. no. 795.
  • ––––, “Arakelov geometry of Abelian varieties” in Proceeding of a Conference on Arithmetic Geometry, technical report, Max-Planck-Institut für Math., Bonn, 1996.
  • ––––, Intrinsic heights of stable varieties and abelian varieties, Duke Math. J. 82 (1996), 21–70.
  • \lccJ.-B. Bost, H. Gillet, and C. Soulé, Heights of projective varieties and positive Green forms, J. Amer. Math. Soc. 7 (1994), 903–1027.
  • \lccL. Breen, Fonctions thêta et théorème du cube, Lecture Notes in Math. 980, Springer, New York, 1983.
  • \lccD. V. Chudnovsky and G. V. Chudnovsky, Padé approximations and Diophantine geometry, Proc. Nat. Acad. Sci. U.S.A. 82 (1985), 2212–2216.
  • \lccS. David, Fonctions thêta et points de torsion des variétés abéliennes, Compositio Math. 78 (1991), 121–160.
  • \lccP. Deligne, “Représentations $l$-adiques” in Séminaire sur les pinceaux arithmétiques; la conjecture de Mordell (Paris, 1983-84), Astérisque 127 (1985), 249–255.
  • \lccC. Deninger and E. Nart, Formal groups and $L$-series, Comment. Math. Helv. 65 (1990), 318–333.
  • \lccL. Denis, Lemmes de multiplicités et intersections, Comment. Math. Helv. 70 (1995), 235–247.
  • \lccG. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlk örpern, Invent. Math. 73 (1983), 349–366.
  • \lccP. Graftieaux, Paramètres algébriques de groupes formels et critères d'isogénie, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998); see also Thèse de Doctorat, Université Paris 6, 1998.
  • \lccT. Honda, On the theory of commutative formal groups, J. Math. Soc. Japan 22 (1970), 213–246.
  • \lccJ. I. Igusa, Theta Functions, Grundlehren Math. Wiss., Springer, New York, 1972.
  • \lccM. Laurent, “Une nouvelle démonstration du théorème d'isogénie, d'après D. V. et G. V. Chudnovsky” in Séminaire de Théorie des Nombres (Paris 1985–86), Progr. Math. 71, Birkhaüser, Boston, 1986, 119–131.
  • \lccD. W. Masser and G. Wüstholz, Endomorphism estimates for abelian varieties, Math. Z. 215 (1994), 641–653.
  • ––––, Factorization estimates for abelian varieties, Inst. Hautes Études Sci. Publ. Math. 81 (1995), 5–24.
  • \lccL. Moret-Bailly, Pinceaux de variétés abéliennes, Astérisque 129, Soc. Math. France, Montrouge, 1985.
  • \lccD. Mumford, On the equations defining abelian varieties, I, Invent. Math. 1 (1966), 287–354.
  • \lccR. Remak, Abschäzungen von Fundamentaleinheiten und Regulator, J. für Math. 165 (1931), 159–179.
  • \lccG. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Kâno Memorial Lectures 1, Iwanami Shoten, Tokyo; Publ. Math. Soc. Japan 11, Princeton Univ. Press, Princeton, N.J., 1971.
  • \lccA. Wiles, Modular elliptic curves and Fermat's last theorem, Ann. of Math. (2) 141 (1995), 443–551.
  • \lccS. Zhang, Positive line bundles on arithmetic varieties, J. Amer. Math. Soc., 8 (1995), 187–221.