Duke Mathematical Journal

Formal groups and the isogeny theorem

Philippe Graftieaux

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Abstract

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

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

Dates
First available in Project Euclid: 13 August 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1092403890

Digital Object Identifier
doi:10.1215/S0012-7094-01-10614-5

Mathematical Reviews number (MathSciNet)
MR1810367

Zentralblatt MATH identifier
1064.14045

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

Citation

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


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