Graftieaux, Philippe Formal groups and the isogeny theorem. (English) Zbl 1064.14045 Duke Math. J. 106, No. 1, 81-121 (2001). Summary: 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 \(\mathbb{Q}\) with real multiplication, we easily deduce from our criterion a new proof of the Tate conjecture which is independent of G. Faltings’s work [Invent. Math. 73, 349–366 (1983; Zbl 0588.14026)] , as well as a bound for the minimal degree of an isogeny between two isogenous abelian varieties, as in the paper of D. W. Masser and G. Wüstholz [Bull. Lond. Math. Soc. 25, No. 3, 247–254 (1993; Zbl 0809.14026)]. To this end, we use C. Deninger’s and E. Nart’s result [Comment. Math. Helv. 65, No. 2, 318– 333 (1990; Zbl 0741.14026)] giving the link between the \(L\)-functions and the formal groups of such varieties. Our method generalizes D. V. Chudnovsky’s and G. V. Chudnovsky’s transcendental proof of the isogeny theorem for elliptic curves over \(\mathbb{Q}\) [Proc. Natl. Acad. Sci. USA 82, 2212–2216 (1985; Zbl 0577.14034)] to the case of abelian varieties, with a systematic use of the Arakelov formalism of J.-B. Bost [Astérisque 237, 115–161 (1996; Zbl 0936.11042)]. Cited in 1 ReviewCited in 10 Documents MSC: 14K02 Isogeny 14L05 Formal groups, \(p\)-divisible groups Citations:Zbl 0588.14026; Zbl 0936.11042; Zbl 0809.14026; Zbl 0741.14026; Zbl 0577.14034 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] J.-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. · Zbl 0936.11042 [2] ——–, “Arakelov geometry of Abelian varieties” in Proceeding of a Conference on Arithmetic Geometry , technical report, Max-Planck-Institut für Math., Bonn, 1996. 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