×

Estimating isogenies on elliptic curves. (English) Zbl 0722.14027

Let d be a positive integer, and let k be a number field of degree at most d. For an elliptic curve E defined by the Weierstrass equation \(y^ 2=4x^ 3-g_ 2x-g_ 3\) with \(g_ 2,g_ 3\in k\), put \(w(E)=\max (1,h(g_ 2),h(g_ 3))\), where h denotes the absolute logarithmic Weil height. Under these notations, the authors show that if \(E'\) is another elliptic curve defined over k isogenous to E, then there exists an isogeny between E and \(E'\) whose degree is at most \(c\cdot w(E)^ 4\), where c is a constant depending effectively on d. To give this estimation, the authors use the transcendence techniques which were used by D. K. and G. V. Chudnovsky [Proc. Natl. Acad. Sci. USA 82, 2212-2216 (1985; Zbl 0577.14034)].

MSC:

14K02 Isogeny
14H52 Elliptic curves

Citations:

Zbl 0577.14034
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] [AM] Anderson, M., Masser, D.W.: Lower bounds for heights on elliptic curves. Math. Z.174, 23-34 (1980) · Zbl 0433.14028 · doi:10.1007/BF01215078
[2] [Ba] Baker, A.: On the periods of the Weierstrass ?-function. Symposia Math. Vol. IV, INDAM Rome 1968, Academic Press, London 1970, pp. 155-174
[3] [BM] Brownawell, W.D., Masser, D.W.: Multiplicity estimates for analytic functions I. J. Reine Angew. Math.314, 200-216 (1979) · Zbl 0417.10027
[4] [Ca] Cassels, J.W.S.: An introduction to the geometry of numbers. Berlin Göttingen Heidelberg: Springer 1959 · Zbl 0086.26203
[5] [CC] Chudnovsky, D.V., Chudnovsky, G.V.: Padé approximations and algebraic geometry. Proc. Natl. Acad. Sci. USA82, 2212-2216 (1985) · Zbl 0577.14034 · doi:10.1073/pnas.82.8.2212
[6] [FP] Faisant, A., Philibert, G.: Quelques résultats de transcendance liés á l’invariant modulairej. J. Number Theory25, 184-200 (1987) · Zbl 0633.10035 · doi:10.1016/0022-314X(87)90024-2
[7] [K] Kolchin, E.R.: Algebraic groups and algebraic dependence. Am. J. Math.90, 1151-1164 (1968) · Zbl 0169.36701 · doi:10.2307/2373294
[8] [La] Laurent, M.: Une nouvelle démonstration du théorème d’isogénie, d’après D.V. et G.V. Choodnovsky, Séminaire de Théorie de Nombres de Paris 1985-6, Boston Basel Stuttgart, Birkhäuser, 1987, pp. 119-131
[9] [Lo] Loxton, J.H.: Some problems involving powers of integers. Acta Arith.46, 113-123 (1986) · Zbl 0549.10012
[10] [M] Masser, D.W.: Counting points of small height on elliptic curves. Bull. Soc. Math. Fr. (to appear) · Zbl 0723.14026
[11] [MW1] Masser, D.W., Wüstholz, G.: Fields of large transcendence degree generated by values of elliptic functions. Invent. Math.72, 407-464 (1983) · Zbl 0516.10027 · doi:10.1007/BF01398396
[12] [MW2] Masser, D.W., Wüstholz, G.: Zero estimates on group varieties II. Invent. Math.80, 233-267 (1985) · Zbl 0564.10041 · doi:10.1007/BF01388605
[13] [MW3] Masser, D.W., Wüstholz, G.: Some effective estimates for elliptic curves, to appear in the Proceedings of the 1988 Erlangen Workshop on Arithmetic Complex Manifolds. (Lect. Notes Math., Berlin Heidelberg New York: Springer, to appear)
[14] [P] Philippon, P.: Lemmes de zéros dans les groupes algébriques. Bull. Soc. Math. Fr.114, 355-383 (1986) · Zbl 0617.14001
[15] [S] Silverman, J.H.: The arithmetic of elliptic curves. New York Berlin Heidelberg: Springer 1986 · Zbl 0585.14026
[16] [V] Vélu, J.: Isogénies entre coubes elliptiques. C.R. Acad. Sci. Paris A273, 238-241 (1973)
[17] [Wa] Waldschmidt, M.: Nombres transcendants et groupes algébriques. Astérisque69-70 (1979)
[18] [Wü 1] Wüstholz, G.: Multiplicity estimates on group varieties. Ann. Math.129, 471-500 (1989) · Zbl 0675.10024 · doi:10.2307/1971514
[19] [Wü 2] Wüstholz, G.: Zum Periodenproblem. Invent. Math.78, 381-391 (1984) · Zbl 0548.10022 · doi:10.1007/BF01388443
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.