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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
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References:
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