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Padé approximations and Diophantine geometry. (English) Zbl 0577.14034
This very interesting paper contains two effective results on isogenies of elliptic curves, proved by methods of classical transcendence theory. (I) Let \(E_ 1, E_ 2\) be elliptic curves defined over \({\mathbb{Q}}\). Then there exist effective constants P, \(C_ I\), depending only on the invariants, such that if \(E_ 1, E_ 2\) have the same number of points modulo p for all primes \(p\leq P\), then they are isogenous, and furthermore the degree of the isogeny is bounded by \(C_ I\). (II) Let E be a Weierstraß elliptic curve defined over a real number field K. Then there exists an effective constant \(C_{II}\), depending only on the invariants and K, such that the order of any cyclic subgroup A of E defined over K does not exceed \(C_{II}\) (in particular the degrees of K-isogenies of E can be bounded). - Thus (I) and (II) can be regarded as effective versions of special cases of Tate’s conjecture and Shafarevich’s conjecture for elliptic curves respectively [see J.-P. Serre, ”Abelian \(\ell\)-adic representations and elliptic curves” (1968; Zbl 0186.257), p. IV-15 and IV-7, respectively].
The proof of (I) uses auxiliary polynomials in certain associated elliptic functions \(z_ 1, z_ 2\) (the functions -x/y on the respective Tate models). The authors note that a result of T. Honda [J. Math. Soc. Japan 22, 213-246 (1970; Zbl 0202.031) implies that \(z_ 1=f(z_ 2)\) for some locally convergent power series \(f(x)=x+O(x^ 2)\) with coefficients in \({\mathbb{Z}}\). The conclusion is that \(z_ 1, z_ 2\) are algebraically dependent, which leads to the desired isogeny. In fact a much more general result is established, too long to state here, giving a criterion for the algebraic dependence of n fairly arbitrary meromorphic functions \(z_ 1(u),...,z_ n(u)\) in g complex variables \(u=(u_ 1,...,u_ g)\) when \(n>g\), provided there is a suitable ”uniformization” by ”reasonably well-behaved” locally convergent power series \(z_ 1=f_ 1(x),...,z_ n=f_ n(x)\) for \(x=(x_ 1,...,x_ g).\)
The proof of (II) follows similar lines, using auxiliary polynomials in z and its composition with multiplication by the other of the cyclic subgroup A.
Also the authors give, without proof, some explicit expressions for P, \(C_ I\) in (I); namely if \(E_ 1, E_ 2\) have invariants in \({\mathbb{Z}}\), and \(\Delta_ 1, \Delta_ 2\) are the areas of the fundamental domains of the period lattices with respect to dx/y, then we can take \(P=cM\), \(C_ I=cM^{1+\epsilon}\), where \(M=\max (1,\Delta_ 1^{-1}\Delta_ 2^{-1})\) and c depends only on the arbitrary \(\epsilon >0\).
Reviewer: D.W.Masser

14K05 Algebraic theory of abelian varieties
11J81 Transcendence (general theory)
32J99 Compact analytic spaces
14H52 Elliptic curves
14H45 Special algebraic curves and curves of low genus
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