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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

##### MSC:
 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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