##
**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 \(\epsilon >0\).

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 |