1985
DOI: 10.1073/pnas.82.8.2212
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Abstract: Using methods of Pade approximations we prove a converse to Eisenstein's theorem on the boundedness of denominators of coefficients in the expansion of an algebraic function, for classes of functions, parametrized by meromorphic functions. This result is applied to the Tate conjecture on the effective description of isogenies for elliptic curves. meromorphic functions of finite order of growth -p. This means that there exist entire functions H(u), Hj(u), ..., HJ(R) of 17 in C9 such that Uj(7) = Hj(77)/H(g) (i … Show more

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Cited by 33 publications
(22 citation statements)
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“…The present work has been motivated by the papers [CC85a] and [CC85b] of D. V. and G. V. Chudnovsky, and by their more recent developments due to André ([And89]; see also [And99]) and Graftieaux ([Gr98] and [Gr01a]). A simple but crucial innovation in our approach is the introduction of foliations, which leads to a general theorem unifying various previous results about either systems of linear differential equations, or particular commutative algebraic groups.…”
Section: Introductionmentioning
confidence: 99%
“…The present work has been motivated by the papers [CC85a] and [CC85b] of D. V. and G. V. Chudnovsky, and by their more recent developments due to André ([And89]; see also [And99]) and Graftieaux ([Gr98] and [Gr01a]). A simple but crucial innovation in our approach is the introduction of foliations, which leads to a general theorem unifying various previous results about either systems of linear differential equations, or particular commutative algebraic groups.…”
Section: Introductionmentioning
confidence: 99%
“…Our algebraicity criteria improve on the ones in [12] and [13], which themselves were inspired by the papers [19] and [20] of D. V. and G. V. Chudnovsky and by the subsequent works by André [2] and Graftieaux [26,27]. As in [12] and [13], our results will be proved by means of a geometric version of "transcendence techniques", which avoids the traditional constructions of "auxiliary polynomials" and the explicit use of Siegel's Lemma, replacing them by a few basic concepts of Arakelov geometry.…”
Section: Introductionmentioning
confidence: 99%
“…As in [12], we prove Theorem 1 by generalizing the work of Chudnovsky [8] (see also [1]), but we get a quite short proof thanks to the general Arakelovian method of Bost [3] for transcendence proofs.…”
Section: Introductionmentioning
confidence: 99%