Dyson's lemma and a theorem of Esnault and Viehweg

Abstract

In the classical theory of diophantine approximation, as encountered in the work of Thue, Siegel, Dyson, Roth, and Schmidt, finiteness results are obtained by constructing an auxilliary polynomial. One knows a priori that the polynomial vanishes to high order at certain approximating points and this contradicts an upper bound on the order of vanishing obtained by other techniques. The difficult part of the argument is always bounding the order of vanishing from above. This technique of diophantine approximation has recently been extended with great success [V3, V4, V5, F1, F2] in order to obtain finiteness results for rational points on certain subvarieties of abelian varieties. An intermediate stage in this development occurred when Esnault and Viehweg [EV1] proved a generalized version of Dyson’s lemma strong enough to imply Roth’s theorem. A few years later, Vojta [V1] built upon this work to prove a variant of Dyson’s lemma valid on a product of curves of arbitrary genus. This played an essential role in his proof [V3] of the Mordell conjecture. Faltings [F1], further extending the work of Vojta, was able to prove finiteness results for certain higher dimensional subvarieties of abelian varieties. Dyson’s lemma, however, plays no immediate role in Faltings’ proof; instead he uses a geometrical result called the product theorem. Though Dyson’s lemma is not used in [F1] there is nonetheless a close relationship between the arguments used in section 4 of [F1] and those occurring in section 5 of [EV1]. In particular, in both instances the proof is by induction on the dimension of product subvarieties with Lemma 2.9 of [EV1] replacing Corollary 4.3 of [F1] and with the Main Lemma 5.3 in [EV1] taking the place of the product theorem in [F1]. In this note, we propose to generalize the main theorems of [EV1] and [V1]. In order to state our results, we need to fix some notation. Let