Diophantine Inequality in Diophantine Geometry and Gap Principle

1)与J. H. EVERTSE合作：研究了定义为DIophantine不等式系统的Wirsing系统，并给出了该系统在有界次代数数上只有有限多个解的若干条件。我们还给出了系统与生成不等式之间的一些关系。大卫：我们证明了椭圆对数线性形式的一个新的下界。就线性形式的高度而言，我们的结果是最好的。因此，我们完全解决了S. Lang追溯到60年代的一个猜想。我们的一般结果包括一个“同时版本”，并且是完全定量的：它考虑了点的高度，椭圆曲线的高度和给定数据的定义域的程度。在这方面，以前最著名的估计是由首席调查员作出的，可追溯到90年代初。3)与Marc HUTTNER合作：我们提出了关于高斯超几何函数值的丢芬图近似。我们的估计依赖于丘德诺夫斯基方法在椭圆对数线性形式定量理论中的自然应用。我们把高斯的超几何函数看作是一个阿贝尔对数函数。4) p进对数：给出了椭圆情况下p进对数的线性形式的估计。我们定义了一个p进椭圆对数函数作为Lutz-Weil p进椭圆函数的局部逆函数。

1) Work with J. H. EVERTSE : We investigate Wirsing system which is defined as a system of DIophantine Inequalities and gave some conditions such that the system has only finitely many solutions in algebraic numbers of bounded degree. We also show some relations between the system and Resultant inequalities.2) Work with Sinnou DAVID : We prove a new lower bound for linear forms in elliptic logarithms. As far as the height of the linear forms is concerned, our result is the best possible. We thus completely solve a conjecture of S. Lang dating back to the sixties. Our general result includes a "simultaneous version" and is totally quantitative : it takes into account the height of the point, the height of the elliptic curve and the degree of the field of definition of the given data. The previously best known estimate in this context was due to the head investigator and goes back to the early nineties.3) Work with Marc HUTTNER : We present Diophantine Approximations concerning values of Gauss' hypergeometric function. Our estimate relies on a natural application of the method of Chudnovsky for the quantitative theory of linear forms in elliptic logarithms. We regard Gauss' hypergeometric function as an abelian logarithmic function.4) p-adic alanogs : We give an estimate of linear forms in p-adic logarithms in elliptic case. We define for this estimate a p-adic elliptic logarithmic function viewed as a local reversed function of the Lutz-Weil p-adic elliptic function.