An approach toward the abc conjecture by means of p-adic Diophantine geometry : a new aspect via the period conjecture

abc猜想蕴涵了数论中几个著名的结果。例如，Mordell猜想（由G. Faltings），Siegel-零问题（A. Granville）等。p-adic图中线性形式的下界是获得非平凡上界的一个工具，其形状类似于abc猜想。该观察结果由R.蒂德曼角Stewart，Kunrui Yu等，虽然这不足以达到几何估计（可能的估计仍是指数估计），但众所周知，p-adic算法中的线性bin是缩短丢番图逼近和abc猜想的一个重要途径。基于这样的原因，我们对线性形式的数学模型产生了兴趣。本文考虑一个椭圆型，并利用G. Chudnovsky在椭圆曲线的形式群上进行了（Sinnou大卫和N.平田光野，” ...更多信息 线性形式在椭圆曲线”，出现在杂志毛皮模具reine和angewandte数学，和预印本：N。Hirata-Kohno，“Linear forms in p-adic elliptic currums”）。在正式文章中，S.朗解决了椭圆对数函数的有用估计也可以用Sinnou大卫的“对数函数和椭圆曲线的形式群”计算，见：丢番图方程，Tata基础研究所，数学研究，Narosa出版社，2008，243-256。本文与M。Huttner提出的超几何级数的积分表示是无穷远处一点上的阿贝尔对数函数。Hirata-Kohno，S.）Laishram，T. N. Shorey和R. Tijdeman“An extension of a theorem of Euler”，Acta Arithmetica vol.129（1），2007，71.102，讨论了一类指数丢番图方程整数解不存在的充分条件.少

The abc conjecture implies several famous results in number theory. For example, Mordell conjecture (solved by G. Faltings), Siegel-zero problem (the connection was made by A. Granville) and so on. A lower bound for linear forms in p-adic logarithms is one tool to obtain a non-trivial upper bound which is in a similar shape of the abc conjecture. The observation was dealt by R. Tijdeman, C. Stewart, Kunrui Yu, etc. Although it is not enough to achieve the conjectural estimate (the possible estimate is still an exponential one), it is well-known that the linear bin in p-adic logarithms is nothing but an important way to abridge Diophantine approximations and the abc conjecture. Being motivated by such a reason, we are interested in linear forms in logarithms. We consider here an elliptic version and obtain an estimate for linear forms in elliptic logarithms, by means of the method of G. Chudnovsky carried out on the formal group of the elliptic curves (Sinnou David and N. Hirata-Kohno," … More Linear forms in elliptic logarithms", to appear in Journal fur die reine and angewandte Mathematik, and a preprint : N. Hirata-Kohno,"Linear forms in p-adic elliptic logarithms"). In the formal article a conjecture of S. Lang is solved. Useful estimates of elliptic logarithmic functions are also calculated with Sinnou David : "Logarithmic Functions and Formal Groups of Elliptic Curves", in : Diophantine Equations, Tata Institute of Fundamental Research, Studies in Mathematics, Narosa Publishing House, 2008, 243-256. We adapt this method to get Diophantine approximations of values of Gauus'hypergeometric series as a joint work with M. Huttner, where we regard the integral representation of the hypergeometric series as an ablelian logarithmic function at one point at the infinity.In N. Hirata-Kohno, S.)Laishram, T. N. Shorey and R. Tijdeman "An extension of a theorem of Euler", Acta Arithmetica vol. 129 (1), 2007, 71.102, sufficient conditions of non-existence of the integer solutions to certain exponential Diophantine equations are discussed. Less