Diophantine problems : modern and classical perspectives

丢番图问题有着超越研究数学的经典吸引力，在某些情况下，如费马大定理，甚至流行文化。对于一个练习数学家，他们提供测试案例的最新定理和代数，实际上，代数几何，理解算术的曲线和曲面，并观察分析和代数技术之间的相互作用，之间的理论和计算。 这个建议的重点是应用三个相当不同的技术，以各种丢番图的问题，有时单独，有时在组合;在大多数情况下，这些都是经典的问题，已被证明特别有弹性，以前的尝试，通过其他手段。我们的前两个技术（下界的线性形式的矩阵和超几何方法的Thue和Siegel）来自丢番图近似，而第三个（Frey-Hellegouarch曲线与模块化阿贝尔品种）产生于伽罗瓦表示和模块化形式之间的深刻互连。我们打算应用这些方法的问题范围从经典的丢番图方程，通过更多的分析问题之间的差距k-自由数，添加剂组合的味道的问题。

Diophantine problems have a classical appeal that reaches beyond research mathematics, in some cases such as Fermat's Last Theorem, even as far as popular culture. For a practising mathematician, they provide test-cases for the latest theorems and conjectures from Arithmetic and, indeed, Algebraic Geometry, for understanding the arithmetic of curves and surfaces, and for observing the interplay between analytic and algebraic techniques, between the theoretical and the computational. This proposal is focussed upon the application of three rather diverse techniques to a variety of Diophantine problems, sometimes solely, sometimes in combination; in most cases, these are classical problems that have proved particularly resilient to prior attempts via other means. Our first two techniques (lower bounds for linear forms in logarithms and the hypergeometric method of Thue and Siegel) derive from Diophantine approximation, while the third (that of Frey-Hellegouarch curves associated with modular abelian varieties) arises from the deep interconnection between Galois representations and modular forms. The problems where we intend to apply these methods range from classical Diophantine equations, through more analytic questions about gaps between k-free numbers, to problems with the flavour of additive combinatorics.