Diophantine problems : modern and classical perspectives

丢番图问题：现代与古典的观点

基本信息

批准号：
250160-2012
负责人：
Bennett, Michael
金额：
$ 2.55万
依托单位：
University of British Columbia
依托单位国家：
加拿大
项目类别：
Discovery Grants Program - Individual
财政年份：
2015
资助国家：
加拿大
起止时间：
2015-01-01 至 2016-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=570832
关键词：
Diophantine problems modern classical perspectives

项目摘要

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.

毒液问题具有经典的吸引力，它超越了研究数学，在某些情况下，例如Fermat的最后一个定理，即使是流行文化也是如此。对于一位实践数学家，他们为算术及代数几何形状的最新定理和猜想提供了测试箱，以理解曲线和表面的算术，并观察分析和代数技术之间的相互作用，理论和代数技术之间的相互作用计算。该提案的重点是将三种相当多样化的技术应用于各种二磷剂问题，有时仅是组合的。在大多数情况下，这些是经典问题，这些问题已被证明是通过其他方式对先前尝试的弹性特别有弹性的。我们的前两种技术（对数中的线性形式的下限和Thue和Siegel的超几何方法）源自二聚体的近似，而第三个（与模块化Abelian品种相关的Frey-Hellegouarch曲线的第三个）是由Galois表示之间的深度互连引起的和模块化形式。我们打算应用这些方法的问题范围从经典的双磷酸方程式，通过有关无K量数之间差距的更多分析问题，到添加剂组合剂的风味的问题。